RSA (cryptosystem)

6826: 2935: 3332: 9497: 238: 6118: 264:, who published this concept in 1976. They also introduced digital signatures and attempted to apply number theory. Their formulation used a shared-secret-key created from exponentiation of some number, modulo a prime number. However, they left open the problem of realizing a one-way function, possibly because the difficulty of factoring was not well-studied at the time. Moreover, like 2601: 2975: 318:(GCHQ), described a similar system in an internal document in 1973. However, given the relatively expensive computers needed to implement it at the time, it was considered to be mostly a curiosity and, as far as is publicly known, was never deployed. His ideas and concepts, were not revealed until 1997 due to its top-secret classification. 5291:(OAEP), which prevents these attacks. As such, OAEP should be used in any new application, and PKCS#1 v1.5 padding should be replaced wherever possible. The PKCS#1 standard also incorporates processing schemes designed to provide additional security for RSA signatures, e.g. the Probabilistic Signature Scheme for RSA ( 356:

The system includes a communications channel coupled to at least one terminal having an encoding device and to at least one terminal having a decoding device. A message-to-be-transferred is enciphered to ciphertext at the encoding terminal by encoding the message as a number M in a predetermined set.

6581:

Heninger says in her blog that the bad keys occurred almost entirely in embedded applications, including "firewalls, routers, VPN devices, remote server administration devices, printers, projectors, and VOIP phones" from more than 30 manufacturers. Heninger explains that the one-shared-prime problem

6005:

The first RSA-512 factorization in 1999 used hundreds of computers and required the equivalent of 8,400 MIPS years, over an elapsed time of about seven months. By 2009, Benjamin Moody could factor an 512-bit RSA key in 73 days using only public software (GGNFS) and his desktop computer (a dual-core

6582:

uncovered by the two groups results from situations where the pseudorandom number generator is poorly seeded initially, and then is reseeded between the generation of the first and second primes. Using seeds of sufficiently high entropy obtained from key stroke timings or electronic diode noise or

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at the house of a student and drank a good deal of wine before returning to their homes at around midnight. Rivest, unable to sleep, lay on the couch with a math textbook and started thinking about their one-way function. He spent the rest of the night formalizing his idea, and he had much of the

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must be somewhat smaller. RSA padding schemes must be carefully designed so as to prevent sophisticated attacks that may be facilitated by a predictable message structure. Early versions of the PKCS#1 standard (up to version 1.5) used a construction that appears to make RSA semantically secure.

6733:

Simple Branch Prediction Analysis (SBPA) claims to improve BPA in a non-statistical way. In their paper, "On the Power of Simple Branch Prediction Analysis", the authors of SBPA (Onur Aciicmez and Cetin Kaya Koc) claim to have discovered 508 out of 512 bits of an RSA key in 10 iterations.

357:

That number is then raised to a first predetermined power (associated with the intended receiver) and finally computed. The remainder or residue, C, is... computed when the exponentiated number is divided by the product of two predetermined prime numbers (associated with the intended receiver).

3349:'s public key to send him an encrypted message. In the message, she can claim to be Alice, but Bob has no way of verifying that the message was from Alice, since anyone can use Bob's public key to send him encrypted messages. In order to verify the origin of a message, RSA can also be used to 2930:{\displaystyle {\begin{aligned}d_{p}&=d{\bmod {(}}p-1)=413{\bmod {(}}61-1)=53,\\d_{q}&=d{\bmod {(}}q-1)=413{\bmod {(}}53-1)=49,\\q_{\text{inv}}&=q^{-1}{\bmod {p}}=53^{-1}{\bmod {6}}1=38\\&\Rightarrow (q_{\text{inv}}\times q){\bmod {p}}=38\times 53{\bmod {6}}1=1.\end{aligned}}} 6411:; this value can be regarded as a compromise between avoiding potential small-exponent attacks and still allowing efficient encryptions (or signature verification). The NIST Special Publication on Computer Security (SP 800-78 Rev. 1 of August 2007) does not allow public exponents 290:

made several attempts over the course of a year to create a function that was hard to invert. Rivest and Shamir, as computer scientists, proposed many potential functions, while Adleman, as a mathematician, was responsible for finding their weaknesses. They tried many approaches, including

3327:{\displaystyle {\begin{aligned}m_{1}&=c^{d_{p}}{\bmod {p}}=2790^{53}{\bmod {6}}1=4,\\m_{2}&=c^{d_{q}}{\bmod {q}}=2790^{49}{\bmod {5}}3=12,\\h&=(q_{\text{inv}}\times (m_{1}-m_{2})){\bmod {p}}=(38\times -8){\bmod {6}}1=1,\\m&=m_{2}+h\times q=12+1\times 53=65.\end{aligned}}} 6703:(a padding scheme randomizes and adds structure to an RSA-encrypted message, so it is possible to determine whether a decrypted message is valid). Due to flaws with the PKCS #1 scheme, Bleichenbacher was able to mount a practical attack against RSA implementations of the 3376:) (as he does when encrypting a message), and compares the resulting hash value with the message's hash value. If the two agree, he knows that the author of the message was in possession of Alice's private key and that the message has not been tampered with since being sent. 6589:

Strong random number generation is important throughout every phase of public-key cryptography. For instance, if a weak generator is used for the symmetric keys that are being distributed by RSA, then an eavesdropper could bypass RSA and guess the symmetric keys directly.

6625:

One way to thwart these attacks is to ensure that the decryption operation takes a constant amount of time for every ciphertext. However, this approach can significantly reduce performance. Instead, most RSA implementations use an alternate technique known as

6080:

was factored by using several hundred computers, and these are now factored in a few weeks using common hardware. Exploits using 512-bit code-signing certificates that may have been factored were reported in 2011. A theoretical hardware device named

6056:

estimated that 1024-bit keys were likely to become crackable by 2010. As of 2020, it is not known whether such keys can be cracked, but minimum recommendations have moved to at least 2048 bits. It is generally presumed that RSA is secure if

2394: 2272: 4861: 3368:) (as she does when decrypting a message), and attaches it as a "signature" to the message. When Bob receives the signed message, he uses the same hash algorithm in conjunction with Alice's public key. He raises the signature to the power of 4383: 4111: 5310:); however, these patents expired on 24 July 2009 and 25 April 2010 respectively. Use of PSS no longer seems to be encumbered by patents. Note that using different RSA key pairs for encryption and signing is potentially more secure. 6532:', totally compromising both keys. Lenstra et al. note that this problem can be minimized by using a strong random seed of bit length twice the intended security level, or by employing a deterministic function to choose 321:

Kid-RSA (KRSA) is a simplified, insecure public-key cipher published in 1997, designed for educational purposes. Some people feel that learning Kid-RSA gives insight into RSA and other public-key ciphers, analogous to

3544: 6601:

described a new attack on RSA in 1995: if the attacker Eve knows Alice's hardware in sufficient detail and is able to measure the decryption times for several known ciphertexts, Eve can deduce the decryption key

1500:). Since the chosen key can be small, whereas the computed key normally is not, the RSA paper's algorithm optimizes decryption compared to encryption, while the modern algorithm optimizes encryption instead. 1986: 1763: 5790: 6737:

A power-fault attack on RSA implementations was described in 2010. The author recovered the key by varying the CPU power voltage outside limits; this caused multiple power faults on the server.

2980: 2606: 2307: 2185: 5562: 4483: 6041:(up to a polynomial time difference). However, Rivest, Shamir, and Adleman noted, in section IX/D of their paper, that they had not found a proof that inverting RSA is as hard as factoring. 3662: 596: 7231: 2516: 2148: 203:, along with an auxiliary value. The prime numbers are kept secret. Messages can be encrypted by anyone, via the public key, but can only be decrypted by someone who knows the private key. 2455: 2302: 5073:

if an attacker cannot distinguish two encryptions from each other, even if the attacker knows (or has chosen) the corresponding plaintexts. RSA without padding is not semantically secure.

2180: 3422: 5710: 5645: 4720: 1641: 8557:

The Original RSA Patent as filed with the U.S. Patent Office by Rivest; Ronald L. (Belmont, MA), Shamir; Adi (Cambridge, MA), Adleman; Leonard M. (Arlington, MA), December 14, 1977,

9477: 9307: 5499: 5440: 7055: 7006: 1894: 3808: 8922: 6462:, James P. Hughes, Maxime Augier, Joppe W. Bos, Thorsten Kleinjung and Christophe Wachter. They were able to factor 0.2% of the keys using only Euclid's algorithm. 7527: 7117: 4213: 3941: 464: 8632: 6052:). Its factorization, by a state-of-the-art distributed implementation, took about 2,700 CPU-years. In practice, RSA keys are typically 1024 to 4096 bits long. In 2003, 5834: 366: 6749:, acceptable public exponent, etc.). This makes the implementation challenging, to the point the book Practical Cryptography With Go suggests avoiding RSA if possible. 6683:

is chosen for each ciphertext. With blinding applied, the decryption time is no longer correlated to the value of the input ciphertext, and so the timing attack fails.

295:-based" and "permutation polynomials". For a time, they thought what they wanted to achieve was impossible due to contradictory requirements. In April 1977, they spent 5843:, even though two modular exponentiations have to be computed. The reason is that these two modular exponentiations both use a smaller exponent and a smaller modulus. 5298:

Secure padding schemes such as RSA-PSS are as essential for the security of message signing as they are for message encryption. Two USA patents on PSS were granted (

5259:

does not fall into the range of insecure plaintexts, and that a given message, once padded, will encrypt to one of a large number of different possible ciphertexts.

5050:

noticed that this attack is possible even if the clear texts are not equal, but the attacker knows a linear relation between them. This attack was later improved by

2097: 4623:

Although the original paper of Rivest, Shamir, and Adleman used Fermat's little theorem to explain why RSA works, it is common to find proofs that rely instead on

2039: 1839: 1813: 9050: 8173: 4226: 3954: 5287:

2000, Coron et al. showed that for some types of messages, this padding does not provide a high enough level of security. Later versions of the standard include

9145: 8107: 5380: 5360: 531: 508: 484: 225:

RSA is a relatively slow algorithm. Because of this, it is not commonly used to directly encrypt user data. More often, RSA is used to transmit shared keys for

3426: 222:. Whether it is as difficult as the factoring problem is an open question. There are no published methods to defeat the system if a large enough key is used. 9045: 7661: 8332: 8372: 5069:

against the cryptosystem, by encrypting likely plaintexts under the public key and test whether they are equal to the ciphertext. A cryptosystem is called

8774: 8136: 6714:

A variant of this attack, dubbed "BERserk", came back in 2014. It impacted the Mozilla NSS Crypto Library, which was used notably by Firefox and Chrome.

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with a 1,900 MHz CPU). Just less than 5 gigabytes of disk storage was required and about 2.5 gigabytes of RAM for the sieving process.

5991:. No polynomial-time method for factoring large integers on a classical computer has yet been found, but it has not been proven that none exists; see 1273: 8953: 8947: 686:

should be chosen at random, be both large and have a large difference. For choosing them the standard method is to choose random integers and use a

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to determine whether a conditional branch in the instruction flow of a program is likely to be taken or not. Often these processors also implement

7694: 7067: 1690: 1516:. If they decide to use RSA, Bob must know Alice's public key to encrypt the message, and Alice must use her private key to decrypt the message. 6707:

protocol and to recover session keys. As a result of this work, cryptographers now recommend the use of provably secure padding schemes such as

9525: 315: 169: 7965:

Coron, Jean-Sébastien; Joye, Marc; Naccache, David; Paillier, Pascal (2000). "New Attacks on PKCS#1 v1.5 Encryption". In Preneel, Bart (ed.).

6730:(SMT). Branch-prediction analysis attacks use a spy process to discover (statistically) the private key when processed with these processors. 9071: 8625: 8541: 8501: 7984: 7941: 7843: 6708: 5288: 339: 287: 8014: 3552:

Decrypt a message only intended for the recipient, which may be encrypted by anyone having the public key (asymmetric encrypted transport).

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RSA has the property that the product of two ciphertexts is equal to the encryption of the product of the respective plaintexts. That is,

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paper ready by daybreak. The algorithm is now known as RSA – the initials of their surnames in same order as their paper.

7601: 6135: 5276: 8689: 9530: 8050: 4420: 8757: 8714: 8679: 3602: 3555:

Encrypt a message which may be decrypted by anyone, but which can only be encrypted by one person; this provides a digital signature.

1932: 536: 9138: 8167:"NIST Special Publication 800-57 Part 3 Revision 1: Recommendation for Key Management: Application-Specific Key Management Guidance" 6696: 6201: 5874: 5280: 2469: 8669: 2408: 7538: 6325: 8618: 3386: 1901: 756: 8747: 8694: 6849: 265: 8833: 6844: 6182: 1594: 5717: 9356: 9287: 8858: 7267: 6786: 6578:

separately, thereby achieving a very significant speedup, since after one large division, the GCD problem is of normal size.

6154: 6139: 6014: 2051: 1470: 1068: 8742: 1351:). Most of the implementations of RSA will accept exponents generated using either method (if they use the private exponent 369:

column. This preceded the patent's filing date of December 1977. Consequently, the patent had no legal standing outside the

6076:, using software already freely available. Keys of 512 bits have been shown to be practically breakable in 1999, when 7627: 6746: 5856: 2521: 9131: 8999: 8932: 6766: 6161: 3742: 1113: 8674: 5505: 5266:

have been carefully designed to securely pad messages prior to RSA encryption. Because these schemes pad the plaintext

9472: 9427: 9230: 9096: 8989: 8838: 8752: 8737: 7477: 6859: 6854: 6761: 6727: 6614:

demonstrated a more practical attack capable of recovering RSA factorizations over a network connection (e.g., from a

5328: 3357: 8313:

Lenstra, Arjen K.; Hughes, James P.; Augier, Maxime; Bos, Joppe W.; Kleinjung, Thorsten; Wachter, Christophe (2012).

8221: 8111: 5873:. Full decryption of an RSA ciphertext is thought to be infeasible on the assumption that both of these problems are 3548:

Thus the keys may be swapped without loss of generality, that is, a private key of a key pair may be used either to:

7126: 2102: 9351: 8848: 8719: 8579: 8527: 8412:

Acıiçmez, Onur; Koç, Çetin Kaya; Seifert, Jean-Pierre (2007). "On the power of simple branch prediction analysis".

6438:(TPM) were shown to be affected. Vulnerable RSA keys are easily identified using a test program the team released. 5840: 4116: 3570: 349: 121: 6168: 9467: 9101: 9081: 7275: 6619: 5336: 5043: 4608: 2555: 2528:. In real-life situations the primes selected would be much larger; in our example it would be trivial to factor 5935:

is an RSA ciphertext. Currently the most promising approach to solving the RSA problem is to factor the modulus

5652: 5587: 399:

A basic principle behind RSA is the observation that it is practical to find three very large positive integers

9457: 9447: 9302: 9040: 8811: 8166: 5110: 3356:

Suppose Alice wishes to send a signed message to Bob. She can use her own private key to do so. She produces a

323: 8263: 8202: 7717:

A Course in Number Theory and Cryptography, Graduate Texts in Math. No. 114, Springer-Verlag, New York, 1987.

6150: 9452: 9442: 9235: 9195: 9188: 9173: 9168: 8994: 8641: 6879: 6627: 6458:. An analysis comparing millions of public keys gathered from the Internet was carried out in early 2012 by 6447: 6435: 6396: 6215: 6128: 5852: 5066: 5062: 5055: 2073: 1132: 957: 226: 149: 7328: 6711:, and RSA Laboratories has released new versions of PKCS #1 that are not vulnerable to these attacks. 5446: 5387: 5247:

To avoid these problems, practical RSA implementations typically embed some form of structured, randomized

9240: 9183: 9076: 8927: 8866: 8801: 8446: 8417: 7877: 7698: 7422: 7284: 7011: 6962: 6839: 6700: 5882: 5248: 2525: 1646: 1563: 1265: 1206: 817: 630: 269: 181: 8144: 1861: 9500: 9346: 9292: 8942: 8699: 8656: 7868: 7759: 6692: 5992: 5866: 3712: 1914: 777: 6825: 6722:

A side-channel attack using branch-prediction analysis (BPA) has been described. Many processors use a

373:. Had Cocks' work been publicly known, a patent in the United States would not have been legal either. 168:, who publicly described the algorithm in 1977. An equivalent system was developed secretly in 1973 at 7732: 2389:{\displaystyle {\begin{aligned}m(c)&=c^{d}{\bmod {n}}\\&=c^{413}{\bmod {3}}233.\end{aligned}}} 9462: 9386: 8853: 8664: 8515: 7969:. Lecture Notes in Computer Science. Vol. 1807. Berlin, Heidelberg: Springer. pp. 369–381. 7802: 7366: 6704: 6615: 6248: 2267:{\displaystyle {\begin{aligned}c(m)&=m^{e}{\bmod {n}}\\&=m^{17}{\bmod {3}}233.\end{aligned}}} 1360: 173: 103: 7882: 7289: 7082: 199:, which is kept secret (private). An RSA user creates and publishes a public key based on two large 9215: 8959: 8533: 8422: 8067: 7703: 7656: 7459: 7427: 6555: 6459: 6101: 5860: 5070: 4896: 4856:{\displaystyle m^{ed}=m^{1+h\varphi (n)}=m(m^{\varphi (n)})^{h}\equiv m(1)^{h}\equiv m{\pmod {n}},} 873: 362: 126: 4198: 3926: 9331: 9315: 9257: 8984: 8806: 8729: 8704: 8684: 8394: 8298: 8244: 7947: 7895: 7390: 7332: 7302: 6884: 6831: 6423: 6419: 4945:

numbers have this property), but even in this case, the desired congruence is still true. Either

722: 511: 487: 429: 393: 8085: 7922:"Probabilistic encryption & how to play mental poker keeping secret all partial information" 6661: 6219: 5797: 4866: 4624: 81: 9391: 9381: 9247: 9066: 9009: 8937: 8823: 8537: 8497: 8046: 8042:

Network security traceback attack and react in the United States Department of Defense network

8040: 7980: 7937: 7839: 7440: 7382: 6889: 6583: 6175: 6085:, described by Shamir and Tromer in 2003, called into question the security of 1024-bit keys. 6073: 3350: 649: 215: 9326: 9178: 8912: 8519: 8511: 8427: 8274: 8236: 8177: 7970: 7929: 7913: 7887: 7831: 7792: 7432: 7374: 7294: 6723: 6598: 6450:, which has been properly seeded with adequate entropy, must be used to generate the primes 6093: 4579:. However, in modern implementations of RSA, it is common to use a reduced private exponent 4378:{\displaystyle m^{ed}=m^{ed-1}m=m^{k(q-1)}m=(m^{q-1})^{k}m\equiv 1^{k}m\equiv m{\pmod {q}}.} 4106:{\displaystyle m^{ed}=m^{ed-1}m=m^{h(p-1)}m=(m^{p-1})^{h}m\equiv 1^{h}m\equiv m{\pmod {p}},} 1378:. Any "oversized" private exponents not meeting this criterion may always be reduced modulo 292: 257: 7860: 7498: 5865:

The security of the RSA cryptosystem is based on two mathematical problems: the problem of

7614: 6097: 5051: 2018: 1818: 1792: 1007:

results in more efficient encryption – the most commonly chosen value for

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Proceedings of the 2nd ACM Symposium on Information, Computer and Communications Security

8351: 7562: 6013:

Rivest, Shamir, and Adleman noted that Miller has shown that – assuming the truth of the

5939:. With the ability to recover prime factors, an attacker can compute the secret exponent 5047: 7788:

Public-Key Cryptography Standards (PKCS) #1: RSA Cryptography Specifications Version 2.1

7370: 6096:– if one could ever be practically created for the purpose – would be able to factor in 9401: 9321: 9277: 9220: 9205: 8523: 8490: 8264:"The Return of Coppersmith's Attack: Practical Factorization of Widely Used RSA Moduli" 8262:

Nemec, Matus; Sys, Marek; Svenda, Petr; Klinec, Dusan; Matyas, Vashek (November 2017).

7680: 7523: 6869: 6551: 6233: 5365: 5345: 5332: 3380: 1004: 687: 645: 516: 493: 469: 377: 311: 303: 261: 196: 192: 177: 8333:"New research: There's no need to panic over factorable keys–just mind your Ps and Qs" 4929:, the argument just given is invalid. This is highly improbable (only a proportion of 9519: 9482: 9437: 9396: 9376: 9267: 9225: 9200: 8464: 8271:

Proceedings of the 2017 ACM SIGSAC Conference on Computer and Communications Security

7917: 7499:"Still Guarding Secrets after Years of Attacks, RSA Earns Accolades for its Founders" 7394: 7354: 6630:. RSA blinding makes use of the multiplicative property of RSA. Instead of computing 6611: 6465:

They exploited a weakness unique to cryptosystems based on integer factorization. If

3346: 3342: 1513: 1509: 1461:

Note: The authors of the original RSA paper carry out the key generation by choosing

370: 307: 207: 114: 17: 8589: 8000: 7951: 7926:

Proceedings of the fourteenth annual ACM symposium on Theory of computing - STOC '82

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quickly. This attack can also be applied against the RSA signature scheme. In 2003,

5998:

Multiple polynomial quadratic sieve (MPQS) can be used to factor the public modulus

725:

for both the public and private keys. Its length, usually expressed in bits, is the

237: 9432: 9272: 9262: 9252: 9210: 9154: 9106: 9086: 8248: 8140: 7718: 7306: 6864: 6618:(SSL)-enabled webserver). This attack takes advantage of information leaked by the 6053: 5877:, i.e., no efficient algorithm exists for solving them. Providing security against 2964:

are used for efficient decryption (encryption is efficient by choice of a suitable

665: 381: 211: 200: 188: 31: 7310: 6745:

There are many details to keep in mind in order to implement RSA securely (strong

5026:

or more recipients in an encrypted way, and the receivers share the same exponent

7588: 7575: 6232:

is usually done by testing random numbers of the correct size with probabilistic

9411: 9004: 8881: 8568: 8181: 7826:

Håstad, Johan (1986). "On using RSA with Low Exponent in a Public Key Network".

7805: 7786: 6117: 5889: 5870: 5113:

is possible. E.g., an attacker who wants to know the decryption of a ciphertext

219: 8373:"'BERserk' Bug Uncovered In Mozilla NSS Crypto Library Impacts Firefox, Chrome" 7861:"Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities" 7631: 7378: 6776: 9371: 9341: 9336: 9297: 9030: 8762: 8573: 8560: 7413:

Diffie, W.; Hellman, M. E. (November 1976). "New directions in cryptography".

7400: 6821: 6431: 6426:, which affects RSA keys generated by an algorithm embodied in a library from 6089: 6045: 5339:. The following values are precomputed and stored as part of the private key: 5306: 5300: 2293: 1531:

to Bob via a reliable, but not necessarily secret, route. Alice's private key

1000: 726: 344: 279: 275: 245: 241: 161: 157: 153: 55: 51: 7975: 7835: 7444: 7436: 1519:

To enable Bob to send his encrypted messages, Alice transmits her public key

9361: 8431: 8278: 7928:. New York, NY, USA: Association for Computing Machinery. pp. 365–377. 6781: 6607: 5284: 2170: 8488:

Menezes, Alfred; van Oorschot, Paul C.; Vanstone, Scott A. (October 1996).

7386: 6399:

has many applications in attacking RSA specifically if the public exponent

5279:

1998, Bleichenbacher showed that this version is vulnerable to a practical

3838:, it suffices (and in fact is equivalent) to check that they are congruent 256:

The idea of an asymmetric public-private key cryptosystem is attributed to

7933: 7891: 7298: 9406: 9366: 9091: 9025: 8896: 8891: 8886: 8767: 8584: 6874: 6811: 6806: 6771: 6566:' they had found (a 729-million-digit number), instead of computing each 6427: 6007: 5335:) use for decryption and signing the following optimization based on the 3539:{\displaystyle (h^{e})^{d}=h^{ed}=h^{de}=(h^{d})^{e}\equiv h{\pmod {n}}.} 361:

A detailed description of the algorithm was published in August 1977, in

296: 93: 7921: 7268:"A Method for Obtaining Digital Signatures and Public-Key Cryptosystems" 6554:

was part of a group that did a similar experiment. They used an idea of

5166:. Hence, if the attacker is successful with the attack, they will learn 1645:

This can be done reasonably quickly, even for very large numbers, using

380:

were 17 years. The patent was about to expire on 21 September 2000, but

8917: 8876: 7355:"Quantum-computing pioneer warns of complacency over Internet security" 6796: 6791: 6586:

from a radio receiver tuned between stations should solve the problem.

6142: in this section. Unsourced material may be challenged and removed. 6077: 6049: 5324: 5292: 5270:

with some number of additional bits, the size of the un-padded message

2001: 1488:, whereas most current implementations of RSA, such as those following 990: 648:, the private and public key can also be swapped, allowing for message 132: 8314: 7830:. Lecture Notes in Computer Science. Vol. 218. pp. 403–408. 6415:

smaller than 65537, but does not state a reason for this restriction.

5959:

using the standard procedure. To accomplish this, an attacker factors

5065:(i.e., has no random component) an attacker can successfully launch a 1458:

have only very small common factors, if any, besides the necessary 2.

9282: 9035: 8240: 7797: 6801: 5042:, then it is easy to decrypt the original clear-text message via the 335: 4980:

There are a number of attacks against plain RSA as described below.

8594: 7628:"RSA Security Releases RSA Encryption Algorithm into Public Domain" 2535:(obtained from the freely available public key) back to the primes 660:

The keys for the RSA algorithm are generated in the following way:

8871: 8828: 8796: 8789: 8784: 8779: 8093:

Proceedings of Seventh Annual ACM Symposium on Theory of Computing

7066:

The parameters used here are artificially small, but one can also

6757:

Some cryptography libraries that provide support for RSA include:

6404: 6082: 5263: 5015:. In this case, ciphertexts can be decrypted easily by taking the 1489: 1336:

will sometimes yield a result that is larger than necessary (i.e.

236: 229:

cryptography, which are then used for bulk encryption–decryption.

152:, one of the oldest widely used for secure data transmission. The 77: 2520:

Both of these calculations can be computed efficiently using the

384:

released the algorithm to the public domain on 6 September 2000.

6388:

There is no known attack against small public exponents such as

5151:

chosen by the attacker. Because of the multiplicative property,

9127: 8614: 8359:

Proceedings of the 12th Conference on USENIX Security Symposium

7068:

OpenSSL can also be used to generate and examine a real keypair

6403:

is small and if the encrypted message is short and not padded.

5567:

These values allow the recipient to compute the exponentiation

1981:{\displaystyle \lambda (3233)=\operatorname {lcm} (60,52)=780.} 1194:

must also be kept secret because they can be used to calculate

8964: 8818: 8569:

RFC 8017: PKCS #1: RSA Cryptography Specifications Version 2.2

6699:

against RSA-encrypted messages using the PKCS #1 v1

6111: 6069: 7679:

Applied Cryptography, John Wiley & Sons, New York, 1996.

7032: 6983: 6679:

can be removed by multiplying by its inverse. A new value of

3239: 3208: 3124: 3101: 3041: 3018: 2905: 2883: 2832: 2806: 2741: 2713: 2658: 2630: 2598:, which are part of the private key are computed as follows: 2490: 2429: 2370: 2340: 2248: 2218: 2128: 1558:(strictly speaking, the un-padded plaintext) into an integer 8445:

Pellegrini, Andrea; Bertacco, Valeria; Austin, Todd (2010).

1547:

After Bob obtains Alice's public key, he can send a message

7576:"The Mathematics of Encryption: An Elementary Introduction" 4967:, and these cases can be treated using the previous proof. 4500:

We cannot trivially break RSA by applying the theorem (mod

1758:{\displaystyle c^{d}\equiv (m^{e})^{d}\equiv m{\pmod {n}}.} 30:

This article is about a cryptosystem. For the company, see

4611:, although it is not the significant part of that theorem. 2558:

to speed up the calculation using modulus of factors (mod

1356: 1355:

at all, rather than using the optimized decryption method

396:

generation, key distribution, encryption, and decryption.

206:

The security of RSA relies on the practical difficulty of

4583:

that only satisfies the weaker, but sufficient condition

7589:"Introduction to Cryptography with Open-Source Software" 7563:"From Private to Public Key Ciphers in Three Easy Steps" 5785:{\displaystyle h=q_{\text{inv}}(m_{1}-m_{2}){\pmod {p}}} 1264:, the algorithm works as well. The possibility of using 641:

corresponds to encryption and decryption, respectively.

6347:

be large enough. Michael J. Wiener showed that if

6236:

that quickly eliminate virtually all of the nonprimes.

5323:

For efficiency, many popular crypto libraries (such as

1577:

by using an agreed-upon reversible protocol known as a

348:"Cryptographic communications system and method". From 9308:

Cryptographically secure pseudorandom number generator

5382: – the primes from the key generation, 7266:

Rivest, R.; Shamir, A.; Adleman, L. (February 1978).

7129: 7085: 7014: 6965: 6061:

is sufficiently large, outside of quantum computing.

5800: 5720: 5655: 5590: 5508: 5449: 5390: 5368: 5348: 4984:

When encrypting with low encryption exponents (e.g.,

4723: 4423: 4229: 4201: 3957: 3929: 3745: 3605: 3429: 3389: 2978: 2604: 2472: 2411: 2305: 2183: 2105: 2076: 2021: 1935: 1864: 1821: 1795: 1783:

Here is an example of RSA encryption and decryption:

1693: 1597: 539: 519: 496: 472: 432: 8602: 8595:

Onur Aciicmez, Cetin Kaya Koc, Jean-Pierre Seifert:

8590:

Prime Number Hide-And-Seek: How the RSA Cipher Works

8532:(2nd ed.). MIT Press and McGraw-Hill. pp. 2547:, also from the public key, is then inverted to get 9420: 9161: 9059: 9018: 8977: 8905: 8847: 8728: 8655: 8648: 7695:"Further Attacks on Server-Aided RSA Cryptosystems" 6072:or shorter, it can be factored in a few hours on a 112: 102: 92: 87: 73: 65: 47: 42: 8489: 7478:"The RSA Cryptosystem: History, Algorithm, Primes" 7225: 7111: 7049: 7000: 5828: 5784: 5704: 5639: 5556: 5493: 5434: 5374: 5354: 4855: 4477: 4377: 4207: 4105: 3935: 3802: 3656: 3538: 3416: 3326: 2929: 2510: 2449: 2388: 2266: 2142: 2091: 2033: 1980: 1888: 1833: 1807: 1757: 1665:, but this is very unlikely to occur in practice. 1635: 1029:has been shown to be less secure in some settings. 590: 525: 502: 478: 458: 8597:On the Power of Simple Branch Prediction Analysis 7657:"Twenty Years of attacks on the RSA Cryptosystem" 6660:. The result of this computation, after applying 5557:{\displaystyle q_{\text{inv}}=q^{-1}{\pmod {p}}.} 4655:is a product of two different prime numbers, and 4519:In particular, the statement above holds for any 1171:consists of the private (or decryption) exponent 6044:As of 2020, the largest publicly known factored 5881:decryption may require the addition of a secure 5255:before encrypting it. This padding ensures that 4478:{\displaystyle (m^{e})^{d}\equiv m{\pmod {pq}}.} 3569:The proof of the correctness of RSA is based on 1021:. The smallest (and fastest) possible value for 7828:Advances in Cryptology – CRYPTO '85 Proceedings 6622:optimization used by many RSA implementations. 4390:This completes the proof that, for any integer 3657:{\displaystyle (m^{e})^{d}\equiv m{\pmod {pq}}} 591:{\displaystyle (m^{e})^{d}\equiv m{\pmod {n}}.} 354: 8174:National Institute of Standards and Technology 8086:"Riemann's Hypothesis and Tests for Primality" 7226:{\displaystyle q_{\text{inv}}\left{\pmod {p}}} 5132:to decrypt an unsuspicious-looking ciphertext 4897:Carmichael's generalization of Euler's theorem 4865:where the second-last congruence follows from 2511:{\displaystyle m=2790^{413}{\bmod {3}}233=65.} 2143:{\displaystyle 1=(17\times 413){\bmod {7}}80.} 9139: 8626: 8222:"Cryptanalysis of short RSA secret exponents" 7460:"The Early Days of RSA – History and Lessons" 6442:Importance of strong random number generation 5109:. Because of this multiplicative property, a 2450:{\displaystyle c=65^{17}{\bmod {3}}233=2790.} 1359:described below), but some standards such as 8: 8066:Lenstra, Arjen; et al. (Group) (2000). 7662:Notices of the American Mathematical Society 6955:besides 2 because this gives more values of 6641:, Alice first chooses a secret random value 6418:In October 2017, a team of researchers from 6395:, provided that the proper padding is used. 6286:, which even for "small" 1024-bit values of 5019:th root of the ciphertext over the integers. 1653:to Alice. Note that at least nine values of 338:describing the RSA algorithm was granted to 37: 8294:"Flaw Found in an Online Encryption Method" 3417:{\displaystyle h=\operatorname {hash} (m),} 1787:Choose two distinct prime numbers, such as 1198:. In fact, they can all be discarded after 218:". Breaking RSA encryption is known as the 9146: 9132: 9124: 8652: 8633: 8619: 8611: 8607: 8603: 8447:"Fault-Based Attack of RSA Authentication" 8165:Barker, Elaine; Dang, Quynh (2015-01-22). 7785:Johnson, J.; Kaliski, B. (February 2003). 7650: 7648: 6562:against the product of all the other keys 6343:It is important that the private exponent 5705:{\displaystyle m_{2}=c^{d_{Q}}{\pmod {q}}} 5640:{\displaystyle m_{1}=c^{d_{P}}{\pmod {p}}} 5022:If the same clear-text message is sent to 3360:of the message, raises it to the power of 1131:being coprime, said equation is a form of 8421: 8203:"RSA-512 certificates abused in-the-wild" 7974: 7881: 7796: 7702: 7426: 7288: 7207: 7196: 7167: 7154: 7134: 7128: 7103: 7090: 7084: 7035: 7031: 7019: 7013: 6986: 6982: 6970: 6964: 6202:Learn how and when to remove this message 5847:Integer factorization and the RSA problem 5811: 5799: 5766: 5757: 5744: 5731: 5719: 5686: 5678: 5673: 5660: 5654: 5621: 5613: 5608: 5595: 5589: 5535: 5526: 5513: 5507: 5466: 5454: 5448: 5407: 5395: 5389: 5367: 5347: 5212:. And given factorization of the modulus 5202:, one can efficiently factor the modulus 4834: 4822: 4800: 4781: 4744: 4728: 4722: 4453: 4441: 4431: 4422: 4356: 4341: 4325: 4309: 4275: 4250: 4234: 4228: 4200: 4084: 4069: 4053: 4037: 4003: 3978: 3962: 3956: 3928: 3744: 3635: 3623: 3613: 3604: 3517: 3505: 3495: 3476: 3460: 3447: 3437: 3428: 3388: 3278: 3242: 3238: 3211: 3207: 3195: 3182: 3166: 3127: 3123: 3117: 3104: 3100: 3092: 3087: 3070: 3044: 3040: 3034: 3021: 3017: 3009: 3004: 2987: 2979: 2977: 2908: 2904: 2886: 2882: 2867: 2835: 2831: 2822: 2809: 2805: 2796: 2779: 2744: 2740: 2716: 2712: 2696: 2661: 2657: 2633: 2629: 2613: 2605: 2603: 2493: 2489: 2483: 2471: 2432: 2428: 2422: 2410: 2373: 2369: 2363: 2343: 2339: 2333: 2306: 2304: 2251: 2247: 2241: 2221: 2217: 2211: 2184: 2182: 2131: 2127: 2104: 2075: 2020: 1934: 1863: 1820: 1794: 1736: 1724: 1714: 1698: 1692: 1636:{\displaystyle c\equiv m^{e}{\pmod {n}}.} 1614: 1608: 1596: 1389:to obtain a smaller equivalent exponent. 1112:can be computed efficiently by using the 569: 557: 547: 538: 518: 495: 471: 450: 440: 431: 98:variable but 2,048 to 4,096 bit typically 8130: 8128: 8068:"Factorization of a 512-bit RSA Modulus" 4150:proceeds in a completely analogous way: 1274:multiplicative group of integers modulo 1163:and the public (or encryption) exponent 8229:IEEE Transactions on Information Theory 7967:Advances in Cryptology — EUROCRYPT 2000 7574:Margaret Cozzens and Steven J. Miller. 7415:IEEE Transactions on Information Theory 7245: 6916:while also equal to −1, 0, or 1 modulo 6901: 6048:had 829 bits (250 decimal digits, 5177:from which they can derive the message 4493: 3725:is, by construction, divisible by both 1771:, she can recover the original message 644:In addition, because the two exponents 392:The RSA algorithm involves four steps: 8575:Explanation of RSA using colored lamps 7492: 7490: 6912:which are equal to −1, 0, or 1 modulo 5839:This is more efficient than computing 5128:may ask the holder of the private key 3821:To check whether two numbers, such as 1357:based on the Chinese remainder theorem 1035:is released as part of the public key. 735:is released as part of the public key. 316:Government Communications Headquarters 170:Government Communications Headquarters 36: 8352:"Remote timing attacks are practical" 8331:Heninger, Nadia (February 15, 2012). 8201:Sandee, Michael (November 21, 2011). 7693:McKee, James; Pinch, Richard (1998). 7617:. Cryptologia, Vol. 21, No. 4 (1997). 7261: 7259: 7257: 7255: 7253: 7251: 7249: 6709:Optimal Asymmetric Encryption Padding 5494:{\displaystyle d_{Q}=d{\pmod {q-1}},} 5435:{\displaystyle d_{P}=d{\pmod {p-1}},} 5319:Using the Chinese remainder algorithm 5314:Security and practical considerations 5289:Optimal Asymmetric Encryption Padding 1238:for calculating the private exponent 288:Massachusetts Institute of Technology 244:, co-inventor of RSA (the others are 176:agency, by the English mathematician 27:Algorithm for public-key cryptography 7: 8954:Naccache–Stern knapsack cryptosystem 7050:{\displaystyle m^{e-1}{\bmod {q}}=1} 7001:{\displaystyle m^{e-1}{\bmod {p}}=1} 6247:should not be "too close", lest the 6140:adding citations to reliable sources 2004:to 780. Choosing a prime number for 1578: 1408:are present in the factorisation of 606:, it is extremely difficult to find 8350:Brumley, David; Boneh, Dan (2003). 8292:Markoff, John (February 14, 2012). 8039:Machie, Edmond K. (29 March 2013). 7758:Dukhovni, Viktor (August 1, 2015). 7353:Castelvecchi, Davide (2020-10-30). 7215: 6558:to compute the GCD of each RSA key 6430:known as RSALib. A large number of 6306:is trivial. Furthermore, if either 5774: 5694: 5629: 5543: 5474: 5415: 4895:, the same conclusion follows from 4842: 4461: 4364: 4092: 3643: 3565:Proof using Fermat's little theorem 3525: 1889:{\displaystyle n=61\times 53=3233.} 1744: 1622: 577: 8110:. Cado-nfs-discuss. Archived from 7731:Dukhovni, Viktor (July 31, 2015). 7476:Calderbank, Michael (2007-08-20). 7327:Smart, Nigel (February 19, 2008). 6948:has other divisors in common with 6687:Adaptive chosen-ciphertext attacks 6407:is a commonly used value for 5995:for a discussion of this problem. 5230:) generated against a public key ( 5222:, one can obtain any private key ( 5063:deterministic encryption algorithm 5011:is strictly less than the modulus 3803:{\displaystyle ed-1=h(p-1)=k(q-1)} 2554:Practical implementations use the 2551:, thus acquiring the private key. 2398:For instance, in order to encrypt 1681:by using her private key exponent 1581:. He then computes the ciphertext 25: 8416:. ASIACCS '07. pp. 312–320. 7615:"Cryptography As a Teaching Tool" 7528:"A Note on Non-Secret Encryption" 6697:adaptive chosen-ciphertext attack 6509:'), then a simple computation of 6377:can be computed efficiently from 5983:that allows the determination of 5892:is defined as the task of taking 5281:adaptive chosen-ciphertext attack 4663:are positive integers satisfying 3684:are positive integers satisfying 1775:by reversing the padding scheme. 1363:(Section B.3.1) may require that 1025:is 3, but such a small value for 156:"RSA" comes from the surnames of 9496: 9495: 8492:Handbook of Applied Cryptography 8465:"Practical Cryptography With Go" 6824: 6116: 625:represents the private key, and 195:is public and distinct from the 8985:Discrete logarithm cryptography 8220:Wiener, Michael J. (May 1990). 8106:Zimmermann, Paul (2020-02-28). 7208: 6920:. There will be more values of 6845:Computational complexity theory 6127:needs additional citations for 5767: 5687: 5622: 5536: 5467: 5408: 4835: 4454: 4357: 4085: 3676:are distinct prime numbers and 3636: 3518: 1737: 1615: 1205:In the original RSA paper, the 570: 9357:Information-theoretic security 8395:"RSA Signature Forgery in NSS" 8315:"Ron was wrong, Whit is right" 7721:, Second edition, 1994. p. 94. 7219: 7209: 7119:, then some libraries compute 7112:{\displaystyle m_{1}<m_{2}} 6695:described the first practical 6490:is another, then if by chance 6320:has only small prime factors, 5778: 5768: 5763: 5737: 5698: 5688: 5633: 5623: 5547: 5537: 5484: 5468: 5425: 5409: 4846: 4836: 4819: 4812: 4797: 4791: 4785: 4774: 4763: 4757: 4701:for some non-negative integer 4468: 4455: 4438: 4424: 4368: 4358: 4322: 4302: 4291: 4279: 4096: 4086: 4050: 4030: 4019: 4007: 3810:for some nonnegative integers 3797: 3785: 3776: 3764: 3650: 3637: 3620: 3606: 3564: 3529: 3519: 3502: 3488: 3444: 3430: 3408: 3402: 3235: 3220: 3204: 3201: 3175: 3159: 2879: 2860: 2857: 2759: 2745: 2731: 2717: 2676: 2662: 2648: 2634: 2319: 2313: 2197: 2191: 2124: 2112: 2052:modular multiplicative inverse 1969: 1957: 1945: 1939: 1748: 1738: 1721: 1707: 1626: 1616: 1471:modular multiplicative inverse 1069:modular multiplicative inverse 872:may be calculated through the 581: 571: 554: 540: 447: 433: 1: 9526:Public-key encryption schemes 6718:Side-channel analysis attacks 6362:(which is quite typical) and 5857:Integer factorization records 5582:more efficiently as follows: 4565:, and thus trivially also by 2522:square-and-multiply algorithm 2299:, the decryption function is 2177:, the encryption function is 2008:leaves us only to check that 1902:Carmichael's totient function 1512:wants to send information to 1175:, which must be kept secret. 757:Carmichael's totient function 411:, such that for all integers 9000:Non-commutative cryptography 8585:Thorough walk through of RSA 8135:Kaliski, Burt (2003-05-06). 7497:Robinson, Sara (June 2003). 5896:th roots modulo a composite 5187:with the modular inverse of 5061:Because RSA encryption is a 4208:{\displaystyle \not \equiv } 3936:{\displaystyle \not \equiv } 1392:Since any common factors of 1114:extended Euclidean algorithm 629:represents the message. The 376:When the patent was issued, 9473:Message authentication code 9428:Cryptographic hash function 9231:Cryptographic hash function 9097:Identity-based cryptography 8990:Elliptic-curve cryptography 8182:10.6028/NIST.SP.800-57pt3r1 6860:Elliptic-curve cryptography 6855:Digital Signature Algorithm 6850:Diffie–Hellman key exchange 6728:simultaneous multithreading 6446:A cryptographically strong 6332:, and hence such values of 6324:can be factored quickly by 6015:extended Riemann hypothesis 5198:Given the private exponent 4925:is not relatively prime to 4686:are positive, we can write 4619:Proof using Euler's theorem 1585:, using Alice's public key 1139:is one of the coefficients. 690:until two primes are found. 459:{\displaystyle (m^{e})^{d}} 352:'s abstract of the patent: 9547: 9352:Harvest now, decrypt later 8529:Introduction to Algorithms 8108:"Factorization of RSA-250" 7379:10.1038/d41586-020-03068-9 6213: 5931:is an RSA public key, and 5850: 5841:exponentiation by squaring 5829:{\displaystyle m=m_{2}+hq} 4991:) and small values of the 678:To make factoring harder, 652:using the same algorithm. 122:General number field sieve 29: 9531:Digital signature schemes 9491: 9468:Post-quantum cryptography 9123: 9102:Post-quantum cryptography 9051:Post-Quantum Cryptography 8610: 8606: 8207:Fox-IT International blog 8045:. Trafford. p. 167. 7859:Coppersmith, Don (1997). 7791:. Network Working Group. 7602:"Public key Cryptography" 7276:Communications of the ACM 6620:Chinese remainder theorem 6224:Finding the large primes 5337:Chinese remainder theorem 5044:Chinese remainder theorem 4976:Attacks against plain RSA 4609:Chinese remainder theorem 3870:, we consider two cases: 2556:Chinese remainder theorem 2012:is not a divisor of 780. 1554:To do it, he first turns 1492:, do the reverse (choose 1442:, it is recommended that 621:comprise the public key, 598:However, when given only 210:the product of two large 120: 9458:Quantum key distribution 9448:Authenticated encryption 9303:Random number generation 8137:"TWIRL and RSA Key Size" 8084:Miller, Gary L. (1975). 7976:10.1007/3-540-45539-6_25 7836:10.1007/3-540-39799-X_29 7437:10.1109/TIT.1976.1055638 6556:Daniel J. Bernstein 6436:trusted platform modules 6029:is as hard as factoring 5111:chosen-ciphertext attack 4872:More generally, for any 1657:will yield a ciphertext 1562:(strictly speaking, the 1159:consists of the modulus 650:signing and verification 124:for classical computers; 9453:Public-key cryptography 9443:Symmetric-key algorithm 9236:Key derivation function 9196:Cryptographic primitive 9189:Authentication protocol 9174:Outline of cryptography 9169:History of cryptography 8995:Hash-based cryptography 8642:Public-key cryptography 8432:10.1145/1229285.1266999 8279:10.1145/3133956.3133969 6880:Public-key cryptography 6675:, and so the effect of 6475:is one public key, and 6448:random number generator 6151:"RSA" cryptosystem 5867:factoring large numbers 5853:RSA Factoring Challenge 5155:' is the encryption of 5067:chosen plaintext attack 4713:is relatively prime to 4618: 4117:Fermat's little theorem 3571:Fermat's little theorem 1253:is always divisible by 150:public-key cryptosystem 9241:Secure Hash Algorithms 9184:Cryptographic protocol 7329:"Dr Clifford Cocks CB" 7227: 7113: 7051: 7002: 6908:Namely, the values of 6840:Acoustic cryptanalysis 6628:cryptographic blinding 6540:, instead of choosing 5830: 5786: 5706: 5641: 5558: 5495: 5436: 5376: 5356: 4857: 4479: 4379: 4209: 4135:The verification that 4107: 3937: 3804: 3658: 3540: 3418: 3379:This works because of 3328: 2931: 2526:modular exponentiation 2512: 2451: 2390: 2268: 2144: 2093: 2092:{\displaystyle d=413,} 2035: 1982: 1890: 1835: 1809: 1759: 1647:modular exponentiation 1637: 1566:plaintext), such that 1539:is never distributed. 1266:Euler totient function 1207:Euler totient function 1145:is kept secret as the 1089:This means: solve for 700:should be kept secret. 631:modular exponentiation 592: 527: 504: 480: 460: 359: 342:on 20 September 1983: 270:modular exponentiation 253: 129:for quantum computers. 9347:End-to-end encryption 9293:Cryptojacking malware 8657:Integer factorization 8561:U.S. patent 4,405,829 8516:Leiserson, Charles E. 7934:10.1145/800070.802212 7892:10.1007/s001459900030 7869:Journal of Cryptology 7600:Surender R. Chiluka. 7299:10.1145/359340.359342 7228: 7114: 7052: 7003: 6741:Tricky implementation 6693:Daniel Bleichenbacher 6460:Arjen K. Lenstra 6340:should be discarded. 6108:Faulty key generation 5993:integer factorization 5900:: recovering a value 5831: 5794: 5787: 5714: 5707: 5649: 5642: 5584: 5559: 5496: 5437: 5377: 5357: 5307:U.S. patent 7,036,014 5301:U.S. patent 6,266,771 4858: 4630:We want to show that 4480: 4380: 4210: 4108: 3938: 3805: 3659: 3599:We want to show that 3560:Proofs of correctness 3541: 3419: 3329: 2932: 2513: 2452: 2391: 2269: 2145: 2094: 2036: 1983: 1891: 1836: 1810: 1760: 1649:. Bob then transmits 1638: 1321:. However, computing 593: 528: 505: 481: 461: 345:U.S. patent 4,405,829 240: 146:Rivest–Shamir–Adleman 18:Rivest Shamir Adleman 9463:Quantum cryptography 9387:Trusted timestamping 8015:"OpenSSL bn_s390x.c" 7760:"common factors in ( 7733:"common factors in ( 7544:on 28 September 2018 7526:(20 November 1973). 7127: 7083: 7012: 6963: 6705:Secure Sockets Layer 6616:Secure Sockets Layer 6397:Coppersmith's attack 6249:Fermat factorization 6216:Coppersmith's attack 6136:improve this article 6100:, breaking RSA; see 5798: 5718: 5653: 5588: 5506: 5447: 5388: 5366: 5346: 5056:Coppersmith's attack 4914:relatively prime to 4899:, which states that 4721: 4607:This is part of the 4421: 4227: 4199: 3955: 3927: 3743: 3603: 3427: 3387: 2976: 2602: 2470: 2409: 2303: 2181: 2103: 2074: 2034:{\displaystyle e=17} 2019: 1933: 1862: 1834:{\displaystyle q=53} 1819: 1808:{\displaystyle p=61} 1793: 1691: 1595: 1147:private key exponent 537: 517: 494: 470: 430: 314:intelligence agency 174:signals intelligence 172:(GCHQ), the British 9216:Cryptographic nonce 8960:Three-pass protocol 8375:. 25 September 2014 8095:. pp. 234–239. 7655:Boneh, Dan (1999). 7587:Alasdair McAndrew. 7371:2020Natur.587..189C 5071:semantically secure 2292:. For an encrypted 1589:, corresponding to 1465:and then computing 1227:is used instead of 1202:has been computed. 1116:, since, thanks to 874:Euclidean algorithm 363:Scientific American 39: 9332:Subliminal channel 9316:Pseudorandom noise 9258:Key (cryptography) 8730:Discrete logarithm 8299:The New York Times 7399:2020 interview of 7333:Bristol University 7223: 7109: 7047: 6998: 6885:Rabin cryptosystem 6832:Mathematics portal 6424:ROCA vulnerability 6420:Masaryk University 6255:be successful. If 5943:from a public key 5826: 5782: 5702: 5637: 5554: 5491: 5432: 5372: 5352: 5283:. Furthermore, at 5262:Standards such as 4853: 4475: 4375: 4205: 4103: 3933: 3800: 3664:for every integer 3654: 3536: 3414: 3324: 3322: 2927: 2925: 2508: 2447: 2386: 2384: 2264: 2262: 2140: 2089: 2031: 1992:Choose any number 1978: 1904:of the product as 1886: 1831: 1805: 1755: 1673:Alice can recover 1633: 1270:Lagrange's theorem 1268:results also from 935:Choose an integer 588: 523: 500: 476: 456: 367:Mathematical Games 268:, RSA is based on 254: 180:. That system was 9513: 9512: 9509: 9508: 9392:Key-based routing 9382:Trapdoor function 9248:Digital signature 9119: 9118: 9115: 9114: 9067:Digital signature 9010:Trapdoor function 8973: 8972: 8690:Goldwasser–Micali 8543:978-0-262-03293-3 8520:Rivest, Ronald L. 8512:Cormen, Thomas H. 8503:978-0-8493-8523-0 8337:Freedom to Tinker 7986:978-3-540-45539-4 7943:978-0-89791-070-5 7914:Goldwasser, Shafi 7845:978-3-540-16463-0 7175: 7137: 6890:Trapdoor function 6584:atmospheric noise 6212: 6211: 6204: 6186: 6074:personal computer 5734: 5516: 5375:{\displaystyle q} 5355:{\displaystyle p} 5005:), the result of 4169:is a multiple of 3897:is a multiple of 3889:is a multiple of 3169: 2870: 2782: 2466:, one calculates 2405:, one calculates 1133:Bézout's identity 664:Choose two large 526:{\displaystyle n} 503:{\displaystyle n} 479:{\displaystyle m} 216:factoring problem 139: 138: 16:(Redirected from 9538: 9499: 9498: 9327:Insecure channel 9179:Classical cipher 9148: 9141: 9134: 9125: 8956: 8857: 8852: 8812:signature scheme 8715:Okamoto–Uchiyama 8653: 8635: 8628: 8621: 8612: 8608: 8604: 8576: 8563: 8547: 8507: 8495: 8476: 8475: 8473: 8471: 8460: 8454: 8453: 8451: 8442: 8436: 8435: 8425: 8409: 8403: 8402: 8391: 8385: 8384: 8382: 8380: 8369: 8363: 8362: 8356: 8347: 8341: 8340: 8328: 8322: 8321: 8319: 8310: 8304: 8303: 8289: 8283: 8282: 8268: 8259: 8253: 8252: 8241:10.1109/18.54902 8226: 8217: 8211: 8210: 8198: 8192: 8191: 8189: 8188: 8171: 8162: 8156: 8155: 8153: 8152: 8143:. Archived from 8141:RSA Laboratories 8132: 8123: 8122: 8120: 8119: 8103: 8097: 8096: 8090: 8081: 8075: 8074: 8072: 8063: 8057: 8056: 8036: 8030: 8029: 8027: 8025: 8011: 8005: 8004: 7997: 7991: 7990: 7978: 7962: 7956: 7955: 7910: 7904: 7903: 7885: 7865: 7856: 7850: 7849: 7823: 7817: 7816: 7814: 7812: 7800: 7798:10.17487/RFC3447 7782: 7776: 7775: 7755: 7749: 7748: 7728: 7722: 7715: 7709: 7708: 7706: 7690: 7684: 7677: 7671: 7670: 7652: 7643: 7642: 7640: 7639: 7634:on June 21, 2007 7630:. Archived from 7624: 7618: 7611: 7605: 7598: 7592: 7585: 7579: 7572: 7566: 7559: 7553: 7552: 7550: 7549: 7543: 7537:. Archived from 7532: 7520: 7514: 7513: 7503: 7494: 7485: 7484: 7482: 7473: 7467: 7466: 7464: 7458:Rivest, Ronald. 7455: 7449: 7448: 7430: 7410: 7404: 7398: 7350: 7344: 7343: 7341: 7339: 7324: 7318: 7317: 7315: 7309:. Archived from 7292: 7272: 7263: 7234: 7232: 7230: 7229: 7224: 7222: 7206: 7202: 7201: 7200: 7188: 7184: 7180: 7176: 7168: 7159: 7158: 7139: 7138: 7135: 7122: 7118: 7116: 7115: 7110: 7108: 7107: 7095: 7094: 7077: 7071: 7064: 7058: 7056: 7054: 7053: 7048: 7040: 7039: 7030: 7029: 7007: 7005: 7004: 6999: 6991: 6990: 6981: 6980: 6958: 6954: 6947: 6940: 6933: 6923: 6919: 6915: 6911: 6906: 6834: 6829: 6828: 6724:branch predictor 6682: 6678: 6674: 6659: 6644: 6640: 6605: 6577: 6565: 6561: 6547: 6543: 6539: 6535: 6531: 6527: 6523: 6508: 6505:is not equal to 6504: 6500: 6489: 6474: 6457: 6453: 6414: 6410: 6402: 6394: 6384: 6380: 6376: 6372: 6361: 6354: 6350: 6346: 6339: 6335: 6323: 6319: 6312: 6305: 6301: 6297: 6295: 6289: 6285: 6271: 6264: 6254: 6246: 6242: 6231: 6227: 6207: 6200: 6196: 6193: 6187: 6185: 6144: 6120: 6112: 6102:Shor's algorithm 6094:quantum computer 6067: 6060: 6040: 6036: 6032: 6028: 6024: 6020: 6001: 5990: 5986: 5982: 5970: 5966: 5962: 5958: 5954: 5942: 5938: 5934: 5930: 5918: 5903: 5899: 5895: 5861:Shor's algorithm 5835: 5833: 5832: 5827: 5816: 5815: 5791: 5789: 5788: 5783: 5781: 5762: 5761: 5749: 5748: 5736: 5735: 5732: 5711: 5709: 5708: 5703: 5701: 5685: 5684: 5683: 5682: 5665: 5664: 5646: 5644: 5643: 5638: 5636: 5620: 5619: 5618: 5617: 5600: 5599: 5581: 5563: 5561: 5560: 5555: 5550: 5534: 5533: 5518: 5517: 5514: 5500: 5498: 5497: 5492: 5487: 5459: 5458: 5441: 5439: 5438: 5433: 5428: 5400: 5399: 5381: 5379: 5378: 5373: 5361: 5359: 5358: 5353: 5309: 5303: 5273: 5269: 5258: 5254: 5237: 5233: 5229: 5225: 5221: 5211: 5201: 5194: 5190: 5186: 5180: 5176: 5165: 5154: 5150: 5146: 5131: 5127: 5108: 5041: 5038:, and therefore 5037: 5033: 5030:, but different 5029: 5025: 5018: 5014: 5010: 5004: 4994: 4990: 4966: 4955: 4944: 4928: 4924: 4917: 4913: 4909: 4894: 4879: 4875: 4862: 4860: 4859: 4854: 4849: 4827: 4826: 4805: 4804: 4795: 4794: 4767: 4766: 4736: 4735: 4716: 4712: 4704: 4700: 4685: 4681: 4677: 4662: 4658: 4654: 4644: 4612: 4605: 4599: 4597: 4582: 4578: 4571: 4564: 4554:is divisible by 4553: 4541: 4526: 4522: 4517: 4511: 4509: 4498: 4484: 4482: 4481: 4476: 4471: 4446: 4445: 4436: 4435: 4416: 4401: 4397: 4393: 4384: 4382: 4381: 4376: 4371: 4346: 4345: 4330: 4329: 4320: 4319: 4295: 4294: 4264: 4263: 4242: 4241: 4220: 4214: 4212: 4211: 4206: 4187: 4172: 4164: 4149: 4128: 4112: 4110: 4109: 4104: 4099: 4074: 4073: 4058: 4057: 4048: 4047: 4023: 4022: 3992: 3991: 3970: 3969: 3948: 3942: 3940: 3939: 3934: 3915: 3900: 3892: 3888: 3884: 3869: 3851: 3844: 3837: 3831:, are congruent 3830: 3826: 3817: 3813: 3809: 3807: 3806: 3801: 3738: 3731: 3724: 3698: 3683: 3679: 3675: 3671: 3667: 3663: 3661: 3660: 3655: 3653: 3628: 3627: 3618: 3617: 3595: 3591: 3587: 3584:for any integer 3583: 3545: 3543: 3542: 3537: 3532: 3510: 3509: 3500: 3499: 3484: 3483: 3468: 3467: 3452: 3451: 3442: 3441: 3423: 3421: 3420: 3415: 3375: 3371: 3367: 3363: 3337:Signing messages 3333: 3331: 3330: 3325: 3323: 3283: 3282: 3247: 3246: 3216: 3215: 3200: 3199: 3187: 3186: 3171: 3170: 3167: 3132: 3131: 3122: 3121: 3109: 3108: 3099: 3098: 3097: 3096: 3075: 3074: 3049: 3048: 3039: 3038: 3026: 3025: 3016: 3015: 3014: 3013: 2992: 2991: 2971: 2967: 2963: 2956: 2947: 2936: 2934: 2933: 2928: 2926: 2913: 2912: 2891: 2890: 2872: 2871: 2868: 2853: 2840: 2839: 2830: 2829: 2814: 2813: 2804: 2803: 2784: 2783: 2780: 2749: 2748: 2721: 2720: 2701: 2700: 2666: 2665: 2638: 2637: 2618: 2617: 2597: 2590: 2581: 2550: 2546: 2542: 2538: 2534: 2517: 2515: 2514: 2509: 2498: 2497: 2488: 2487: 2465: 2456: 2454: 2453: 2448: 2437: 2436: 2427: 2426: 2404: 2395: 2393: 2392: 2387: 2385: 2378: 2377: 2368: 2367: 2352: 2348: 2347: 2338: 2337: 2298: 2291: 2273: 2271: 2270: 2265: 2263: 2256: 2255: 2246: 2245: 2230: 2226: 2225: 2216: 2215: 2176: 2168: 2149: 2147: 2146: 2141: 2136: 2135: 2098: 2096: 2095: 2090: 2068: 2049: 2040: 2038: 2037: 2032: 2011: 2007: 1999: 1987: 1985: 1984: 1979: 1926: 1895: 1893: 1892: 1887: 1855: 1840: 1838: 1837: 1832: 1814: 1812: 1811: 1806: 1774: 1770: 1764: 1762: 1761: 1756: 1751: 1729: 1728: 1719: 1718: 1703: 1702: 1684: 1680: 1676: 1664: 1660: 1656: 1652: 1642: 1640: 1639: 1634: 1629: 1613: 1612: 1588: 1584: 1576: 1561: 1557: 1550: 1538: 1530: 1504:Key distribution 1499: 1495: 1487: 1476: 1468: 1464: 1457: 1449: 1441: 1421: 1414: 1407: 1399: 1388: 1377: 1354: 1350: 1335: 1324: 1320: 1301: 1282: 1263: 1252: 1241: 1237: 1226: 1201: 1197: 1193: 1182: 1178: 1174: 1166: 1162: 1144: 1138: 1130: 1119: 1111: 1107: 1092: 1085: 1074: 1066: 1062: 1043: 1034: 1028: 1024: 1020: 1019: 1018: 1010: 998: 988: 977: 973: 954: 938: 929: 916: 915: 913: 912: 901: 898: 896: 871: 864: 845: 830: 805: 801: 797: 754: 750: 734: 720: 714: 699: 695: 685: 681: 674: 670: 640: 636: 628: 624: 620: 616: 609: 605: 601: 597: 595: 594: 589: 584: 562: 561: 552: 551: 532: 530: 529: 524: 512:congruent modulo 509: 507: 506: 501: 490:when divided by 485: 483: 482: 477: 465: 463: 462: 457: 455: 454: 445: 444: 425: 414: 410: 406: 402: 347: 310:working for the 258:Whitfield Diffie 187:In a public-key 135:has been broken. 127:Shor's algorithm 40: 21: 9546: 9545: 9541: 9540: 9539: 9537: 9536: 9535: 9516: 9515: 9514: 9505: 9487: 9416: 9157: 9152: 9111: 9055: 9019:Standardization 9014: 8969: 8952: 8901: 8849:Lattice/SVP/CVP 8843: 8724: 8670:Blum–Goldwasser 8644: 8639: 8574: 8559: 8554: 8544: 8524:Stein, Clifford 8510: 8504: 8487: 8484: 8482:Further reading 8479: 8469: 8467: 8462: 8461: 8457: 8449: 8444: 8443: 8439: 8411: 8410: 8406: 8393: 8392: 8388: 8378: 8376: 8371: 8370: 8366: 8354: 8349: 8348: 8344: 8330: 8329: 8325: 8317: 8312: 8311: 8307: 8291: 8290: 8286: 8266: 8261: 8260: 8256: 8224: 8219: 8218: 8214: 8200: 8199: 8195: 8186: 8184: 8169: 8164: 8163: 8159: 8150: 8148: 8134: 8133: 8126: 8117: 8115: 8105: 8104: 8100: 8088: 8083: 8082: 8078: 8070: 8065: 8064: 8060: 8053: 8038: 8037: 8033: 8023: 8021: 8013: 8012: 8008: 8001:"RSA Algorithm" 7999: 7998: 7994: 7987: 7964: 7963: 7959: 7944: 7912: 7911: 7907: 7883:10.1.1.298.4806 7863: 7858: 7857: 7853: 7846: 7825: 7824: 7820: 7810: 7808: 7784: 7783: 7779: 7774:(Mailing list). 7757: 7756: 7752: 7747:(Mailing list). 7730: 7729: 7725: 7716: 7712: 7692: 7691: 7687: 7678: 7674: 7654: 7653: 7646: 7637: 7635: 7626: 7625: 7621: 7612: 7608: 7599: 7595: 7586: 7582: 7573: 7569: 7561:Jim Sauerberg. 7560: 7556: 7547: 7545: 7541: 7535:www.gchq.gov.uk 7530: 7522: 7521: 7517: 7501: 7496: 7495: 7488: 7480: 7475: 7474: 7470: 7462: 7457: 7456: 7452: 7412: 7411: 7407: 7352: 7351: 7347: 7337: 7335: 7326: 7325: 7321: 7313: 7290:10.1.1.607.2677 7270: 7265: 7264: 7247: 7243: 7238: 7237: 7192: 7163: 7150: 7149: 7145: 7144: 7140: 7130: 7125: 7124: 7120: 7099: 7086: 7081: 7080: 7078: 7074: 7065: 7061: 7015: 7010: 7009: 6966: 6961: 6960: 6956: 6949: 6942: 6935: 6925: 6921: 6917: 6913: 6909: 6907: 6903: 6898: 6830: 6823: 6820: 6755: 6753:Implementations 6743: 6720: 6689: 6680: 6676: 6665: 6662:Euler's theorem 6646: 6642: 6631: 6603: 6596: 6567: 6563: 6559: 6548:independently. 6545: 6541: 6537: 6533: 6529: 6525: 6510: 6506: 6502: 6491: 6476: 6466: 6455: 6451: 6444: 6412: 6408: 6400: 6389: 6382: 6378: 6374: 6363: 6356: 6352: 6348: 6344: 6337: 6333: 6321: 6314: 6307: 6303: 6299: 6298:), solving for 6293: 6291: 6287: 6273: 6266: 6256: 6252: 6244: 6240: 6234:primality tests 6229: 6225: 6222: 6220:Wiener's attack 6208: 6197: 6191: 6188: 6145: 6143: 6133: 6121: 6110: 6098:polynomial time 6065: 6058: 6038: 6034: 6030: 6026: 6022: 6018: 5999: 5988: 5984: 5972: 5971:, and computes 5968: 5964: 5960: 5956: 5955:, then decrypt 5944: 5940: 5936: 5932: 5920: 5905: 5901: 5897: 5893: 5863: 5849: 5807: 5796: 5795: 5793: 5753: 5740: 5727: 5716: 5715: 5713: 5674: 5669: 5656: 5651: 5650: 5648: 5609: 5604: 5591: 5586: 5585: 5583: 5568: 5522: 5509: 5504: 5503: 5450: 5445: 5444: 5391: 5386: 5385: 5364: 5363: 5344: 5343: 5321: 5316: 5305: 5299: 5271: 5267: 5256: 5252: 5251:into the value 5245: 5243:Padding schemes 5235: 5231: 5227: 5223: 5213: 5203: 5199: 5192: 5188: 5182: 5181:by multiplying 5178: 5167: 5156: 5152: 5148: 5147:for some value 5133: 5129: 5114: 5102: 5096: 5089: 5083: 5077: 5052:Don Coppersmith 5039: 5035: 5031: 5027: 5023: 5016: 5012: 5006: 4996: 4992: 4985: 4978: 4973: 4957: 4946: 4930: 4926: 4922: 4915: 4911: 4900: 4881: 4877: 4873: 4867:Euler's theorem 4818: 4796: 4777: 4740: 4724: 4719: 4718: 4714: 4710: 4702: 4687: 4683: 4679: 4664: 4660: 4656: 4646: 4631: 4625:Euler's theorem 4621: 4616: 4615: 4606: 4602: 4584: 4580: 4573: 4566: 4555: 4543: 4528: 4524: 4520: 4518: 4514: 4505: 4499: 4495: 4490: 4437: 4427: 4419: 4418: 4403: 4399: 4395: 4394:, and integers 4391: 4337: 4321: 4305: 4271: 4246: 4230: 4225: 4224: 4197: 4196: 4192: 4174: 4170: 4155: 4136: 4120: 4065: 4049: 4033: 3999: 3974: 3958: 3953: 3952: 3925: 3924: 3920: 3902: 3898: 3890: 3886: 3875: 3856: 3846: 3839: 3832: 3828: 3822: 3815: 3811: 3741: 3740: 3739:, we can write 3733: 3726: 3703: 3685: 3681: 3677: 3673: 3669: 3665: 3619: 3609: 3601: 3600: 3593: 3592:, not dividing 3589: 3585: 3574: 3573:, stating that 3567: 3562: 3501: 3491: 3472: 3456: 3443: 3433: 3425: 3424: 3385: 3384: 3373: 3369: 3365: 3361: 3339: 3321: 3320: 3274: 3267: 3261: 3260: 3191: 3178: 3162: 3152: 3146: 3145: 3113: 3088: 3083: 3076: 3066: 3063: 3062: 3030: 3005: 3000: 2993: 2983: 2974: 2973: 2969: 2965: 2962: 2958: 2955: 2949: 2946: 2940: 2924: 2923: 2863: 2851: 2850: 2818: 2792: 2785: 2775: 2772: 2771: 2702: 2692: 2689: 2688: 2619: 2609: 2600: 2599: 2596: 2592: 2589: 2583: 2580: 2574: 2548: 2544: 2540: 2536: 2529: 2479: 2468: 2467: 2460: 2418: 2407: 2406: 2399: 2383: 2382: 2359: 2350: 2349: 2329: 2322: 2301: 2300: 2296: 2281: 2261: 2260: 2237: 2228: 2227: 2207: 2200: 2179: 2178: 2174: 2169:. For a padded 2158: 2101: 2100: 2072: 2071: 2070: 2055: 2047: 2017: 2016: 2009: 2005: 1993: 1931: 1930: 1905: 1860: 1859: 1847: 1817: 1816: 1791: 1790: 1781: 1772: 1768: 1720: 1710: 1694: 1689: 1688: 1682: 1678: 1674: 1671: 1662: 1658: 1654: 1650: 1604: 1593: 1592: 1586: 1582: 1567: 1559: 1555: 1548: 1545: 1532: 1520: 1506: 1497: 1493: 1478: 1474: 1466: 1462: 1451: 1443: 1423: 1416: 1409: 1401: 1393: 1379: 1364: 1361:FIPS 186-4 1352: 1337: 1326: 1322: 1303: 1302:also satisfies 1284: 1280: 1272:applied to the 1254: 1243: 1239: 1228: 1209: 1199: 1195: 1184: 1180: 1176: 1172: 1164: 1160: 1142: 1136: 1121: 1117: 1109: 1094: 1090: 1076: 1072: 1064: 1045: 1041: 1032: 1026: 1022: 1016: 1014: 1012: 1008: 999:having a short 996: 979: 975: 956: 940: 936: 930:is kept secret. 920: 902: 899: 892: 890: 889: 887: 877: 869: 847: 832: 831:, and likewise 807: 803: 799: 760: 752: 741: 732: 721:is used as the 718: 706: 697: 693: 683: 679: 672: 668: 658: 638: 634: 626: 622: 618: 614: 607: 603: 599: 553: 543: 535: 534: 515: 514: 492: 491: 468: 467: 446: 436: 428: 427: 416: 412: 408: 404: 400: 390: 378:terms of patent 343: 332: 284:Leonard Adleman 250:Leonard Adleman 235: 166:Leonard Adleman 130: 125: 66:First published 60:Leonard Adleman 35: 28: 23: 22: 15: 12: 11: 5: 9544: 9542: 9534: 9533: 9528: 9518: 9517: 9511: 9510: 9507: 9506: 9504: 9503: 9492: 9489: 9488: 9486: 9485: 9480: 9478:Random numbers 9475: 9470: 9465: 9460: 9455: 9450: 9445: 9440: 9435: 9430: 9424: 9422: 9418: 9417: 9415: 9414: 9409: 9404: 9402:Garlic routing 9399: 9394: 9389: 9384: 9379: 9374: 9369: 9364: 9359: 9354: 9349: 9344: 9339: 9334: 9329: 9324: 9322:Secure channel 9319: 9313: 9312: 9311: 9300: 9295: 9290: 9285: 9280: 9278:Key stretching 9275: 9270: 9265: 9260: 9255: 9250: 9245: 9244: 9243: 9238: 9233: 9223: 9221:Cryptovirology 9218: 9213: 9208: 9206:Cryptocurrency 9203: 9198: 9193: 9192: 9191: 9181: 9176: 9171: 9165: 9163: 9159: 9158: 9153: 9151: 9150: 9143: 9136: 9128: 9121: 9120: 9117: 9116: 9113: 9112: 9110: 9109: 9104: 9099: 9094: 9089: 9084: 9079: 9074: 9069: 9063: 9061: 9057: 9056: 9054: 9053: 9048: 9043: 9038: 9033: 9028: 9022: 9020: 9016: 9015: 9013: 9012: 9007: 9002: 8997: 8992: 8987: 8981: 8979: 8975: 8974: 8971: 8970: 8968: 8967: 8962: 8957: 8950: 8948:Merkle–Hellman 8945: 8940: 8935: 8930: 8925: 8920: 8915: 8909: 8907: 8903: 8902: 8900: 8899: 8894: 8889: 8884: 8879: 8874: 8869: 8863: 8861: 8845: 8844: 8842: 8841: 8836: 8831: 8826: 8821: 8816: 8815: 8814: 8804: 8799: 8794: 8793: 8792: 8787: 8777: 8772: 8771: 8770: 8765: 8755: 8750: 8745: 8740: 8734: 8732: 8726: 8725: 8723: 8722: 8717: 8712: 8707: 8702: 8697: 8695:Naccache–Stern 8692: 8687: 8682: 8677: 8672: 8667: 8661: 8659: 8650: 8646: 8645: 8640: 8638: 8637: 8630: 8623: 8615: 8601: 8600: 8592: 8587: 8582: 8571: 8566: 8553: 8552:External links 8550: 8549: 8548: 8542: 8508: 8502: 8483: 8480: 8478: 8477: 8455: 8437: 8423:10.1.1.80.1438 8404: 8386: 8364: 8342: 8323: 8305: 8284: 8254: 8235:(3): 553–558. 8212: 8193: 8176:. p. 12. 8157: 8124: 8098: 8076: 8058: 8052:978-1466985742 8051: 8031: 8006: 7992: 7985: 7957: 7942: 7920:(1982-05-05). 7918:Micali, Silvio 7905: 7876:(4): 233–260. 7851: 7844: 7818: 7777: 7750: 7723: 7710: 7704:10.1.1.33.1333 7685: 7681:Bruce Schneier 7672: 7644: 7619: 7613:Neal Koblitz. 7606: 7593: 7580: 7567: 7554: 7515: 7486: 7468: 7450: 7428:10.1.1.37.9720 7421:(6): 644–654. 7405: 7345: 7319: 7316:on 2023-01-27. 7283:(2): 120–126. 7244: 7242: 7239: 7236: 7235: 7221: 7218: 7214: 7211: 7205: 7199: 7195: 7191: 7187: 7183: 7179: 7174: 7171: 7166: 7162: 7157: 7153: 7148: 7143: 7133: 7106: 7102: 7098: 7093: 7089: 7072: 7059: 7046: 7043: 7038: 7034: 7028: 7025: 7022: 7018: 6997: 6994: 6989: 6985: 6979: 6976: 6973: 6969: 6953: − 1 6946: − 1 6939: − 1 6900: 6899: 6897: 6894: 6893: 6892: 6887: 6882: 6877: 6872: 6870:Key management 6867: 6862: 6857: 6852: 6847: 6842: 6836: 6835: 6819: 6816: 6815: 6814: 6809: 6804: 6799: 6794: 6789: 6784: 6779: 6774: 6769: 6764: 6754: 6751: 6742: 6739: 6719: 6716: 6701:padding scheme 6688: 6685: 6595: 6594:Timing attacks 6592: 6552:Nadia Heninger 6443: 6440: 6422:announced the 6210: 6209: 6124: 6122: 6115: 6109: 6106: 6092:showed that a 5883:padding scheme 5848: 5845: 5825: 5822: 5819: 5814: 5810: 5806: 5803: 5780: 5777: 5773: 5770: 5765: 5760: 5756: 5752: 5747: 5743: 5739: 5730: 5726: 5723: 5700: 5697: 5693: 5690: 5681: 5677: 5672: 5668: 5663: 5659: 5635: 5632: 5628: 5625: 5616: 5612: 5607: 5603: 5598: 5594: 5565: 5564: 5553: 5549: 5546: 5542: 5539: 5532: 5529: 5525: 5521: 5512: 5501: 5490: 5486: 5483: 5480: 5477: 5473: 5470: 5465: 5462: 5457: 5453: 5442: 5431: 5427: 5424: 5421: 5418: 5414: 5411: 5406: 5403: 5398: 5394: 5383: 5371: 5351: 5320: 5317: 5315: 5312: 5244: 5241: 5240: 5239: 5196: 5100: 5094: 5087: 5081: 5074: 5059: 5020: 4977: 4974: 4972: 4969: 4852: 4848: 4845: 4841: 4838: 4833: 4830: 4825: 4821: 4817: 4814: 4811: 4808: 4803: 4799: 4793: 4790: 4787: 4784: 4780: 4776: 4773: 4770: 4765: 4762: 4759: 4756: 4753: 4750: 4747: 4743: 4739: 4734: 4731: 4727: 4620: 4617: 4614: 4613: 4600: 4512: 4492: 4491: 4489: 4486: 4474: 4470: 4467: 4464: 4460: 4457: 4452: 4449: 4444: 4440: 4434: 4430: 4426: 4388: 4387: 4386: 4385: 4374: 4370: 4367: 4363: 4360: 4355: 4352: 4349: 4344: 4340: 4336: 4333: 4328: 4324: 4318: 4315: 4312: 4308: 4304: 4301: 4298: 4293: 4290: 4287: 4284: 4281: 4278: 4274: 4270: 4267: 4262: 4259: 4256: 4253: 4249: 4245: 4240: 4237: 4233: 4204: 4189: 4133: 4132: 4131: 4130: 4115:where we used 4113: 4102: 4098: 4095: 4091: 4088: 4083: 4080: 4077: 4072: 4068: 4064: 4061: 4056: 4052: 4046: 4043: 4040: 4036: 4032: 4029: 4026: 4021: 4018: 4015: 4012: 4009: 4006: 4002: 3998: 3995: 3990: 3987: 3984: 3981: 3977: 3973: 3968: 3965: 3961: 3932: 3917: 3799: 3796: 3793: 3790: 3787: 3784: 3781: 3778: 3775: 3772: 3769: 3766: 3763: 3760: 3757: 3754: 3751: 3748: 3652: 3649: 3646: 3642: 3639: 3634: 3631: 3626: 3622: 3616: 3612: 3608: 3566: 3563: 3561: 3558: 3557: 3556: 3553: 3535: 3531: 3528: 3524: 3521: 3516: 3513: 3508: 3504: 3498: 3494: 3490: 3487: 3482: 3479: 3475: 3471: 3466: 3463: 3459: 3455: 3450: 3446: 3440: 3436: 3432: 3413: 3410: 3407: 3404: 3401: 3398: 3395: 3392: 3381:exponentiation 3338: 3335: 3319: 3316: 3313: 3310: 3307: 3304: 3301: 3298: 3295: 3292: 3289: 3286: 3281: 3277: 3273: 3270: 3268: 3266: 3263: 3262: 3259: 3256: 3253: 3250: 3245: 3241: 3237: 3234: 3231: 3228: 3225: 3222: 3219: 3214: 3210: 3206: 3203: 3198: 3194: 3190: 3185: 3181: 3177: 3174: 3165: 3161: 3158: 3155: 3153: 3151: 3148: 3147: 3144: 3141: 3138: 3135: 3130: 3126: 3120: 3116: 3112: 3107: 3103: 3095: 3091: 3086: 3082: 3079: 3077: 3073: 3069: 3065: 3064: 3061: 3058: 3055: 3052: 3047: 3043: 3037: 3033: 3029: 3024: 3020: 3012: 3008: 3003: 2999: 2996: 2994: 2990: 2986: 2982: 2981: 2960: 2951: 2942: 2922: 2919: 2916: 2911: 2907: 2903: 2900: 2897: 2894: 2889: 2885: 2881: 2878: 2875: 2866: 2862: 2859: 2856: 2854: 2852: 2849: 2846: 2843: 2838: 2834: 2828: 2825: 2821: 2817: 2812: 2808: 2802: 2799: 2795: 2791: 2788: 2786: 2778: 2774: 2773: 2770: 2767: 2764: 2761: 2758: 2755: 2752: 2747: 2743: 2739: 2736: 2733: 2730: 2727: 2724: 2719: 2715: 2711: 2708: 2705: 2703: 2699: 2695: 2691: 2690: 2687: 2684: 2681: 2678: 2675: 2672: 2669: 2664: 2660: 2656: 2653: 2650: 2647: 2644: 2641: 2636: 2632: 2628: 2625: 2622: 2620: 2616: 2612: 2608: 2607: 2594: 2585: 2576: 2507: 2504: 2501: 2496: 2492: 2486: 2482: 2478: 2475: 2446: 2443: 2440: 2435: 2431: 2425: 2421: 2417: 2414: 2381: 2376: 2372: 2366: 2362: 2358: 2355: 2353: 2351: 2346: 2342: 2336: 2332: 2328: 2325: 2323: 2321: 2318: 2315: 2312: 2309: 2308: 2259: 2254: 2250: 2244: 2240: 2236: 2233: 2231: 2229: 2224: 2220: 2214: 2210: 2206: 2203: 2201: 2199: 2196: 2193: 2190: 2187: 2186: 2151: 2150: 2139: 2134: 2130: 2126: 2123: 2120: 2117: 2114: 2111: 2108: 2088: 2085: 2082: 2079: 2044: 2043: 2042: 2030: 2027: 2024: 1990: 1989: 1988: 1977: 1974: 1971: 1968: 1965: 1962: 1959: 1956: 1953: 1950: 1947: 1944: 1941: 1938: 1898: 1897: 1896: 1885: 1882: 1879: 1876: 1873: 1870: 1867: 1844: 1843: 1842: 1830: 1827: 1824: 1804: 1801: 1798: 1780: 1777: 1754: 1750: 1747: 1743: 1740: 1735: 1732: 1727: 1723: 1717: 1713: 1709: 1706: 1701: 1697: 1670: 1667: 1632: 1628: 1625: 1621: 1618: 1611: 1607: 1603: 1600: 1579:padding scheme 1544: 1541: 1505: 1502: 1153: 1152: 1151: 1150: 1140: 1038: 1037: 1036: 1030: 1005:Hamming weight 933: 932: 931: 918: 738: 737: 736: 730: 703: 702: 701: 691: 688:primality test 657: 656:Key generation 654: 646:can be swapped 587: 583: 580: 576: 573: 568: 565: 560: 556: 550: 546: 542: 522: 499: 486:have the same 475: 453: 449: 443: 439: 435: 389: 386: 331: 328: 324:simplified DES 304:Clifford Cocks 266:Diffie-Hellman 262:Martin Hellman 234: 231: 197:decryption key 193:encryption key 178:Clifford Cocks 137: 136: 118: 117: 110: 109: 106: 100: 99: 96: 90: 89: 85: 84: 75: 71: 70: 67: 63: 62: 49: 45: 44: 26: 24: 14: 13: 10: 9: 6: 4: 3: 2: 9543: 9532: 9529: 9527: 9524: 9523: 9521: 9502: 9494: 9493: 9490: 9484: 9483:Steganography 9481: 9479: 9476: 9474: 9471: 9469: 9466: 9464: 9461: 9459: 9456: 9454: 9451: 9449: 9446: 9444: 9441: 9439: 9438:Stream cipher 9436: 9434: 9431: 9429: 9426: 9425: 9423: 9419: 9413: 9410: 9408: 9405: 9403: 9400: 9398: 9397:Onion routing 9395: 9393: 9390: 9388: 9385: 9383: 9380: 9378: 9377:Shared secret 9375: 9373: 9370: 9368: 9365: 9363: 9360: 9358: 9355: 9353: 9350: 9348: 9345: 9343: 9340: 9338: 9335: 9333: 9330: 9328: 9325: 9323: 9320: 9317: 9314: 9309: 9306: 9305: 9304: 9301: 9299: 9296: 9294: 9291: 9289: 9286: 9284: 9281: 9279: 9276: 9274: 9271: 9269: 9268:Key generator 9266: 9264: 9261: 9259: 9256: 9254: 9251: 9249: 9246: 9242: 9239: 9237: 9234: 9232: 9229: 9228: 9227: 9226:Hash function 9224: 9222: 9219: 9217: 9214: 9212: 9209: 9207: 9204: 9202: 9201:Cryptanalysis 9199: 9197: 9194: 9190: 9187: 9186: 9185: 9182: 9180: 9177: 9175: 9172: 9170: 9167: 9166: 9164: 9160: 9156: 9149: 9144: 9142: 9137: 9135: 9130: 9129: 9126: 9122: 9108: 9105: 9103: 9100: 9098: 9095: 9093: 9090: 9088: 9085: 9083: 9080: 9078: 9075: 9073: 9070: 9068: 9065: 9064: 9062: 9058: 9052: 9049: 9047: 9044: 9042: 9039: 9037: 9034: 9032: 9029: 9027: 9024: 9023: 9021: 9017: 9011: 9008: 9006: 9003: 9001: 8998: 8996: 8993: 8991: 8988: 8986: 8983: 8982: 8980: 8976: 8966: 8963: 8961: 8958: 8955: 8951: 8949: 8946: 8944: 8941: 8939: 8936: 8934: 8931: 8929: 8926: 8924: 8921: 8919: 8916: 8914: 8911: 8910: 8908: 8904: 8898: 8895: 8893: 8890: 8888: 8885: 8883: 8880: 8878: 8875: 8873: 8870: 8868: 8865: 8864: 8862: 8860: 8855: 8850: 8846: 8840: 8837: 8835: 8832: 8830: 8827: 8825: 8822: 8820: 8817: 8813: 8810: 8809: 8808: 8805: 8803: 8800: 8798: 8795: 8791: 8788: 8786: 8783: 8782: 8781: 8778: 8776: 8773: 8769: 8766: 8764: 8761: 8760: 8759: 8756: 8754: 8751: 8749: 8746: 8744: 8741: 8739: 8736: 8735: 8733: 8731: 8727: 8721: 8720:Schmidt–Samoa 8718: 8716: 8713: 8711: 8708: 8706: 8703: 8701: 8698: 8696: 8693: 8691: 8688: 8686: 8683: 8681: 8680:Damgård–Jurik 8678: 8676: 8675:Cayley–Purser 8673: 8671: 8668: 8666: 8663: 8662: 8660: 8658: 8654: 8651: 8647: 8643: 8636: 8631: 8629: 8624: 8622: 8617: 8616: 8613: 8609: 8605: 8599: 8598: 8593: 8591: 8588: 8586: 8583: 8581: 8577: 8572: 8570: 8567: 8564: 8562: 8556: 8555: 8551: 8545: 8539: 8535: 8531: 8530: 8525: 8521: 8517: 8513: 8509: 8505: 8499: 8496:. CRC Press. 8494: 8493: 8486: 8485: 8481: 8466: 8459: 8456: 8448: 8441: 8438: 8433: 8429: 8424: 8419: 8415: 8408: 8405: 8400: 8396: 8390: 8387: 8374: 8368: 8365: 8360: 8353: 8346: 8343: 8338: 8334: 8327: 8324: 8316: 8309: 8306: 8301: 8300: 8295: 8288: 8285: 8280: 8276: 8272: 8265: 8258: 8255: 8250: 8246: 8242: 8238: 8234: 8230: 8223: 8216: 8213: 8208: 8204: 8197: 8194: 8183: 8179: 8175: 8168: 8161: 8158: 8147:on 2017-04-17 8146: 8142: 8138: 8131: 8129: 8125: 8114:on 2020-02-28 8113: 8109: 8102: 8099: 8094: 8087: 8080: 8077: 8069: 8062: 8059: 8054: 8048: 8044: 8043: 8035: 8032: 8020: 8016: 8010: 8007: 8002: 7996: 7993: 7988: 7982: 7977: 7972: 7968: 7961: 7958: 7953: 7949: 7945: 7939: 7935: 7931: 7927: 7923: 7919: 7915: 7909: 7906: 7901: 7897: 7893: 7889: 7884: 7879: 7875: 7871: 7870: 7862: 7855: 7852: 7847: 7841: 7837: 7833: 7829: 7822: 7819: 7807: 7804: 7799: 7794: 7790: 7789: 7781: 7778: 7773: 7769: 7767: 7763: 7754: 7751: 7746: 7742: 7740: 7736: 7727: 7724: 7720: 7714: 7711: 7705: 7700: 7696: 7689: 7686: 7682: 7676: 7673: 7669:(2): 203–213. 7668: 7664: 7663: 7658: 7651: 7649: 7645: 7633: 7629: 7623: 7620: 7616: 7610: 7607: 7603: 7597: 7594: 7590: 7584: 7581: 7577: 7571: 7568: 7564: 7558: 7555: 7540: 7536: 7529: 7525: 7519: 7516: 7511: 7507: 7500: 7493: 7491: 7487: 7479: 7472: 7469: 7461: 7454: 7451: 7446: 7442: 7438: 7434: 7429: 7424: 7420: 7416: 7409: 7406: 7402: 7396: 7392: 7388: 7384: 7380: 7376: 7372: 7368: 7365:(7833): 189. 7364: 7360: 7356: 7349: 7346: 7334: 7330: 7323: 7320: 7312: 7308: 7304: 7300: 7296: 7291: 7286: 7282: 7278: 7277: 7269: 7262: 7260: 7258: 7256: 7254: 7252: 7250: 7246: 7240: 7216: 7212: 7203: 7197: 7193: 7189: 7185: 7181: 7177: 7172: 7169: 7164: 7160: 7155: 7151: 7146: 7141: 7131: 7104: 7100: 7096: 7091: 7087: 7076: 7073: 7069: 7063: 7060: 7057:respectively. 7044: 7041: 7036: 7026: 7023: 7020: 7016: 6995: 6992: 6987: 6977: 6974: 6971: 6967: 6952: 6945: 6938: 6932: 6928: 6905: 6902: 6895: 6891: 6888: 6886: 6883: 6881: 6878: 6876: 6873: 6871: 6868: 6866: 6863: 6861: 6858: 6856: 6853: 6851: 6848: 6846: 6843: 6841: 6838: 6837: 6833: 6827: 6822: 6817: 6813: 6810: 6808: 6805: 6803: 6800: 6798: 6795: 6793: 6790: 6788: 6785: 6783: 6780: 6778: 6775: 6773: 6770: 6768: 6767:Bouncy Castle 6765: 6763: 6760: 6759: 6758: 6752: 6750: 6748: 6740: 6738: 6735: 6731: 6729: 6725: 6717: 6715: 6712: 6710: 6706: 6702: 6698: 6694: 6686: 6684: 6672: 6668: 6663: 6657: 6653: 6650: 6645:and computes 6638: 6634: 6629: 6623: 6621: 6617: 6613: 6609: 6600: 6593: 6591: 6587: 6585: 6579: 6575: 6571: 6557: 6553: 6549: 6524:factors both 6522: 6518: 6514: 6498: 6494: 6487: 6483: 6479: 6473: 6469: 6463: 6461: 6449: 6441: 6439: 6437: 6433: 6429: 6425: 6421: 6416: 6406: 6398: 6392: 6386: 6370: 6366: 6360: 6341: 6331: 6330:− 1 algorithm 6329: 6317: 6310: 6284: 6280: 6276: 6270: 6265:is less than 6263: 6259: 6250: 6237: 6235: 6221: 6217: 6206: 6203: 6195: 6184: 6181: 6177: 6174: 6170: 6167: 6163: 6160: 6156: 6153: – 6152: 6148: 6147:Find sources: 6141: 6137: 6131: 6130: 6125:This section 6123: 6119: 6114: 6113: 6107: 6105: 6103: 6099: 6095: 6091: 6086: 6084: 6079: 6075: 6071: 6062: 6055: 6051: 6047: 6042: 6016: 6011: 6009: 6003: 5996: 5994: 5980: 5976: 5952: 5948: 5928: 5924: 5916: 5912: 5908: 5891: 5886: 5884: 5880: 5876: 5872: 5868: 5862: 5858: 5854: 5846: 5844: 5842: 5837: 5823: 5820: 5817: 5812: 5808: 5804: 5801: 5775: 5771: 5758: 5754: 5750: 5745: 5741: 5728: 5724: 5721: 5695: 5691: 5679: 5675: 5670: 5666: 5661: 5657: 5630: 5626: 5614: 5610: 5605: 5601: 5596: 5592: 5579: 5575: 5571: 5551: 5544: 5540: 5530: 5527: 5523: 5519: 5510: 5502: 5488: 5481: 5478: 5475: 5471: 5463: 5460: 5455: 5451: 5443: 5429: 5422: 5419: 5416: 5412: 5404: 5401: 5396: 5392: 5384: 5369: 5349: 5342: 5341: 5340: 5338: 5334: 5330: 5326: 5318: 5313: 5311: 5308: 5302: 5296: 5294: 5290: 5286: 5282: 5278: 5265: 5260: 5250: 5242: 5220: 5216: 5210: 5206: 5197: 5185: 5174: 5170: 5163: 5159: 5144: 5140: 5136: 5125: 5121: 5117: 5112: 5106: 5099: 5093: 5086: 5080: 5075: 5072: 5068: 5064: 5060: 5057: 5053: 5049: 5045: 5021: 5009: 5003: 4999: 4988: 4983: 4982: 4981: 4975: 4970: 4968: 4964: 4960: 4953: 4949: 4942: 4938: 4934: 4919: 4907: 4903: 4898: 4892: 4888: 4884: 4870: 4868: 4863: 4850: 4843: 4839: 4831: 4828: 4823: 4815: 4809: 4806: 4801: 4788: 4782: 4778: 4771: 4768: 4760: 4754: 4751: 4748: 4745: 4741: 4737: 4732: 4729: 4725: 4708: 4698: 4694: 4690: 4675: 4671: 4667: 4653: 4649: 4642: 4638: 4634: 4628: 4626: 4610: 4604: 4601: 4595: 4591: 4587: 4576: 4569: 4562: 4558: 4551: 4547: 4539: 4535: 4531: 4527:that satisfy 4516: 4513: 4510:is not prime. 4508: 4503: 4497: 4494: 4487: 4485: 4472: 4465: 4462: 4458: 4450: 4447: 4442: 4432: 4428: 4414: 4410: 4406: 4372: 4365: 4361: 4353: 4350: 4347: 4342: 4338: 4334: 4331: 4326: 4316: 4313: 4310: 4306: 4299: 4296: 4288: 4285: 4282: 4276: 4272: 4268: 4265: 4260: 4257: 4254: 4251: 4247: 4243: 4238: 4235: 4231: 4223: 4222: 4218: 4202: 4195: 4190: 4185: 4181: 4177: 4168: 4162: 4158: 4153: 4152: 4151: 4147: 4143: 4139: 4127: 4123: 4118: 4114: 4100: 4093: 4089: 4081: 4078: 4075: 4070: 4066: 4062: 4059: 4054: 4044: 4041: 4038: 4034: 4027: 4024: 4016: 4013: 4010: 4004: 4000: 3996: 3993: 3988: 3985: 3982: 3979: 3975: 3971: 3966: 3963: 3959: 3951: 3950: 3946: 3930: 3923: 3918: 3913: 3909: 3905: 3896: 3882: 3878: 3873: 3872: 3871: 3867: 3863: 3859: 3853: 3850: 3843: 3836: 3825: 3819: 3794: 3791: 3788: 3782: 3779: 3773: 3770: 3767: 3761: 3758: 3755: 3752: 3749: 3746: 3736: 3729: 3722: 3718: 3714: 3710: 3706: 3700: 3696: 3692: 3688: 3647: 3644: 3640: 3632: 3629: 3624: 3614: 3610: 3597: 3581: 3577: 3572: 3559: 3554: 3551: 3550: 3549: 3546: 3533: 3526: 3522: 3514: 3511: 3506: 3496: 3492: 3485: 3480: 3477: 3473: 3469: 3464: 3461: 3457: 3453: 3448: 3438: 3434: 3411: 3405: 3399: 3396: 3393: 3390: 3382: 3377: 3359: 3354: 3352: 3348: 3344: 3336: 3334: 3317: 3314: 3311: 3308: 3305: 3302: 3299: 3296: 3293: 3290: 3287: 3284: 3279: 3275: 3271: 3269: 3264: 3257: 3254: 3251: 3248: 3243: 3232: 3229: 3226: 3223: 3217: 3212: 3196: 3192: 3188: 3183: 3179: 3172: 3163: 3156: 3154: 3149: 3142: 3139: 3136: 3133: 3128: 3118: 3114: 3110: 3105: 3093: 3089: 3084: 3080: 3078: 3071: 3067: 3059: 3056: 3053: 3050: 3045: 3035: 3031: 3027: 3022: 3010: 3006: 3001: 2997: 2995: 2988: 2984: 2954: 2945: 2937: 2920: 2917: 2914: 2909: 2901: 2898: 2895: 2892: 2887: 2876: 2873: 2864: 2855: 2847: 2844: 2841: 2836: 2826: 2823: 2819: 2815: 2810: 2800: 2797: 2793: 2789: 2787: 2776: 2768: 2765: 2762: 2756: 2753: 2750: 2737: 2734: 2728: 2725: 2722: 2709: 2706: 2704: 2697: 2693: 2685: 2682: 2679: 2673: 2670: 2667: 2654: 2651: 2645: 2642: 2639: 2626: 2623: 2621: 2614: 2610: 2588: 2579: 2571: 2569: 2565: 2561: 2557: 2552: 2532: 2527: 2523: 2518: 2505: 2502: 2499: 2494: 2484: 2480: 2476: 2473: 2463: 2457: 2444: 2441: 2438: 2433: 2423: 2419: 2415: 2412: 2402: 2396: 2379: 2374: 2364: 2360: 2356: 2354: 2344: 2334: 2330: 2326: 2324: 2316: 2310: 2295: 2289: 2285: 2279: 2274: 2257: 2252: 2242: 2238: 2234: 2232: 2222: 2212: 2208: 2204: 2202: 2194: 2188: 2172: 2166: 2162: 2156: 2137: 2132: 2121: 2118: 2115: 2109: 2106: 2086: 2083: 2080: 2077: 2066: 2062: 2058: 2053: 2045: 2028: 2025: 2022: 2014: 2013: 2003: 1997: 1991: 1975: 1972: 1966: 1963: 1960: 1954: 1951: 1948: 1942: 1936: 1929: 1928: 1924: 1920: 1916: 1912: 1908: 1903: 1899: 1883: 1880: 1877: 1874: 1871: 1868: 1865: 1858: 1857: 1854: 1850: 1845: 1828: 1825: 1822: 1802: 1799: 1796: 1789: 1788: 1786: 1785: 1784: 1778: 1776: 1765: 1752: 1745: 1741: 1733: 1730: 1725: 1715: 1711: 1704: 1699: 1695: 1686: 1685:by computing 1668: 1666: 1648: 1643: 1630: 1623: 1619: 1609: 1605: 1601: 1598: 1590: 1580: 1575: 1571: 1565: 1552: 1542: 1540: 1536: 1528: 1524: 1517: 1515: 1511: 1508:Suppose that 1503: 1501: 1491: 1485: 1481: 1472: 1459: 1455: 1447: 1439: 1435: 1431: 1427: 1419: 1412: 1405: 1397: 1390: 1386: 1382: 1375: 1371: 1367: 1362: 1358: 1348: 1344: 1340: 1333: 1329: 1318: 1314: 1310: 1306: 1299: 1295: 1291: 1287: 1278: 1277: 1271: 1267: 1261: 1257: 1250: 1246: 1235: 1231: 1224: 1220: 1216: 1212: 1208: 1203: 1191: 1187: 1170: 1158: 1148: 1141: 1134: 1128: 1124: 1115: 1105: 1101: 1097: 1093:the equation 1088: 1087: 1083: 1079: 1070: 1060: 1056: 1052: 1048: 1039: 1031: 1006: 1002: 995: 994: 992: 986: 982: 971: 967: 963: 959: 952: 948: 944: 934: 927: 923: 919: 910: 906: 895: 885: 881: 875: 867: 866: 862: 858: 854: 850: 843: 839: 835: 828: 824: 820: 819: 814: 810: 795: 791: 787: 783: 779: 775: 771: 767: 763: 758: 748: 744: 739: 731: 728: 724: 717: 716: 713: 709: 704: 692: 689: 677: 676: 667: 666:prime numbers 663: 662: 661: 655: 653: 651: 647: 642: 632: 613:The integers 611: 585: 578: 574: 566: 563: 558: 548: 544: 520: 513: 497: 489: 473: 451: 441: 437: 424: 420: 397: 395: 387: 385: 383: 379: 374: 372: 371:United States 368: 364: 358: 353: 351: 346: 341: 337: 329: 327: 325: 319: 317: 313: 309: 308:mathematician 306:, an English 305: 301: 298: 294: 289: 285: 281: 277: 273: 271: 267: 263: 259: 251: 247: 243: 239: 232: 230: 228: 227:symmetric-key 223: 221: 217: 213: 212:prime numbers 209: 204: 202: 201:prime numbers 198: 194: 190: 185: 183: 179: 175: 171: 167: 163: 159: 155: 151: 147: 143: 134: 128: 123: 119: 116: 115:cryptanalysis 111: 107: 105: 101: 97: 95: 91: 88:Cipher detail 86: 83: 79: 76: 74:Certification 72: 68: 64: 61: 57: 53: 50: 46: 41: 33: 19: 9433:Block cipher 9273:Key schedule 9263:Key exchange 9253:Kleptography 9211:Cryptosystem 9155:Cryptography 9107:OpenPGP card 9087:Web of trust 8743:Cramer–Shoup 8709: 8596: 8558: 8528: 8491: 8468:. Retrieved 8463:Isom, Kyle. 8458: 8440: 8413: 8407: 8398: 8389: 8377:. Retrieved 8367: 8358: 8345: 8336: 8326: 8308: 8297: 8287: 8270: 8257: 8232: 8228: 8215: 8206: 8196: 8185:. Retrieved 8160: 8149:. Retrieved 8145:the original 8116:. Retrieved 8112:the original 8101: 8092: 8079: 8073:. Eurocrypt. 8061: 8041: 8034: 8022:. Retrieved 8018: 8009: 7995: 7966: 7960: 7925: 7908: 7873: 7867: 7854: 7827: 7821: 7809:. Retrieved 7787: 7780: 7771: 7765: 7761: 7753: 7744: 7738: 7734: 7726: 7719:Neal Koblitz 7713: 7688: 7675: 7666: 7660: 7636:. Retrieved 7632:the original 7622: 7609: 7596: 7583: 7570: 7557: 7546:. Retrieved 7539:the original 7534: 7524:Cocks, C. C. 7518: 7509: 7505: 7471: 7453: 7418: 7414: 7408: 7362: 7358: 7348: 7336:. Retrieved 7322: 7311:the original 7280: 7274: 7075: 7062: 6950: 6943: 6936: 6930: 6926: 6904: 6865:Key exchange 6756: 6744: 6736: 6732: 6721: 6713: 6690: 6670: 6666: 6655: 6651: 6648: 6636: 6632: 6624: 6597: 6588: 6580: 6573: 6569: 6550: 6520: 6516: 6512: 6496: 6492: 6485: 6481: 6477: 6471: 6467: 6464: 6445: 6417: 6390: 6387: 6368: 6364: 6358: 6342: 6327: 6315: 6308: 6282: 6278: 6274: 6268: 6261: 6257: 6239:The numbers 6238: 6223: 6198: 6192:October 2017 6189: 6179: 6172: 6165: 6158: 6146: 6134:Please help 6129:verification 6126: 6087: 6068:is 300 6063: 6054:RSA Security 6043: 6012: 6004: 5997: 5978: 5974: 5950: 5946: 5926: 5922: 5914: 5910: 5906: 5887: 5878: 5864: 5838: 5577: 5573: 5569: 5566: 5322: 5297: 5275:However, at 5261: 5246: 5218: 5214: 5208: 5204: 5183: 5172: 5168: 5161: 5157: 5142: 5138: 5134: 5123: 5119: 5115: 5104: 5097: 5091: 5084: 5078: 5048:Johan Håstad 5007: 5001: 4997: 4986: 4979: 4962: 4958: 4951: 4947: 4940: 4936: 4932: 4920: 4905: 4901: 4890: 4886: 4882: 4871: 4864: 4706: 4696: 4692: 4688: 4673: 4669: 4665: 4651: 4647: 4640: 4636: 4632: 4629: 4622: 4603: 4593: 4589: 4585: 4574: 4567: 4560: 4556: 4549: 4545: 4537: 4533: 4529: 4515: 4506: 4501: 4496: 4412: 4408: 4404: 4389: 4216: 4193: 4183: 4179: 4175: 4166: 4160: 4156: 4145: 4141: 4137: 4134: 4125: 4121: 3944: 3921: 3911: 3907: 3903: 3894: 3880: 3876: 3865: 3861: 3857: 3854: 3852:separately. 3848: 3841: 3834: 3824: 3820: 3734: 3727: 3720: 3716: 3708: 3704: 3701: 3694: 3690: 3686: 3598: 3579: 3575: 3568: 3547: 3378: 3355: 3340: 2952: 2943: 2939:Here is how 2938: 2586: 2577: 2572: 2567: 2563: 2559: 2553: 2530: 2519: 2461: 2458: 2400: 2397: 2287: 2283: 2277: 2275: 2164: 2160: 2154: 2152: 2064: 2060: 2056: 1995: 1922: 1918: 1910: 1906: 1900:Compute the 1852: 1848: 1782: 1766: 1687: 1672: 1644: 1591: 1573: 1569: 1553: 1546: 1534: 1526: 1522: 1518: 1507: 1496:and compute 1483: 1479: 1460: 1453: 1445: 1437: 1433: 1429: 1425: 1417: 1410: 1403: 1395: 1391: 1384: 1380: 1373: 1369: 1365: 1346: 1342: 1338: 1331: 1327: 1316: 1312: 1308: 1304: 1297: 1293: 1289: 1285: 1275: 1259: 1255: 1248: 1244: 1233: 1229: 1222: 1218: 1214: 1210: 1204: 1189: 1185: 1168: 1156: 1154: 1146: 1126: 1122: 1103: 1099: 1095: 1081: 1077: 1058: 1054: 1050: 1046: 984: 980: 969: 965: 961: 950: 946: 942: 925: 921: 908: 904: 893: 883: 879: 860: 856: 852: 848: 841: 837: 833: 826: 822: 816: 812: 808: 798:, and since 793: 789: 785: 781: 773: 769: 765: 761: 746: 742: 711: 707: 659: 643: 612: 422: 418: 398: 391: 382:RSA Security 375: 360: 355: 333: 320: 302: 274: 255: 224: 205: 189:cryptosystem 186: 182:declassified 145: 141: 140: 113:Best public 32:RSA Security 9421:Mathematics 9412:Mix network 9077:Fingerprint 9041:NSA Suite B 9005:RSA problem 8882:NTRUEncrypt 8273:. CCS '17. 7772:openssl-dev 7745:openssl-dev 6432:smart cards 6351:is between 5890:RSA problem 5871:RSA problem 4880:satisfying 4119:to replace 3353:a message. 2573:The values 2459:To decrypt 2278:private key 1283:satisfying 1279:. Thus any 1169:private key 1063:; that is, 974:; that is, 806:are prime, 220:RSA problem 133:829-bit key 9520:Categories 9372:Ciphertext 9342:Decryption 9337:Encryption 9298:Ransomware 9031:IEEE P1363 8649:Algorithms 8361:. SSYM'03. 8187:2017-11-24 8151:2017-11-24 8118:2020-07-12 7764:− 1) and ( 7737:− 1) and ( 7638:2010-03-03 7548:2017-05-30 7401:Peter Shor 7338:August 14, 7241:References 6959:such that 6326:Pollard's 6214:See also: 6162:newspapers 6090:Peter Shor 6046:RSA number 6017:– finding 5904:such that 5851:See also: 4717:, we have 4532:≡ 1 (mod ( 4504:) because 4402:such that 3588:and prime 3358:hash value 2562:using mod 2294:ciphertext 2155:public key 2069:, yielding 1669:Decryption 1568:0 ≤ 1551:to Alice. 1543:Encryption 1157:public key 1040:Determine 1003:and small 1001:bit-length 939:such that 727:key length 510:(they are 417:0 ≤ 280:Adi Shamir 276:Ron Rivest 246:Ron Rivest 242:Adi Shamir 162:Adi Shamir 158:Ron Rivest 154:initialism 82:ANSI X9.31 56:Adi Shamir 52:Ron Rivest 9362:Plaintext 8470:4 January 8418:CiteSeerX 8379:4 January 7878:CiteSeerX 7699:CiteSeerX 7683:, p. 467. 7578:. p. 180. 7506:SIAM News 7445:0018-9448 7423:CiteSeerX 7395:226243008 7285:CiteSeerX 7190:− 7024:− 6975:− 6797:wolfCrypt 6782:Libgcrypt 6691:In 1998, 6381:and 6088:In 1994, 5751:− 5528:− 5479:− 5420:− 5285:Eurocrypt 4961:≡ 0 (mod 4950:≡ 0 (mod 4904:≡ 1 (mod 4885:≡ 1 (mod 4829:≡ 4807:≡ 4783:φ 4755:φ 4668:≡ 1 (mod 4588:≡ 1 (mod 4448:≡ 4407:≡ 1 (mod 4351:≡ 4335:≡ 4314:− 4286:− 4258:− 4159:≡ 0 (mod 4079:≡ 4063:≡ 4042:− 4014:− 3986:− 3879:≡ 0 (mod 3847:mod 3840:mod 3833:mod 3792:− 3771:− 3753:− 3689:≡ 1 (mod 3630:≡ 3578:≡ 1 (mod 3512:≡ 3400:⁡ 3309:× 3291:× 3230:− 3227:× 3189:− 3173:× 2899:× 2874:× 2858:⇒ 2824:− 2798:− 2754:− 2726:− 2671:− 2643:− 2171:plaintext 2119:× 1955:⁡ 1937:λ 1875:× 1731:≡ 1705:≡ 1661:equal to 1602:≡ 1311:≡ 1 (mod 1292:≡ 1 (mod 1098:≡ 1 (mod 564:≡ 488:remainder 388:Operation 208:factoring 184:in 1997. 94:Key sizes 48:Designers 9501:Category 9407:Kademlia 9367:Codetext 9310:(CSPRNG) 9288:Machines 9092:Key size 9026:CRYPTREC 8943:McEliece 8897:RLWE-SIG 8892:RLWE-KEX 8887:NTRUSign 8700:Paillier 8526:(2001). 8024:2 August 7952:10316867 7900:15726802 7591:. p. 12. 7387:33139910 7178:⌉ 7165:⌈ 6875:Key size 6818:See also 6812:LibreSSL 6807:mbed TLS 6777:Crypto++ 6772:cryptlib 6428:Infineon 6008:Athlon64 5919:, where 5869:and the 5234:', 5226:', 4910:for all 4707:Assuming 4678:. Since 4645:, where 4542:, since 4203:≢ 3931:≢ 3855:To show 3372:(modulo 3364:(modulo 3341:Suppose 2566:and mod 2286:= 3233, 2173:message 2163:= 3233, 2046:Compute 2000:that is 1998:< 780 1846:Compute 1436:− 1) + ( 1432:− 1) + ( 1242:. Since 1135:, where 1013:2 + 1 = 876:, since 855:) = lcm( 846:. Hence 788:), 759:. Since 751:, where 740:Compute 705:Compute 426:), both 297:Passover 293:knapsack 9162:General 8938:Lamport 8918:CEILIDH 8877:NewHope 8824:Schnorr 8807:ElGamal 8785:Ed25519 8665:Benaloh 8580:YouTube 8399:Mozilla 8249:7120331 7811:9 March 7367:Bibcode 7307:2873616 6924:having 6792:OpenSSL 6654:) (mod 6612:Brumley 6373:, then 6176:scholar 6078:RSA-155 6050:RSA-250 5879:partial 5325:OpenSSL 5293:RSA-PSS 5249:padding 5191:modulo 5103:) (mod 4995:(i.e., 4971:Padding 4693:hφ 4215:0 (mod 4129:with 1. 3943:0 (mod 3893:. Thus 3383:rules: 2972:pair): 2002:coprime 1994:2 < 1927:giving 1856:giving 1779:Example 1477:modulo 1469:as the 1325:modulo 1075:modulo 1067:is the 991:coprime 941:1 < 914:⁠ 907:, 888:⁠ 882:, 723:modulus 312:British 286:at the 233:History 214:, the " 148:) is a 43:General 9283:Keygen 9060:Topics 9036:NESSIE 8978:Theory 8906:Others 8763:X25519 8540: 8536:–887. 8500: 8420: 8247: 8049: 8019:Github 7983: 7950: 7940: 7898: 7880: 7842: 7701: 7443: 7425: 7393: 7385: 7359:Nature 7305: 7287: 6802:GnuTLS 6787:Nettle 6599:Kocher 6536:given 6178: 6171: 6164: 6157: 6149: 5859:, and 5277:Crypto 5264:PKCS#1 4887:λ 4691:= 1 + 4670:φ 4590:λ 4557:λ 4409:λ 4178:≡ 0 ≡ 3906:≡ 0 ≡ 3705:λ 3702:Since 3691:λ 2533:= 3233 2464:= 2790 2290:= 413) 2061:λ 2050:, the 1907:λ 1767:Given 1564:padded 1490:PKCS#1 1480:φ 1381:λ 1370:λ 1343:λ 1328:φ 1313:λ 1294:φ 1256:λ 1245:φ 1230:λ 1211:φ 1186:λ 1183:, and 1167:. The 1123:λ 1100:λ 1078:λ 1055:λ 981:λ 972:)) = 1 966:λ 947:λ 922:λ 897:| 891:| 849:λ 834:λ 818:φ 809:λ 790:λ 782:λ 770:λ 753:λ 743:λ 407:, and 336:patent 330:Patent 282:, and 191:, the 104:Rounds 78:PKCS#1 58:, and 9318:(PRN) 8872:Kyber 8867:BLISS 8829:SPEKE 8797:ECMQV 8790:Ed448 8780:EdDSA 8775:ECDSA 8705:Rabin 8450:(PDF) 8355:(PDF) 8318:(PDF) 8267:(PDF) 8245:S2CID 8225:(PDF) 8170:(PDF) 8089:(PDF) 8071:(PDF) 7948:S2CID 7896:S2CID 7864:(PDF) 7768:− 1)" 7741:− 1)" 7542:(PDF) 7531:(PDF) 7502:(PDF) 7481:(PDF) 7463:(PDF) 7391:S2CID 7314:(PDF) 7303:S2CID 7271:(PDF) 6896:Notes 6762:Botan 6669:(mod 6664:, is 6635:(mod 6608:Boneh 6519:′) = 6501:(but 6405:65537 6367:< 6183:JSTOR 6169:books 6083:TWIRL 6033:into 6021:from 5987:from 5977:− 1, 5963:into 5913:(mod 5576:(mod 5171:(mod 5160:(mod 5141:(mod 5122:(mod 5054:(see 5000:< 4939:− 1/( 4921:When 4709:that 4639:(mod 4548:− 1)( 4540:− 1)) 4536:− 1)( 4488:Notes 4182:(mod 4173:. So 4144:(mod 3910:(mod 3901:. So 3864:(mod 3719:− 1, 3668:when 3345:uses 3343:Alice 2445:2790. 2167:= 17) 2059:(mod 1921:− 1, 1884:3233. 1677:from 1572:< 1514:Alice 1428:− 1)( 1368:< 1341:> 1221:− 1)( 1217:) = ( 1053:(mod 945:< 859:− 1, 421:< 9072:OAEP 9046:CNSA 8923:EPOC 8768:X448 8758:ECDH 8538:ISBN 8498:ISBN 8472:2022 8381:2022 8047:ISBN 8026:2024 7981:ISBN 7938:ISBN 7840:ISBN 7813:2016 7806:3447 7512:(5). 7441:ISSN 7383:PMID 7340:2011 7097:< 6747:PRNG 6610:and 6568:gcd( 6544:and 6528:and 6511:gcd( 6480:′ = 6454:and 6434:and 6355:and 6302:and 6251:for 6243:and 6228:and 6218:and 6155:news 6070:bits 6037:and 6025:and 5981:− 1) 5973:lcm( 5967:and 5888:The 5875:hard 5362:and 5333:.NET 5331:and 5329:Java 5304:and 5137:′ ≡ 4935:+ 1/ 4876:and 4682:and 4659:and 4572:and 4552:− 1) 4523:and 4124:mod 3845:and 3827:and 3814:and 3732:and 3723:− 1) 3711:) = 3680:and 3672:and 3397:hash 3351:sign 3115:2790 3032:2790 2968:and 2957:and 2591:and 2539:and 2524:for 2481:2790 2403:= 65 2380:233. 2276:The 2258:233. 2153:The 2015:Let 1976:780. 1943:3233 1925:− 1) 1913:) = 1815:and 1456:− 1) 1450:and 1448:− 1) 1440:− 1) 1406:− 1) 1400:and 1398:− 1) 1225:− 1) 1155:The 1120:and 989:are 978:and 955:and 903:gcd( 886:) = 878:lcm( 868:The 863:− 1) 840:) = 825:) = 815:) = 802:and 776:) = 696:and 682:and 671:and 637:and 617:and 602:and 466:and 350:DWPI 260:and 248:and 164:and 69:1977 9082:PKI 8965:XTR 8933:IES 8928:HFE 8859:SIS 8854:LWE 8839:STS 8834:SRP 8819:MQV 8802:EKE 8753:DSA 8738:BLS 8710:RSA 8685:GMR 8578:on 8534:881 8428:doi 8275:doi 8237:doi 8178:doi 7971:doi 7930:doi 7888:doi 7832:doi 7803:RFC 7793:doi 7433:doi 7375:doi 7363:587 7295:doi 7213:mod 7136:inv 7123:as 7079:If 7033:mod 7008:or 6984:mod 6941:or 6934:if 6393:= 3 6336:or 6318:− 1 6313:or 6311:− 1 6290:is 6138:by 6064:If 5772:mod 5733:inv 5692:mod 5627:mod 5541:mod 5515:inv 5472:mod 5413:mod 5295:). 5090:≡ ( 4989:= 3 4956:or 4840:mod 4577:− 1 4570:− 1 4459:mod 4362:mod 4191:If 4154:If 4090:mod 3919:If 3874:If 3737:− 1 3730:− 1 3713:lcm 3641:mod 3523:mod 3347:Bob 3318:65. 3240:mod 3209:mod 3168:inv 3125:mod 3102:mod 3042:mod 3019:mod 2961:inv 2906:mod 2884:mod 2869:inv 2833:mod 2807:mod 2781:inv 2742:mod 2738:413 2714:mod 2659:mod 2655:413 2631:mod 2595:inv 2570:). 2506:65. 2500:233 2491:mod 2485:413 2439:233 2430:mod 2371:mod 2365:413 2341:mod 2280:is 2249:mod 2219:mod 2157:is 2138:80. 2129:mod 2122:413 2099:as 2084:413 2054:of 1952:lcm 1915:lcm 1742:mod 1620:mod 1510:Bob 1473:of 1420:− 1 1413:− 1 1071:of 1044:as 1017:537 1011:is 958:gcd 870:lcm 844:− 1 829:− 1 778:lcm 755:is 633:to 575:mod 394:key 365:'s 340:MIT 142:RSA 131:An 38:RSA 9522:: 8913:AE 8748:DH 8522:; 8518:; 8514:; 8426:. 8397:. 8357:. 8335:. 8296:. 8269:. 8243:. 8233:36 8231:. 8227:. 8205:. 8172:. 8139:. 8127:^ 8091:. 8017:. 7979:. 7946:. 7936:. 7924:. 7916:; 7894:. 7886:. 7874:10 7872:. 7866:. 7838:. 7801:. 7770:. 7743:. 7697:. 7667:46 7665:. 7659:. 7647:^ 7533:. 7510:36 7508:. 7504:. 7489:^ 7439:. 7431:. 7419:22 7417:. 7389:. 7381:. 7373:. 7361:. 7357:. 7331:. 7301:. 7293:. 7281:21 7279:. 7273:. 7248:^ 6929:= 6667:rc 6576:′) 6572:, 6515:, 6495:= 6472:pq 6470:= 6385:. 6371:/3 6296:10 6277:= 6260:− 6104:. 6002:. 5949:, 5925:, 5909:≡ 5885:. 5855:, 5836:. 5792:, 5712:, 5647:, 5578:pq 5572:= 5327:, 5238:). 5219:pq 5217:= 5209:pq 5207:= 5184:mr 5175:), 5169:mr 5158:mr 5139:cr 5118:≡ 5058:). 5046:. 5034:, 4941:pq 4931:1/ 4918:. 4893:)) 4883:ed 4869:. 4705:. 4689:ed 4676:)) 4666:ed 4652:pq 4650:= 4635:≡ 4627:. 4596:)) 4594:pq 4586:ed 4561:pq 4530:ed 4507:pq 4502:pq 4417:, 4415:)) 4413:pq 4405:ed 4398:, 4221:, 4165:, 4140:≡ 3949:, 3885:, 3860:≡ 3835:pq 3818:. 3709:pq 3699:. 3697:)) 3695:pq 3687:ed 3596:. 3312:53 3300:12 3224:38 3140:12 3119:49 3036:53 2948:, 2921:1. 2902:53 2896:38 2848:38 2820:53 2766:49 2751:53 2683:53 2668:61 2582:, 2560:pq 2543:. 2424:17 2420:65 2243:17 2116:17 2067:)) 2029:17 1967:52 1961:60 1878:53 1872:61 1853:pq 1851:= 1829:53 1803:61 1525:, 1422:= 1418:pq 1415:= 1319:)) 1300:)) 1276:pq 1179:, 1108:; 1106:)) 1096:de 1086:. 1061:)) 1049:≡ 1015:65 993:. 964:, 894:ab 865:. 796:)) 768:, 766:pq 764:= 715:. 712:pq 710:= 675:. 610:. 533:): 403:, 334:A 326:. 278:, 272:. 160:, 80:, 54:, 9147:e 9140:t 9133:v 8856:/ 8851:/ 8634:e 8627:t 8620:v 8565:. 8546:. 8506:. 8474:. 8452:. 8434:. 8430:: 8401:. 8383:. 8339:. 8320:. 8302:. 8281:. 8277:: 8251:. 8239:: 8209:. 8190:. 8180:: 8154:. 8121:. 8055:. 8028:. 8003:. 7989:. 7973:: 7954:. 7932:: 7902:. 7890:: 7848:. 7834:: 7815:. 7795:: 7766:q 7762:p 7739:q 7735:p 7707:. 7641:. 7604:. 7565:. 7551:. 7483:. 7465:. 7447:. 7435:: 7403:. 7397:. 7377:: 7369:: 7342:. 7297:: 7233:. 7220:) 7217:p 7210:( 7204:] 7198:2 7194:m 7186:) 7182:p 7173:p 7170:q 7161:+ 7156:1 7152:m 7147:( 7142:[ 7132:q 7121:h 7105:2 7101:m 7092:1 7088:m 7070:. 7045:1 7042:= 7037:q 7027:1 7021:e 7017:m 6996:1 6993:= 6988:p 6978:1 6972:e 6968:m 6957:m 6951:e 6944:q 6937:p 6931:m 6927:c 6922:m 6918:q 6914:p 6910:m 6681:r 6677:r 6673:) 6671:n 6658:) 6656:n 6652:c 6649:r 6647:( 6643:r 6639:) 6637:n 6633:c 6604:d 6574:n 6570:n 6564:n 6560:n 6546:q 6542:p 6538:p 6534:q 6530:n 6526:n 6521:p 6517:n 6513:n 6507:q 6503:q 6499:′ 6497:p 6493:p 6488:′ 6486:q 6484:′ 6482:p 6478:n 6468:n 6456:q 6452:p 6413:e 6409:e 6401:e 6391:e 6383:e 6379:n 6375:d 6369:n 6365:d 6359:q 6357:2 6353:q 6349:p 6345:d 6338:q 6334:p 6328:p 6322:n 6316:q 6309:p 6304:q 6300:p 6294:× 6292:3 6288:n 6283:q 6281:⋅ 6279:p 6275:n 6272:( 6269:n 6267:2 6262:q 6258:p 6253:n 6245:q 6241:p 6230:q 6226:p 6205:) 6199:( 6194:) 6190:( 6180:· 6173:· 6166:· 6159:· 6132:. 6066:n 6059:n 6039:q 6035:p 6031:n 6027:e 6023:n 6019:d 6000:n 5989:e 5985:d 5979:q 5975:p 5969:q 5965:p 5961:n 5957:c 5953:) 5951:e 5947:n 5945:( 5941:d 5937:n 5933:c 5929:) 5927:e 5923:n 5921:( 5917:) 5915:n 5911:m 5907:c 5902:m 5898:n 5894:e 5824:q 5821:h 5818:+ 5813:2 5809:m 5805:= 5802:m 5779:) 5776:p 5769:( 5764:) 5759:2 5755:m 5746:1 5742:m 5738:( 5729:q 5725:= 5722:h 5699:) 5696:q 5689:( 5680:Q 5676:d 5671:c 5667:= 5662:2 5658:m 5634:) 5631:p 5624:( 5615:P 5611:d 5606:c 5602:= 5597:1 5593:m 5580:) 5574:c 5570:m 5552:. 5548:) 5545:p 5538:( 5531:1 5524:q 5520:= 5511:q 5489:, 5485:) 5482:1 5476:q 5469:( 5464:d 5461:= 5456:Q 5452:d 5430:, 5426:) 5423:1 5417:p 5410:( 5405:d 5402:= 5397:P 5393:d 5370:q 5350:p 5272:M 5268:m 5257:m 5253:m 5236:n 5232:e 5228:n 5224:d 5215:n 5205:n 5200:d 5195:. 5193:n 5189:r 5179:m 5173:n 5164:) 5162:n 5153:c 5149:r 5145:) 5143:n 5135:c 5130:d 5126:) 5124:n 5120:m 5116:c 5107:) 5105:n 5101:2 5098:m 5095:1 5092:m 5088:2 5085:m 5082:1 5079:m 5040:n 5036:q 5032:p 5028:e 5024:e 5017:e 5013:n 5008:m 5002:n 4998:m 4993:m 4987:e 4965:) 4963:q 4959:m 4954:) 4952:p 4948:m 4943:) 4937:q 4933:p 4927:n 4923:m 4916:n 4912:m 4908:) 4906:n 4902:m 4891:n 4889:( 4878:d 4874:e 4851:, 4847:) 4844:n 4837:( 4832:m 4824:h 4820:) 4816:1 4813:( 4810:m 4802:h 4798:) 4792:) 4789:n 4786:( 4779:m 4775:( 4772:m 4769:= 4764:) 4761:n 4758:( 4752:h 4749:+ 4746:1 4742:m 4738:= 4733:d 4730:e 4726:m 4715:n 4711:m 4703:h 4699:) 4697:n 4695:( 4684:d 4680:e 4674:n 4672:( 4661:d 4657:e 4648:n 4643:) 4641:n 4637:m 4633:m 4598:. 4592:( 4581:d 4575:q 4568:p 4563:) 4559:( 4550:q 4546:p 4544:( 4538:q 4534:p 4525:d 4521:e 4473:. 4469:) 4466:q 4463:p 4456:( 4451:m 4443:d 4439:) 4433:e 4429:m 4425:( 4411:( 4400:d 4396:e 4392:m 4373:. 4369:) 4366:q 4359:( 4354:m 4348:m 4343:k 4339:1 4332:m 4327:k 4323:) 4317:1 4311:q 4307:m 4303:( 4300:= 4297:m 4292:) 4289:1 4283:q 4280:( 4277:k 4273:m 4269:= 4266:m 4261:1 4255:d 4252:e 4248:m 4244:= 4239:d 4236:e 4232:m 4219:) 4217:q 4194:m 4188:. 4186:) 4184:q 4180:m 4176:m 4171:q 4167:m 4163:) 4161:q 4157:m 4148:) 4146:q 4142:m 4138:m 4126:p 4122:m 4101:, 4097:) 4094:p 4087:( 4082:m 4076:m 4071:h 4067:1 4060:m 4055:h 4051:) 4045:1 4039:p 4035:m 4031:( 4028:= 4025:m 4020:) 4017:1 4011:p 4008:( 4005:h 4001:m 3997:= 3994:m 3989:1 3983:d 3980:e 3976:m 3972:= 3967:d 3964:e 3960:m 3947:) 3945:p 3922:m 3916:. 3914:) 3912:p 3908:m 3904:m 3899:p 3895:m 3891:p 3887:m 3883:) 3881:p 3877:m 3868:) 3866:p 3862:m 3858:m 3849:q 3842:p 3829:m 3823:m 3816:k 3812:h 3798:) 3795:1 3789:q 3786:( 3783:k 3780:= 3777:) 3774:1 3768:p 3765:( 3762:h 3759:= 3756:1 3750:d 3747:e 3735:q 3728:p 3721:q 3717:p 3715:( 3707:( 3693:( 3682:d 3678:e 3674:q 3670:p 3666:m 3651:) 3648:q 3645:p 3638:( 3633:m 3625:d 3621:) 3615:e 3611:m 3607:( 3594:a 3590:p 3586:a 3582:) 3580:p 3576:a 3534:. 3530:) 3527:n 3520:( 3515:h 3507:e 3503:) 3497:d 3493:h 3489:( 3486:= 3481:e 3478:d 3474:h 3470:= 3465:d 3462:e 3458:h 3454:= 3449:d 3445:) 3439:e 3435:h 3431:( 3412:, 3409:) 3406:m 3403:( 3394:= 3391:h 3374:n 3370:e 3366:n 3362:d 3315:= 3306:1 3303:+ 3297:= 3294:q 3288:h 3285:+ 3280:2 3276:m 3272:= 3265:m 3258:, 3255:1 3252:= 3249:1 3244:6 3236:) 3233:8 3221:( 3218:= 3213:p 3205:) 3202:) 3197:2 3193:m 3184:1 3180:m 3176:( 3164:q 3160:( 3157:= 3150:h 3143:, 3137:= 3134:3 3129:5 3111:= 3106:q 3094:q 3090:d 3085:c 3081:= 3072:2 3068:m 3060:, 3057:4 3054:= 3051:1 3046:6 3028:= 3023:p 3011:p 3007:d 3002:c 2998:= 2989:1 2985:m 2970:e 2966:d 2959:q 2953:q 2950:d 2944:p 2941:d 2918:= 2915:1 2910:6 2893:= 2888:p 2880:) 2877:q 2865:q 2861:( 2845:= 2842:1 2837:6 2827:1 2816:= 2811:p 2801:1 2794:q 2790:= 2777:q 2769:, 2763:= 2760:) 2757:1 2746:( 2735:= 2732:) 2729:1 2723:q 2718:( 2710:d 2707:= 2698:q 2694:d 2686:, 2680:= 2677:) 2674:1 2663:( 2652:= 2649:) 2646:1 2640:p 2635:( 2627:d 2624:= 2615:p 2611:d 2593:q 2587:q 2584:d 2578:p 2575:d 2568:q 2564:p 2549:d 2545:e 2541:q 2537:p 2531:n 2503:= 2495:3 2477:= 2474:m 2462:c 2442:= 2434:3 2416:= 2413:c 2401:m 2375:3 2361:c 2357:= 2345:n 2335:d 2331:c 2327:= 2320:) 2317:c 2314:( 2311:m 2297:c 2288:d 2284:n 2282:( 2253:3 2239:m 2235:= 2223:n 2213:e 2209:m 2205:= 2198:) 2195:m 2192:( 2189:c 2175:m 2165:e 2161:n 2159:( 2133:7 2125:) 2113:( 2110:= 2107:1 2087:, 2081:= 2078:d 2065:n 2063:( 2057:e 2048:d 2041:. 2026:= 2023:e 2010:e 2006:e 1996:e 1973:= 1970:) 1964:, 1958:( 1949:= 1946:) 1940:( 1923:q 1919:p 1917:( 1911:n 1909:( 1881:= 1869:= 1866:n 1849:n 1841:. 1826:= 1823:q 1800:= 1797:p 1773:M 1769:m 1753:. 1749:) 1746:n 1739:( 1734:m 1726:d 1722:) 1716:e 1712:m 1708:( 1700:d 1696:c 1683:d 1679:c 1675:m 1663:m 1659:c 1655:m 1651:c 1631:. 1627:) 1624:n 1617:( 1610:e 1606:m 1599:c 1587:e 1583:c 1574:n 1570:m 1560:m 1556:M 1549:M 1537:) 1535:d 1533:( 1529:) 1527:e 1523:n 1521:( 1498:d 1494:e 1486:) 1484:n 1482:( 1475:d 1467:e 1463:d 1454:q 1452:( 1446:p 1444:( 1438:q 1434:p 1430:q 1426:p 1424:( 1411:n 1404:q 1402:( 1396:p 1394:( 1387:) 1385:n 1383:( 1376:) 1374:n 1372:( 1366:d 1353:d 1349:) 1347:n 1345:( 1339:d 1334:) 1332:n 1330:( 1323:d 1317:n 1315:( 1309:e 1307:⋅ 1305:d 1298:n 1296:( 1290:e 1288:⋅ 1286:d 1281:d 1262:) 1260:n 1258:( 1251:) 1249:n 1247:( 1240:d 1236:) 1234:n 1232:( 1223:q 1219:p 1215:n 1213:( 1200:d 1196:d 1192:) 1190:n 1188:( 1181:q 1177:p 1173:d 1165:e 1161:n 1149:. 1143:d 1137:d 1129:) 1127:n 1125:( 1118:e 1110:d 1104:n 1102:( 1091:d 1084:) 1082:n 1080:( 1073:e 1065:d 1059:n 1057:( 1051:e 1047:d 1042:d 1033:e 1027:e 1023:e 1009:e 997:e 987:) 985:n 983:( 976:e 970:n 968:( 962:e 960:( 953:) 951:n 949:( 943:e 937:e 928:) 926:n 924:( 917:. 911:) 909:b 905:a 900:/ 884:b 880:a 861:q 857:p 853:n 851:( 842:q 838:q 836:( 827:p 823:p 821:( 813:p 811:( 804:q 800:p 794:q 792:( 786:p 784:( 780:( 774:n 772:( 762:n 749:) 747:n 745:( 733:n 729:. 719:n 708:n 698:q 694:p 684:q 680:p 673:q 669:p 639:d 635:e 627:m 623:d 619:e 615:n 608:d 604:n 600:e 586:. 582:) 579:n 572:( 567:m 559:d 555:) 549:e 545:m 541:( 521:n 498:n 474:m 452:d 448:) 442:e 438:m 434:( 423:n 419:m 415:( 413:m 409:n 405:d 401:e 291:" 252:) 144:( 108:1 34:. 20:)

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Knowledge

RSA (cryptosystem)

Index