287:, i.e., capable of evaluating its own decryption circuit and then at least one more operation. Finally, he shows that any bootstrappable somewhat homomorphic encryption scheme can be converted into a fully homomorphic encryption through a recursive self-embedding. For Gentry's "noisy" scheme, the bootstrapping procedure effectively "refreshes" the ciphertext by applying to it the decryption procedure homomorphically, thereby obtaining a new ciphertext that encrypts the same value as before but has lower noise. By "refreshing" the ciphertext periodically whenever the noise grows too large, it is possible to compute an arbitrary number of additions and multiplications without increasing the noise too much.
3364:-like packing of data; they are typically used to compute on encrypted integers or real/complex numbers. Third-generation FHE scheme implementations often bootstrap after each operation but have limited support for packing; they were initially used to compute Boolean circuits over encrypted bits, but have been extended to support integer arithmetics and univariate function evaluation. The choice of using a second-generation vs. third-generation vs fourth-generation scheme depends on the input data types and the desired computation.
555:
HElib and PALISADE) and report that it is possible to recover the secret key from decryption results in several parameter configurations. The authors also propose mitigation strategies for these attacks, and include a
Responsible Disclosure in the paper suggesting that the homomorphic encryption libraries already implemented mitigations for the attacks before the article became publicly available. Further information on the mitigation strategies implemented in the homomorphic encryption libraries has also been published.
309:. Instead, they show that the somewhat homomorphic component of Gentry's ideal lattice-based scheme can be replaced with a very simple somewhat homomorphic scheme that uses integers. The scheme is therefore conceptually simpler than Gentry's ideal lattice scheme, but has similar properties with regards to homomorphic operations and efficiency. The somewhat homomorphic component in the work of Van Dijk et al. is similar to an encryption scheme proposed by Levieil and
530:(GSW) proposed a new technique for building FHE schemes that avoids an expensive "relinearization" step in homomorphic multiplication. Zvika Brakerski and Vinod Vaikuntanathan observed that for certain types of circuits, the GSW cryptosystem features an even slower growth rate of noise, and hence better efficiency and stronger security. Jacob Alperin-Sheriff and Chris Peikert then described a very efficient bootstrapping technique based on this observation.
1634:
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294:, and the sparse (or low-weight) subset sum problem. Gentry's Ph.D. thesis provides additional details. The Gentry-Halevi implementation of Gentry's original cryptosystem reported a timing of about 30 minutes per basic bit operation. Extensive design and implementation work in subsequent years have improved upon these early implementations by many orders of magnitude runtime performance.
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second. FHEW introduced a new method to compute
Boolean gates on encrypted data that greatly simplifies bootstrapping and implemented a variant of the bootstrapping procedure. The efficiency of FHEW was further improved by the TFHE scheme, which implements a ring variant of the bootstrapping procedure using a method similar to the one in FHEW.
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is not even additively homomorphic, however. The
Levieil–Naccache scheme supports only additions, but it can be modified to also support a small number of multiplications. Many refinements and optimizations of the scheme of Van Dijk et al. were proposed in a sequence of works by Jean-Sébastien Coron,
79:
that allows computations to be performed on encrypted data without first having to decrypt it. The resulting computations are left in an encrypted form which, when decrypted, result in an output that is identical to that produced had the operations been performed on the unencrypted data. Homomorphic
554:
A 2020 article by Baiyu Li and
Daniele Micciancio discusses passive attacks against CKKS, suggesting that the standard IND-CPA definition may not be sufficient in scenarios where decryption results are shared. The authors apply the attack to four modern homomorphic encryption libraries (HEAAN, SEAL,
533:
These techniques were further improved to develop efficient ring variants of the GSW cryptosystem: FHEW (2014) and TFHE (2016). The FHEW scheme was the first to show that by refreshing the ciphertexts after every single operation, it is possible to reduce the bootstrapping time to a fraction of a
275:, described the first plausible construction for a fully homomorphic encryption scheme in 2009. Gentry's scheme supports both addition and multiplication operations on ciphertexts, from which it is possible to construct circuits for performing arbitrary computation. The construction starts from a
546:
arithmetic. The CKKS scheme includes an efficient rescaling operation that scales down an encrypted message after a multiplication. For comparison, such rescaling requires bootstrapping in the BGV and BFV schemes. The rescaling operation makes CKKS scheme the most efficient method for evaluating
3241:
on ciphertexts is known as fully homomorphic encryption (FHE). Such a scheme enables the construction of programs for any desirable functionality, which can be run on encrypted inputs to produce an encryption of the result. Since such a program need never decrypt its inputs, it can be run by an
279:
encryption scheme, which is limited to evaluating low-degree polynomials over encrypted data; it is limited because each ciphertext is noisy in some sense, and this noise grows as one adds and multiplies ciphertexts, until ultimately the noise makes the resulting ciphertext indecipherable.
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There are several open-source implementations of fully homomorphic encryption schemes. Second-generation and fourth-generation FHE scheme implementations typically operate in the leveled FHE mode (though bootstrapping is still available in some libraries) and support efficient
210:
The problem of constructing a fully homomorphic encryption scheme was first proposed in 1978, within a year of publishing of the RSA scheme. For more than 30 years, it was unclear whether a solution existed. During that period, partial results included the following schemes:
107:
concerns. But if the predictive-analytics service provider could operate on encrypted data instead, without having the decryption keys, these privacy concerns are diminished. Moreover, even if the service provider's system is compromised, the data would remain secure.
139:
Homomorphic encryption includes multiple types of encryption schemes that can perform different classes of computations over encrypted data. The computations are represented as either
Boolean or arithmetic circuits. Some common types of homomorphic encryption are
1629:{\displaystyle {\begin{aligned}{\mathcal {E}}(m_{1})\cdot {\mathcal {E}}(m_{2})&=(g^{r_{1}},m_{1}\cdot h^{r_{1}})(g^{r_{2}},m_{2}\cdot h^{r_{2}})\\&=(g^{r_{1}+r_{2}},(m_{1}\cdot m_{2})h^{r_{1}+r_{2}})\\&={\mathcal {E}}(m_{1}\cdot m_{2}).\end{aligned}}}
2668:{\displaystyle {\begin{aligned}{\mathcal {E}}(m_{1})\cdot {\mathcal {E}}(m_{2})&=(g^{m_{1}}r_{1}^{c})(g^{m_{2}}r_{2}^{c})\;{\bmod {\;}}n\\&=g^{m_{1}+m_{2}}(r_{1}r_{2})^{c}\;{\bmod {\;}}n\\&={\mathcal {E}}(m_{1}+m_{2}\;{\bmod {\;}}c).\end{aligned}}}
3185:{\displaystyle {\begin{aligned}{\mathcal {E}}(m_{1})\cdot {\mathcal {E}}(m_{2})&=(g^{m_{1}}r_{1}^{n})(g^{m_{2}}r_{2}^{n})\;{\bmod {\;}}n^{2}\\&=g^{m_{1}+m_{2}}(r_{1}r_{2})^{n}\;{\bmod {\;}}n^{2}\\&={\mathcal {E}}(m_{1}+m_{2}).\end{aligned}}}
392:
All the second-generation cryptosystems still follow the basic blueprint of Gentry's original construction, namely they first construct a somewhat homomorphic cryptosystem and then convert it to a fully homomorphic cryptosystem using bootstrapping.
2113:{\displaystyle {\begin{aligned}{\mathcal {E}}(b_{1})\cdot {\mathcal {E}}(b_{2})&=x^{b_{1}}r_{1}^{2}x^{b_{2}}r_{2}^{2}\;{\bmod {\;}}n\\&=x^{b_{1}+b_{2}}(r_{1}r_{2})^{2}\;{\bmod {\;}}n\\&={\mathcal {E}}(b_{1}\oplus b_{2}).\end{aligned}}}
98:
For sensitive data, such as healthcare information, homomorphic encryption can be used to enable new services by removing privacy barriers inhibiting data sharing or increasing security to existing services. For example,
188:
For the majority of homomorphic encryption schemes, the multiplicative depth of circuits is the main practical limitation in performing computations over encrypted data. Homomorphic encryption schemes are inherently
966:{\displaystyle {\begin{aligned}{\mathcal {E}}(m_{1})\cdot {\mathcal {E}}(m_{2})&=m_{1}^{e}m_{2}^{e}\;{\bmod {\;}}n\\&=(m_{1}m_{2})^{e}\;{\bmod {\;}}n\\&={\mathcal {E}}(m_{1}\cdot m_{2})\end{aligned}}}
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untrusted party without revealing its inputs and internal state. Fully homomorphic cryptosystems have great practical implications in the outsourcing of private computations, for instance, in the context of
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Homomorphic encryption schemes have been developed using different approaches. Specifically, fully homomorphic encryption schemes are often grouped into generations corresponding to the underlying approach.
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Another distinguishing feature of second-generation schemes is that they are efficient enough for many applications even without invoking bootstrapping, instead operating in the leveled FHE mode.
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A distinguishing characteristic of the second-generation cryptosystems is that they all feature a much slower growth of the noise during the homomorphic computations. Additional optimizations by
501:
91:
Homomorphic encryption eliminates the need for processing data in the clear, thereby preventing attacks that would enable a hacker to access that data while it is being processed, using
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In 2016, Cheon, Kim, Kim and Song (CKKS) proposed an approximate homomorphic encryption scheme that supports a special kind of fixed-point arithmetic that is commonly referred to as
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2355:
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503:. These optimizations build on the Smart-Vercauteren techniques that enable packing of many plaintext values in a single ciphertext and operating on all these plaintext values in a
4521:
Coron, Jean-SĂ©bastien; Mandal, Avradip; Naccache, David; Tibouchi, Mehdi (2011). "Fully
Homomorphic Encryption over the Integers with Shorter Public Keys". In Rogaway, P. (ed.).
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A list of open-source FHE libraries implementing second-generation (BGV/BFV), third-generation (FHEW/TFHE), and/or fourth-generation (CKKS) FHE schemes is provided below.
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computational problem. This NTRU variant was subsequently shown vulnerable to subfield lattice attacks, which is why these two schemes are no longer used in practice.
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and Vinod
Vaikuntanathan presented a second fully homomorphic encryption scheme, which uses many of the tools of Gentry's construction, but which does not require
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184:(FHE) allows the evaluation of arbitrary circuits composed of multiple types of gates of unbounded depth and is the strongest notion of homomorphic encryption.
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Rust implementation of TFHE-extended. Supporting boolean, integer operation and univariate function evaluation (via
Programmable Bootstrapping).
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Topics in
Cryptology – CT-RSA 2015, The Cryptographer's Track at the RSA Conference 2015, San Francisco, CA, USA, April 20–24, 2015. Proceedings
5532:
5019:
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348:, and others. These innovations led to the development of much more efficient somewhat and fully homomorphic cryptosystems. These include:
551:. The scheme introduces several approximation errors, both nondeterministic and deterministic, that require special handling in practice.
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encompasses schemes that support the evaluation of circuits consisting of only one type of gate, e.g., addition or multiplication.
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Cheon, Jung Hee; Kim, Andrey; Kim, Miran; Song, Yongsoo (2017). "Homomorphic encryption for arithmetic of approximate numbers".
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in algebra: the encryption and decryption functions can be thought of as homomorphisms between plaintext and ciphertext spaces.
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fashion. Many of the advances in these second-generation cryptosystems were also ported to the cryptosystem over the integers.
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373:-based scheme by Bos, Lauter, Loftus, and Naehrig (BLLN, 2013), building on LTV and Brakerski's scale-invariant cryptosystem;
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336:
The homomorphic cryptosystems of this generation are derived from techniques that were developed starting in 2011–2012 by
4484:
193:. In terms of malleability, homomorphic encryption schemes have weaker security properties than non-homomorphic schemes.
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3310:
88:. This allows data to be encrypted and outsourced to commercial cloud environments for processing, all while encrypted.
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supports the evaluation of arbitrary circuits composed of multiple types of gates of bounded (pre-determined) depth.
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4736:"An algorithm for NTRU problems and cryptanalysis of the GGH multilinear map without a low-level encoding of zero"
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Gentry based the security of his scheme on the assumed hardness of two problems: certain worst-case problems over
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Armknecht, Frederik; Boyd, Colin; Gjøsteen, Kristian; Jäschke, Angela; Reuter, Christian; Strand, Martin (2015).
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49:
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Homomorphic
Encryption from Learning with Errors: Conceptually-Simpler, Asymptotically-Faster, Attribute-Based
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124:. The result of such a computation remains encrypted. Homomorphic encryption can be viewed as an extension of
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Structural Lattice Reduction: Generalized Worst-Case to Average-Case Reductions and Homomorphic Cryptosystems
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The Brakerski-Gentry-Vaikuntanathan (BGV, 2011) scheme, building on techniques of Brakerski-Vaikuntanathan;
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5006:. ASIACRYPT 2017. Lecture Notes in Computer Science. Vol. 10624. Springer, Cham. pp. 409–437.
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Boneh–Goh–Nissim cryptosystem (unlimited number of addition operations but at most one multiplication)
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2156:
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345:
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Sander-Young-Yung system (after more than 20 years solved the problem for logarithmic depth circuits)
237:
100:
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45:
4836:
3317:
4318:
Sander, Tomas; Young, Adam L.; Yung, Moti (1999). "Non-interactive cryptocomputing for NC/Sup 1/".
4197:"Council Post: Everything You Wanted To Know About Homomorphic Encryption (But Were Afraid To Ask)"
3528:
3520:
5269:
328:, and Mehdi Tibouchi. Some of these works included also implementations of the resulting schemes.
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4849:
4504:"Public Key Compression and Modulus Switching for Fully Homomorphic Encryption over the Integers"
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with an additional evaluation capability for computing over encrypted data without access to the
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5838:"Intel, Microsoft Research and Duality Technologies Convene AI Community for Privacy Standards"
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5510:"Programmable Bootstrapping Enables Efficient Homomorphic Inference of Deep Neural Networks"
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4222:"A systematic review of homomorphic encryption and its contributions in healthcare industry"
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17:
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R. L. Rivest, L. Adleman, and M. L. Dertouzos. On data banks and privacy homomorphisms. In
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Christian Mouchet, Juan Troncoso-Pastoriza, Jean-Philippe Bossuat and Jean-Pierre Hubaux.
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Jean-Philippe Bossuat, Christian Mouchet, Juan Troncoso-Pastoriza and Jean-Pierre Hubaux.
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337:
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85:
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Verifiable encrypted computations based on Rinocchio ZKP and BGV homomorphic Encryption.
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On-the-Fly Multiparty Computation on the Cloud via Multikey Fully Homomorphic Encryption
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Efficient Bootstrapping for Approximate Homomorphic Encryption with Non-Sparse Keys
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5178:. Lecture Notes in Computer Science. Vol. 9048. Springer. pp. 487–505.
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resulted in cryptosystems with nearly optimal asymptotic complexity: Performing
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5859:"Intel, Microsoft join DARPA effort to accelerate fully homomorphic encryption"
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D. Boneh, E. Goh, and K. Nissim. Evaluating 2-DNF Formulas on Ciphertexts. In
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117:
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4962:"Faster Fully Homomorphic Encryption: Bootstrapping in less than 0.1 Seconds"
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Van Dijk, Marten; Gentry, Craig; Halevi, Shai; Vinod, Vaikuntanathan (2009).
103:
in healthcare can be hard to apply via a third-party service provider due to
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5274:
4681:
Fully Homomorphic Encryption without Modulus Switching from Classical GapSVP
4371:
Y. Ishai and A. Paskin. Evaluating branching programs on encrypted data. In
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40th Annual Symposium on Foundations of Computer Science (Cat. No.99CB37039)
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schemes can evaluate two types of gates, but only for a subset of circuits.
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polynomial approximations, and is the preferred approach for implementing
362:
The Brakerski/Fan-Vercauteren (BFV, 2012) scheme, building on Brakerski's
5909:
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5430:
5117:
Benhamouda, Fabrice; Herranz, Javier; Joye, Marc; Libert, Benoît (2017).
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5222:
Jung Hee Cheon, Kyoohyung Han, Andrey Kim, Miran Kim and Yongsoo Song.
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4527:. Lecture Notes in Computer Science. Vol. 6841. pp. 487–504.
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3467:
5712:"T2: A cross compiler and standardized benchmarks for FHE computation"
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4698:
Improved Security for a Ring-Based Fully Homomorphic Encryption Scheme
3639:
Ilaria Chillotti, Nicolas Gama, Mariya Georgieva and Malika Izabachene
377:
The security of most of these schemes is based on the hardness of the
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5519:. Lecture Notes in Computer Science. Vol. 12716. pp. 1–19.
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4960:
Ilaria Chillotti; Nicolas Gama; Mariya Georgieva; Malika Izabachene.
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5374:
5317:
Jung Hee Cheon; Kyoohyung Han; Andrey Kim; Miran Kim; Yongsoo Song.
5035:
4403:
4272:
381:(RLWE) problem, except for the LTV and BLLN schemes that rely on an
5740:"HELM: Navigating Homomorphic Evaluation through Gates and Lookups"
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5223:
5036:
Approximate Homomorphic Encryption with Reduced Approximation Error
4982:
4717:
4697:
4659:
4638:
359:-based scheme by Lopez-Alt, Tromer, and Vaikuntanathan (LTV, 2012);
5768:"Juliet: A Configurable Processor for Computing on Encrypted Data"
5241:
5054:"On the Security of Homomorphic Encryption on Approximate Numbers"
5004:
Takagi T., Peyrin T. (eds) Advances in Cryptology – ASIACRYPT 2017
4903:
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4621:
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Coron, Jean-Sébastien; Lepoint, Tancrède; Tibouchi, Mehdi (2014).
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Coron, Jean-Sébastien; Lepoint, Tancrède; Tibouchi, Mehdi (2013).
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3487:
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5625:
5475:
Multiparty Homomorphic Encryption from Ring-Learning-With-Errors
4585:"Scale-Invariant Fully Homomorphic Encryption over the Integers"
4502:
Coron, Jean-SĂ©bastien; Naccache, David; Tibouchi, Mehdi (2011).
3361:
504:
386:
370:
356:
283:
Gentry then shows how to slightly modify this scheme to make it
5375:"A GPU implementation of fully homomorphic encryption on torus"
2815:{\displaystyle {\mathcal {E}}(m)=g^{m}r^{n}\;{\bmod {\;}}n^{2}}
5319:"Homomorphic Encryption for Arithmetic of Approximate Numbers"
4961:
4094:
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General-purpose lattice cryptography library. Predecessor of
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FHE.org Community (conference, meetup and discussion group)
4718:
A subfield lattice attack on overstretched NTRU assumptions
4422:"Implementing Gentry's fully-homomorphic encryption scheme"
2300:{\displaystyle {\mathcal {E}}(m)=g^{m}r^{c}\;{\bmod {\;}}n}
1769:{\displaystyle {\mathcal {E}}(b)=x^{b}r^{2}\;{\bmod {\;}}n}
4622:
Efficient Fully Homomorphic Encryption from (Standard) LWE
218:
cryptosystem (unbounded number of modular multiplications)
5119:"Efficient cryptosystems from 2-th power residue symbols"
80:
encryption can be used for privacy-preserving outsourced
27:
Form of encryption that allows computation on ciphertexts
5508:
Chillotti, Ilaria; Joye, Marc; Paillier, Pascal (2021).
5074:
Cheon, Jung Hee; Hong, Seungwan; Kim, Duhyeong (2020).
5640:
MoMA Lab, New York University Abu Dhabi (2019-07-24).
4563:"Batch Fully Homomorphic Encryption over the Integers"
4404:"A Fully Homomorphic Encryption Scheme (Ph.D. thesis)"
1226:{\displaystyle {\mathcal {E}}(m)=(g^{r},m\cdot h^{r})}
739:{\displaystyle {\mathcal {E}}(m)=m^{e}\;{\bmod {\;}}n}
5670:"SHEEP, a Homomorphic Encryption Evaluation Platform"
5403:"A Multi-GPU Implementation of the CGGI Cryptosystem"
4142:
Verifiable computing using a fully homomorphic scheme
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operations on data encrypted with security parameter
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5242:"HElib: An Implementation of homomorphic encryption"
5224:
Bootstrapping for Approximate Homomorphic Encryption
4785:
Better Bootstrapping in Fully Homomorphic Encryption
4390:
the 41st ACM Symposium on Theory of Computing (STOC)
1142:
is the secret key, then the encryption of a message
5076:"Remark on the Security of CKKS Scheme in Practice"
60:
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4768:Fully Homomorphic Encryption with Polylog Overhead
4606:Fully Homomorphic Encryption without Bootstrapping
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5817:"Homomorphic Encryption Standardization Workshop"
5167:Castagnos, Guilhem; Laguillaumie, Fabien (2015).
4660:"Somewhat Practical Fully Homomorphic Encryption"
4386:Fully Homomorphic Encryption Using Ideal Lattices
4111:Homomorphic Encryption Standardization Consortium
5766:Trustworthy Computing (TwC) Group (2024-06-25).
5738:Trustworthy Computing (TwC) Group (2024-07-29).
5710:Trustworthy Computing (TwC) Group (2023-03-02).
5668:Alan Turing Institute, London, UK (2019-11-01).
5517:Cyber Security Cryptography and Machine Learning
4981:N. Gama, M. Izabachène, P.Q. Nguyen, and X. Xie
4637:A. Lopez-Alt, E. Tromer, and V. Vaikuntanathan.
4604:Z. Brakerski, C. Gentry, and V. Vaikuntanathan.
4441:"Fully Homomorphic Encryption over the Integers"
596:is used to denote the encryption of the message
549:privacy-preserving machine learning applications
4818:Smart, Nigel P.; Vercauteren, Frederik (2014).
3912:TFHE-extended compiler with a Python Frontend.
5910:list of homomorphic encryption implementations
5199:"A First Glimpse of Cryptography's Holy Grail"
4929:"FHEW: A Fully Homomorphic Encryption library"
4696:J. Bos, K. Lauter, J. Loftus, and M. Naehrig.
313:in 2008, and also to one that was proposed by
224:(unbounded number of modular multiplications)
8:
5695:: CS1 maint: multiple names: authors list (
4658:Fan, Junfeng; Vercauteren, Frederik (2012).
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1246:
496:{\displaystyle T\cdot \mathrm {polylog} (k)}
30:
255:Ishai-Paskin cryptosystem (polynomial-size
5169:"Linearly Homomorphic Encryption from DDH"
4904:Faster Bootstrapping with Polynomial Error
4740:LMS Journal of Computation and Mathematics
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3110:
3024:
3018:
2799:
2793:
2649:
2643:
2594:
2588:
2509:
2503:
2291:
2285:
2051:
2045:
1966:
1960:
1760:
1754:
907:
901:
852:
846:
730:
724:
4997:
4995:
4835:
4802:Homomorphic Evaluation of the AES Circuit
4751:
4532:
4273:"A Guide to Fully Homomorphic Encryption"
4247:
4237:
4132:Homomorphic signatures for network coding
3344:Learn how and when to remove this message
3196:Other partially homomorphic cryptosystems
3166:
3153:
3140:
3139:
3123:
3115:
3111:
3104:
3094:
3084:
3069:
3056:
3051:
3031:
3023:
3019:
3009:
3004:
2992:
2987:
2971:
2966:
2954:
2949:
2926:
2913:
2912:
2900:
2887:
2886:
2882:
2880:
2827:
2806:
2798:
2794:
2787:
2777:
2755:
2754:
2752:
2732:
2712:
2692:
2648:
2644:
2637:
2624:
2611:
2610:
2593:
2589:
2582:
2572:
2562:
2547:
2534:
2529:
2508:
2504:
2494:
2489:
2477:
2472:
2456:
2451:
2439:
2434:
2411:
2398:
2397:
2385:
2372:
2371:
2367:
2365:
2312:
2290:
2286:
2279:
2269:
2247:
2246:
2244:
2224:
2204:
2184:
2164:
2128:
2094:
2081:
2068:
2067:
2050:
2046:
2039:
2029:
2019:
2004:
1991:
1986:
1965:
1961:
1954:
1949:
1937:
1932:
1922:
1917:
1905:
1900:
1880:
1867:
1866:
1854:
1841:
1840:
1836:
1834:
1781:
1759:
1755:
1748:
1738:
1716:
1715:
1713:
1693:
1673:
1653:
1610:
1597:
1584:
1583:
1562:
1549:
1544:
1531:
1518:
1500:
1487:
1482:
1454:
1449:
1436:
1421:
1416:
1398:
1393:
1380:
1365:
1360:
1337:
1324:
1323:
1311:
1298:
1297:
1293:
1291:
1238:
1214:
1195:
1170:
1169:
1167:
1147:
1127:
1106:
1094:
1050:
1030:
1010:
990:
950:
937:
924:
923:
906:
902:
895:
885:
875:
851:
847:
840:
835:
825:
820:
800:
787:
786:
774:
761:
760:
756:
754:
729:
725:
718:
696:
695:
693:
673:
653:
633:
601:
571:
570:
568:
461:
453:
433:
413:
4734:Cheon, J. H.; Jeong, J; Lee, C. (2016).
4712:
4710:
4675:
4673:
4600:
4598:
4578:
4576:
4556:
4554:
3851:
3477:Samsung Advanced Institute of Technology
3366:
563:In the following examples, the notation
5797:TrustworthyComputing/PEEV-verifiableFHE
5297:"PALISADE Lattice Cryptography Library"
5052:Li, Baily; Micciancio, Daniele (2020).
4955:
4953:
4951:
4922:
4920:
4918:
4916:
4898:
4896:
4800:C. Gentry, S. Halevi, and N. P. Smart.
4783:C. Gentry, S. Halevi, and N. P. Smart.
4766:C. Gentry, S. Halevi, and N. P. Smart.
4653:
4651:
4187:
4113:, which maintains a community security
3716:Provides a GPU implementation of TFHE.
3430:BGV scheme with the GHS optimizations.
246:(unbounded number of modular additions)
240:(unbounded number of modular additions)
5688:
5034:Kim A., Papadimitriou A., Polyakov Y.
4220:Munjal, Kundan; Bhatia, Rekha (2022).
3280:Please improve this section by adding
2865:{\displaystyle r\in \{0,\ldots ,n-1\}}
2350:{\displaystyle r\in \{0,\ldots ,n-1\}}
1819:{\displaystyle r\in \{0,\ldots ,n-1\}}
1276:{\displaystyle r\in \{0,\ldots ,q-1\}}
29:
5885:Vinod Vaikuntanathan's FHE references
7:
5642:"Encrypt-Everything-Everywhere (E3)"
4883:Z. Brakerski and V. Vaikuntanathan.
4866:C. Gentry, A. Sahai, and B. Waters.
4620:Z. Brakerski and V. Vaikuntanathan.
4524:Advances in Cryptology – CRYPTO 2011
4420:Gentry, Craig; Halevi, Shai (2010).
3846:A multi-GPU implementation of CKKS.
3745:A multi-GPU implementation of TFHE.
176:Leveled fully homomorphic encryption
116:Homomorphic encryption is a form of
5880:Daniele Micciancio's FHE references
5794:TrustworthyComputing (2024-07-18),
5401:Trustworthy Computing (TwC) Group.
5203:Association for Computing Machinery
4902:J. Alperin-Sheriff and C. Peikert.
4820:"Fully Homomorphic SIMD Operations"
4716:M. Albrecht, S. Bai, and L. Ducas.
4292:"Homomorphic Encryption References"
3546:University of California, San Diego
3496:University of California, San Diego
3228:Castagnos–Laguillaumie cryptosystem
3216:Sander–Young–Yung encryption scheme
2872:. The homomorphic property is then
2727:, then the encryption of a message
2687:, if the public key is the modulus
2357:. The homomorphic property is then
2219:, then the encryption of a message
2159:, if the public key is the modulus
1826:. The homomorphic property is then
1648:, if the public key is the modulus
1283:. The homomorphic property is then
746:. The homomorphic property is then
668:, then the encryption of a message
559:Partially homomorphic cryptosystems
4885:Lattice-Based FHE as Secure as PKE
3534:New Jersey Institute of Technology
2143:denotes addition modulo 2, (i.e.,
480:
477:
474:
471:
468:
465:
462:
324:Tancrède Lepoint, Avradip Mandal,
25:
5197:Daniele Micciancio (2010-03-01).
4373:Theory of Cryptography Conference
4360:Theory of Cryptography Conference
4306:Foundations of Secure Computation
4226:Complex & Intelligent Systems
589:{\displaystyle {\mathcal {E}}(x)}
3611:Leo Ducas and Daniele Micciancio
3258:
164:Partially homomorphic encryption
4927:Leo Ducas; Daniele Micciancio.
4824:Designs, Codes and Cryptography
4464:"Cryptographic Test Correction"
4157:Searchable symmetric encryption
4115:Homomorphic Encryption Standard
1688:, then the encryption of a bit
170:Somewhat homomorphic encryption
44:Various assumptions, including
5891:"Alice and Bob in Cipherspace"
4485:"Simple Public Key Encryption"
4162:Secure multi-party computation
3784:Secure multi-party computation
3172:
3146:
3101:
3077:
3015:
2980:
2977:
2942:
2932:
2919:
2906:
2893:
2767:
2761:
2655:
2617:
2579:
2555:
2500:
2465:
2462:
2427:
2417:
2404:
2391:
2378:
2259:
2253:
2100:
2074:
2036:
2012:
1886:
1873:
1860:
1847:
1728:
1722:
1646:Goldwasser–Micali cryptosystem
1616:
1590:
1570:
1537:
1511:
1475:
1462:
1409:
1406:
1353:
1343:
1330:
1317:
1304:
1220:
1188:
1182:
1176:
1076:
1052:
956:
930:
892:
868:
806:
793:
780:
767:
708:
702:
583:
577:
490:
484:
228:Goldwasser–Micali cryptosystem
1:
5083:IACR ePrint Archive 2020/1581
5061:IACR ePrint Archive 2020/1533
4109:, and others formed the open
3282:secondary or tertiary sources
3237:A cryptosystem that supports
3219:Boneh–Goh–Nissim cryptosystem
3202:Okamoto–Uchiyama cryptosystem
5840:. Intel Newsroom. 2019-08-16
5184:10.1007/978-3-319-16715-2_26
5012:10.1007/978-3-319-70694-8_15
4534:10.1007/978-3-642-22792-9_28
4167:Format-preserving encryption
3233:Fully homomorphic encryption
182:Fully homomorphic encryption
18:Fully homomorphic encryption
5525:10.1007/978-3-030-78086-9_1
5240:Shai Halevi; Victor Shoup.
3207:Naccache–Stern cryptosystem
379:(Ring) Learning With Errors
56:(multiplicative) and others
5960:
5174:. In Nyberg, Kaisa (ed.).
4239:10.1007/s40747-022-00756-z
4127:Homomorphic secret sharing
4093:In 2017, researchers from
3212:Damgård–Jurik cryptosystem
1668:and quadratic non-residue
297:In 2010, Marten van Dijk,
273:lattice-based cryptography
5462:EUROCRYPT 2021 (Springer)
5228:EUROCRYPT 2018 (Springer)
5138:10.1007/s00145-016-9229-5
4846:10.1007/s10623-012-9720-4
4753:10.1112/S1461157016000371
4664:Cryptology ePrint Archive
4328:10.1109/SFFCS.1999.814630
4277:Cryptology ePrint Archive
3921:MoMA Lab at NYU Abu Dhabi
3538:Raytheon BBN Technologies
3222:Ishai–Paskin cryptosystem
1082:{\displaystyle (G,q,g,h)}
50:Ring learning with errors
5934:Cryptographic primitives
4177:Private set intersection
3536:, Duality Technologies,
3293:"Homomorphic encryption"
3225:Joye-Libert cryptosystem
648:and encryption exponent
159:homomorphic encryption:
5939:Public-key cryptography
5819:. Microsoft. 2017-07-13
2136:{\displaystyle \oplus }
1115:{\displaystyle h=g^{x}}
1045:, if the public key is
628:public key has modulus
448:has complexity of only
126:public-key cryptography
5929:Homomorphic encryption
4290:Vinod Vaikuntanathan.
4152:Confidential computing
4147:Client-side encryption
3753:EPFL-LDS, Tune Insight
3269:relies excessively on
3186:
2866:
2816:
2741:
2721:
2701:
2669:
2351:
2301:
2233:
2213:
2193:
2173:
2137:
2114:
1820:
1770:
1702:
1682:
1662:
1630:
1277:
1227:
1156:
1136:
1116:
1083:
1039:
1019:
999:
967:
740:
682:
662:
642:
610:
590:
497:
442:
422:
73:Homomorphic encryption
31:Homomorphic encryption
5619:Zama (15 June 2023).
5487:Zama (15 June 2023).
5126:Journal of Cryptology
3949:Alan Turing Institute
3239:arbitrary computation
3187:
2867:
2817:
2742:
2722:
2702:
2685:Paillier cryptosystem
2670:
2352:
2302:
2234:
2214:
2194:
2174:
2138:
2115:
1821:
1771:
1703:
1683:
1663:
1631:
1278:
1228:
1157:
1137:
1117:
1084:
1040:
1020:
1000:
968:
741:
683:
663:
643:
611:
591:
538:Fourth-generation FHE
498:
443:
423:
332:Second-generation FHE
244:Paillier cryptosystem
230:(unbounded number of
65:Functional encryption
5268:Microsoft Research.
4322:. pp. 554–566.
3473:Duality Technologies
2879:
2826:
2751:
2731:
2711:
2691:
2364:
2311:
2243:
2223:
2203:
2199:with a blocksize of
2183:
2163:
2157:Benaloh cryptosystem
2127:
1833:
1780:
1712:
1692:
1672:
1652:
1290:
1237:
1166:
1146:
1126:
1093:
1049:
1029:
1009:
989:
985:, in a cyclic group
983:ElGamal cryptosystem
753:
692:
672:
652:
632:
600:
567:
544:block floating point
514:Third-generation FHE
452:
432:
412:
346:Vinod Vaikuntanathan
277:somewhat homomorphic
264:First-generation FHE
238:Benaloh cryptosystem
222:ElGamal cryptosystem
105:medical data privacy
101:predictive analytics
93:privilege escalation
46:learning with errors
5944:Information privacy
3854:
3369:
3014:
2976:
2499:
2461:
1959:
1927:
845:
830:
32:
5895:American Scientist
5097:"Security of CKKS"
4137:Private biometrics
3852:
3782:variants enabling
3774:Implementation in
3391:CKKS Bootstrapping
3367:
3182:
3180:
3000:
2962:
2862:
2822:, for some random
2812:
2737:
2717:
2697:
2665:
2663:
2485:
2447:
2347:
2307:, for some random
2297:
2229:
2209:
2189:
2169:
2133:
2110:
2108:
1945:
1913:
1816:
1776:, for some random
1766:
1698:
1678:
1658:
1626:
1624:
1273:
1233:, for some random
1223:
1152:
1132:
1112:
1079:
1035:
1015:
995:
963:
961:
831:
816:
736:
678:
658:
638:
606:
586:
493:
438:
418:
257:branching programs
5621:"Concrete Python"
5534:978-3-030-78085-2
5021:978-3-319-70693-1
4544:978-3-642-22791-2
4337:978-0-7695-0409-4
4195:Sellers, Andrew.
4086:
4085:
3850:
3849:
3778:along with their
3354:
3353:
3346:
3328:
2740:{\displaystyle m}
2720:{\displaystyle g}
2700:{\displaystyle n}
2232:{\displaystyle m}
2212:{\displaystyle c}
2192:{\displaystyle g}
2172:{\displaystyle n}
1701:{\displaystyle b}
1681:{\displaystyle x}
1661:{\displaystyle n}
1640:Goldwasser–Micali
1155:{\displaystyle m}
1135:{\displaystyle x}
1038:{\displaystyle g}
1018:{\displaystyle q}
998:{\displaystyle G}
681:{\displaystyle m}
661:{\displaystyle e}
641:{\displaystyle n}
609:{\displaystyle x}
441:{\displaystyle k}
421:{\displaystyle T}
155:homomorphic, and
70:
69:
16:(Redirected from
5951:
5905:
5903:
5902:
5897:. September 2012
5863:
5862:
5855:
5849:
5848:
5846:
5845:
5834:
5828:
5827:
5825:
5824:
5813:
5807:
5806:
5805:
5804:
5791:
5785:
5784:
5782:
5780:
5763:
5757:
5756:
5754:
5752:
5735:
5729:
5728:
5726:
5724:
5707:
5701:
5700:
5694:
5686:
5684:
5682:
5665:
5659:
5658:
5656:
5654:
5637:
5631:
5630:
5616:
5610:
5609:
5607:
5605:
5588:
5582:
5581:
5579:
5577:
5560:
5554:
5553:
5551:
5549:
5514:
5505:
5499:
5498:
5484:
5478:
5471:
5465:
5454:
5448:
5447:
5445:
5443:
5431:"Lattigo v3.0.5"
5426:
5420:
5419:
5417:
5415:
5398:
5392:
5391:
5389:
5387:
5370:
5364:
5363:
5361:
5359:
5345:Crypto Experts.
5342:
5336:
5335:
5333:
5331:
5314:
5308:
5307:
5305:
5303:
5293:
5287:
5286:
5284:
5282:
5270:"Microsoft SEAL"
5265:
5259:
5258:
5256:
5254:
5237:
5231:
5220:
5214:
5213:
5211:
5210:
5194:
5188:
5187:
5173:
5164:
5158:
5157:
5123:
5114:
5108:
5107:
5105:
5103:
5093:
5087:
5086:
5080:
5071:
5065:
5064:
5058:
5049:
5043:
5032:
5026:
5025:
4999:
4990:
4979:
4973:
4972:
4970:
4968:
4957:
4946:
4945:
4943:
4941:
4924:
4911:
4900:
4891:
4881:
4875:
4864:
4858:
4857:
4839:
4815:
4809:
4798:
4792:
4781:
4775:
4764:
4758:
4757:
4755:
4731:
4725:
4714:
4705:
4694:
4688:
4677:
4668:
4667:
4655:
4646:
4635:
4629:
4618:
4612:
4602:
4593:
4592:
4580:
4571:
4570:
4558:
4549:
4548:
4536:
4518:
4512:
4511:
4499:
4493:
4492:
4487:. Archived from
4477:
4471:
4470:
4468:
4455:
4449:
4448:
4436:
4430:
4429:
4417:
4411:
4410:
4408:
4399:
4393:
4382:
4376:
4369:
4363:
4356:
4350:
4349:
4315:
4309:
4302:
4296:
4295:
4287:
4281:
4280:
4268:
4262:
4261:
4251:
4241:
4232:(4): 3759–3786.
4217:
4211:
4210:
4208:
4207:
4192:
4172:Polymorphic code
3855:
3485:
3370:
3349:
3342:
3338:
3335:
3329:
3327:
3286:
3262:
3254:
3191:
3189:
3188:
3183:
3181:
3171:
3170:
3158:
3157:
3145:
3144:
3132:
3128:
3127:
3118:
3117:
3109:
3108:
3099:
3098:
3089:
3088:
3076:
3075:
3074:
3073:
3061:
3060:
3040:
3036:
3035:
3026:
3025:
3013:
3008:
2999:
2998:
2997:
2996:
2975:
2970:
2961:
2960:
2959:
2958:
2931:
2930:
2918:
2917:
2905:
2904:
2892:
2891:
2871:
2869:
2868:
2863:
2821:
2819:
2818:
2813:
2811:
2810:
2801:
2800:
2792:
2791:
2782:
2781:
2760:
2759:
2746:
2744:
2743:
2738:
2726:
2724:
2723:
2718:
2706:
2704:
2703:
2698:
2674:
2672:
2671:
2666:
2664:
2651:
2650:
2642:
2641:
2629:
2628:
2616:
2615:
2603:
2596:
2595:
2587:
2586:
2577:
2576:
2567:
2566:
2554:
2553:
2552:
2551:
2539:
2538:
2518:
2511:
2510:
2498:
2493:
2484:
2483:
2482:
2481:
2460:
2455:
2446:
2445:
2444:
2443:
2416:
2415:
2403:
2402:
2390:
2389:
2377:
2376:
2356:
2354:
2353:
2348:
2306:
2304:
2303:
2298:
2293:
2292:
2284:
2283:
2274:
2273:
2252:
2251:
2238:
2236:
2235:
2230:
2218:
2216:
2215:
2210:
2198:
2196:
2195:
2190:
2178:
2176:
2175:
2170:
2142:
2140:
2139:
2134:
2119:
2117:
2116:
2111:
2109:
2099:
2098:
2086:
2085:
2073:
2072:
2060:
2053:
2052:
2044:
2043:
2034:
2033:
2024:
2023:
2011:
2010:
2009:
2008:
1996:
1995:
1975:
1968:
1967:
1958:
1953:
1944:
1943:
1942:
1941:
1926:
1921:
1912:
1911:
1910:
1909:
1885:
1884:
1872:
1871:
1859:
1858:
1846:
1845:
1825:
1823:
1822:
1817:
1775:
1773:
1772:
1767:
1762:
1761:
1753:
1752:
1743:
1742:
1721:
1720:
1707:
1705:
1704:
1699:
1687:
1685:
1684:
1679:
1667:
1665:
1664:
1659:
1635:
1633:
1632:
1627:
1625:
1615:
1614:
1602:
1601:
1589:
1588:
1576:
1569:
1568:
1567:
1566:
1554:
1553:
1536:
1535:
1523:
1522:
1507:
1506:
1505:
1504:
1492:
1491:
1468:
1461:
1460:
1459:
1458:
1441:
1440:
1428:
1427:
1426:
1425:
1405:
1404:
1403:
1402:
1385:
1384:
1372:
1371:
1370:
1369:
1342:
1341:
1329:
1328:
1316:
1315:
1303:
1302:
1282:
1280:
1279:
1274:
1232:
1230:
1229:
1224:
1219:
1218:
1200:
1199:
1175:
1174:
1161:
1159:
1158:
1153:
1141:
1139:
1138:
1133:
1121:
1119:
1118:
1113:
1111:
1110:
1088:
1086:
1085:
1080:
1044:
1042:
1041:
1036:
1024:
1022:
1021:
1016:
1004:
1002:
1001:
996:
972:
970:
969:
964:
962:
955:
954:
942:
941:
929:
928:
916:
909:
908:
900:
899:
890:
889:
880:
879:
861:
854:
853:
844:
839:
829:
824:
805:
804:
792:
791:
779:
778:
766:
765:
745:
743:
742:
737:
732:
731:
723:
722:
701:
700:
687:
685:
684:
679:
667:
665:
664:
659:
647:
645:
644:
639:
615:
613:
612:
607:
595:
593:
592:
587:
576:
575:
502:
500:
499:
494:
483:
447:
445:
444:
439:
427:
425:
424:
419:
33:
21:
5959:
5958:
5954:
5953:
5952:
5950:
5949:
5948:
5919:
5918:
5900:
5898:
5889:
5871:
5866:
5861:. 8 March 2021.
5857:
5856:
5852:
5843:
5841:
5836:
5835:
5831:
5822:
5820:
5815:
5814:
5810:
5802:
5800:
5793:
5792:
5788:
5778:
5776:
5765:
5764:
5760:
5750:
5748:
5737:
5736:
5732:
5722:
5720:
5709:
5708:
5704:
5687:
5680:
5678:
5667:
5666:
5662:
5652:
5650:
5639:
5638:
5634:
5618:
5617:
5613:
5603:
5601:
5590:
5589:
5585:
5575:
5573:
5562:
5561:
5557:
5547:
5545:
5535:
5512:
5507:
5506:
5502:
5486:
5485:
5481:
5472:
5468:
5455:
5451:
5441:
5439:
5428:
5427:
5423:
5413:
5411:
5400:
5399:
5395:
5385:
5383:
5372:
5371:
5367:
5357:
5355:
5344:
5343:
5339:
5329:
5327:
5316:
5315:
5311:
5301:
5299:
5295:
5294:
5290:
5280:
5278:
5267:
5266:
5262:
5252:
5250:
5239:
5238:
5234:
5221:
5217:
5208:
5206:
5196:
5195:
5191:
5171:
5166:
5165:
5161:
5121:
5116:
5115:
5111:
5101:
5099:
5095:
5094:
5090:
5078:
5073:
5072:
5068:
5056:
5051:
5050:
5046:
5033:
5029:
5022:
5001:
5000:
4993:
4980:
4976:
4966:
4964:
4959:
4958:
4949:
4939:
4937:
4926:
4925:
4914:
4901:
4894:
4882:
4878:
4865:
4861:
4837:10.1.1.294.4088
4817:
4816:
4812:
4799:
4795:
4782:
4778:
4765:
4761:
4733:
4732:
4728:
4715:
4708:
4695:
4691:
4678:
4671:
4657:
4656:
4649:
4636:
4632:
4619:
4615:
4603:
4596:
4582:
4581:
4574:
4560:
4559:
4552:
4545:
4520:
4519:
4515:
4501:
4500:
4496:
4479:
4478:
4474:
4466:
4460:Naccache, David
4458:Levieil, Eric;
4457:
4456:
4452:
4438:
4437:
4433:
4419:
4418:
4414:
4406:
4401:
4400:
4396:
4383:
4379:
4370:
4366:
4357:
4353:
4338:
4317:
4316:
4312:
4303:
4299:
4289:
4288:
4284:
4270:
4269:
4265:
4219:
4218:
4214:
4205:
4203:
4194:
4193:
4189:
4185:
4123:
4091:
4089:Standardization
3853:FHE frameworks
3479:
3350:
3339:
3333:
3330:
3287:
3285:
3279:
3275:primary sources
3263:
3252:
3250:Implementations
3244:cloud computing
3235:
3179:
3178:
3162:
3149:
3130:
3129:
3119:
3100:
3090:
3080:
3065:
3052:
3047:
3038:
3037:
3027:
2988:
2983:
2950:
2945:
2935:
2922:
2896:
2877:
2876:
2824:
2823:
2802:
2783:
2773:
2749:
2748:
2729:
2728:
2709:
2708:
2689:
2688:
2662:
2661:
2633:
2620:
2601:
2600:
2578:
2568:
2558:
2543:
2530:
2525:
2516:
2515:
2473:
2468:
2435:
2430:
2420:
2407:
2381:
2362:
2361:
2309:
2308:
2275:
2265:
2241:
2240:
2221:
2220:
2201:
2200:
2181:
2180:
2161:
2160:
2125:
2124:
2107:
2106:
2090:
2077:
2058:
2057:
2035:
2025:
2015:
2000:
1987:
1982:
1973:
1972:
1933:
1928:
1901:
1896:
1889:
1876:
1850:
1831:
1830:
1778:
1777:
1744:
1734:
1710:
1709:
1690:
1689:
1670:
1669:
1650:
1649:
1623:
1622:
1606:
1593:
1574:
1573:
1558:
1545:
1540:
1527:
1514:
1496:
1483:
1478:
1466:
1465:
1450:
1445:
1432:
1417:
1412:
1394:
1389:
1376:
1361:
1356:
1346:
1333:
1307:
1288:
1287:
1235:
1234:
1210:
1191:
1164:
1163:
1144:
1143:
1124:
1123:
1102:
1091:
1090:
1047:
1046:
1027:
1026:
1025:with generator
1007:
1006:
987:
986:
960:
959:
946:
933:
914:
913:
891:
881:
871:
859:
858:
809:
796:
770:
751:
750:
714:
690:
689:
670:
669:
650:
649:
630:
629:
598:
597:
565:
564:
561:
540:
516:
450:
449:
430:
429:
410:
409:
385:variant of the
364:scale-invariant
338:Zvika Brakerski
334:
266:
208:
199:
114:
61:Related to
28:
23:
22:
15:
12:
11:
5:
5957:
5955:
5947:
5946:
5941:
5936:
5931:
5921:
5920:
5917:
5916:
5912:maintained on
5906:
5887:
5882:
5877:
5870:
5869:External links
5867:
5865:
5864:
5850:
5829:
5808:
5786:
5758:
5730:
5702:
5660:
5632:
5611:
5583:
5565:"Liberate.FHE"
5555:
5533:
5500:
5479:
5466:
5449:
5421:
5393:
5365:
5337:
5309:
5288:
5260:
5232:
5215:
5189:
5159:
5132:(2): 519–549.
5109:
5088:
5066:
5044:
5027:
5020:
4991:
4987:EUROCRYPT 2016
4974:
4947:
4912:
4892:
4876:
4859:
4810:
4793:
4776:
4772:EUROCRYPT 2012
4759:
4746:(1): 255–266.
4726:
4706:
4689:
4679:Z. Brakerski.
4669:
4647:
4630:
4613:
4594:
4572:
4567:Eurocrypt 2013
4550:
4543:
4513:
4508:Eurocrypt 2012
4494:
4491:on 2011-10-07.
4472:
4450:
4445:Eurocrypt 2010
4431:
4426:Eurocrypt 2011
4412:
4402:Craig Gentry.
4394:
4384:Craig Gentry.
4377:
4364:
4351:
4336:
4310:
4297:
4282:
4263:
4212:
4186:
4184:
4181:
4180:
4179:
4174:
4169:
4164:
4159:
4154:
4149:
4144:
4139:
4134:
4129:
4122:
4119:
4090:
4087:
4084:
4083:
4080:
4077:
4074:
4071:
4068:
4065:
4062:
4059:
4055:
4054:
4052:
4049:
4046:
4043:
4040:
4037:
4034:
4031:
4027:
4026:
4024:
4021:
4018:
4015:
4012:
4009:
4006:
4003:
3999:
3998:
3996:
3993:
3990:
3987:
3984:
3981:
3978:
3975:
3971:
3970:
3968:
3965:
3962:
3959:
3956:
3953:
3950:
3947:
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3940:
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3934:
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3928:
3925:
3922:
3919:
3915:
3914:
3909:
3906:
3903:
3900:
3897:
3894:
3891:
3888:
3884:
3883:
3880:
3877:
3874:
3871:
3868:
3865:
3862:
3859:
3848:
3847:
3844:
3841:
3838:
3835:
3832:
3829:
3826:
3823:
3819:
3818:
3813:
3810:
3807:
3804:
3801:
3798:
3795:
3792:
3788:
3787:
3772:
3769:
3766:
3763:
3760:
3757:
3754:
3751:
3747:
3746:
3743:
3740:
3737:
3734:
3731:
3728:
3725:
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3718:
3717:
3714:
3711:
3708:
3705:
3702:
3699:
3696:
3693:
3689:
3688:
3686:
3683:
3680:
3677:
3674:
3671:
3668:
3665:
3661:
3660:
3658:
3655:
3652:
3649:
3646:
3643:
3640:
3637:
3633:
3632:
3630:
3627:
3624:
3621:
3618:
3615:
3612:
3609:
3605:
3604:
3602:
3599:
3596:
3593:
3590:
3587:
3584:
3581:
3575:
3574:
3567:
3564:
3561:
3558:
3555:
3552:
3549:
3531:
3525:
3524:
3517:
3514:
3511:
3508:
3505:
3502:
3499:
3470:
3464:
3463:
3461:
3458:
3455:
3452:
3449:
3446:
3443:
3438:
3436:Microsoft SEAL
3432:
3431:
3428:
3425:
3422:
3419:
3416:
3413:
3410:
3405:
3399:
3398:
3395:
3392:
3389:
3386:
3383:
3380:
3377:
3374:
3368:FHE libraries
3352:
3351:
3266:
3264:
3257:
3251:
3248:
3240:
3234:
3231:
3230:
3229:
3226:
3223:
3220:
3217:
3214:
3209:
3204:
3193:
3192:
3177:
3174:
3169:
3165:
3161:
3156:
3152:
3148:
3143:
3138:
3135:
3133:
3131:
3126:
3122:
3114:
3107:
3103:
3097:
3093:
3087:
3083:
3079:
3072:
3068:
3064:
3059:
3055:
3050:
3046:
3043:
3041:
3039:
3034:
3030:
3022:
3017:
3012:
3007:
3003:
2995:
2991:
2986:
2982:
2979:
2974:
2969:
2965:
2957:
2953:
2948:
2944:
2941:
2938:
2936:
2934:
2929:
2925:
2921:
2916:
2911:
2908:
2903:
2899:
2895:
2890:
2885:
2884:
2861:
2858:
2855:
2852:
2849:
2846:
2843:
2840:
2837:
2834:
2831:
2809:
2805:
2797:
2790:
2786:
2780:
2776:
2772:
2769:
2766:
2763:
2758:
2736:
2716:
2696:
2676:
2675:
2660:
2657:
2654:
2647:
2640:
2636:
2632:
2627:
2623:
2619:
2614:
2609:
2606:
2604:
2602:
2599:
2592:
2585:
2581:
2575:
2571:
2565:
2561:
2557:
2550:
2546:
2542:
2537:
2533:
2528:
2524:
2521:
2519:
2517:
2514:
2507:
2502:
2497:
2492:
2488:
2480:
2476:
2471:
2467:
2464:
2459:
2454:
2450:
2442:
2438:
2433:
2429:
2426:
2423:
2421:
2419:
2414:
2410:
2406:
2401:
2396:
2393:
2388:
2384:
2380:
2375:
2370:
2369:
2346:
2343:
2340:
2337:
2334:
2331:
2328:
2325:
2322:
2319:
2316:
2296:
2289:
2282:
2278:
2272:
2268:
2264:
2261:
2258:
2255:
2250:
2228:
2208:
2188:
2168:
2132:
2121:
2120:
2105:
2102:
2097:
2093:
2089:
2084:
2080:
2076:
2071:
2066:
2063:
2061:
2059:
2056:
2049:
2042:
2038:
2032:
2028:
2022:
2018:
2014:
2007:
2003:
1999:
1994:
1990:
1985:
1981:
1978:
1976:
1974:
1971:
1964:
1957:
1952:
1948:
1940:
1936:
1931:
1925:
1920:
1916:
1908:
1904:
1899:
1895:
1892:
1890:
1888:
1883:
1879:
1875:
1870:
1865:
1862:
1857:
1853:
1849:
1844:
1839:
1838:
1815:
1812:
1809:
1806:
1803:
1800:
1797:
1794:
1791:
1788:
1785:
1765:
1758:
1751:
1747:
1741:
1737:
1733:
1730:
1727:
1724:
1719:
1697:
1677:
1657:
1637:
1636:
1621:
1618:
1613:
1609:
1605:
1600:
1596:
1592:
1587:
1582:
1579:
1577:
1575:
1572:
1565:
1561:
1557:
1552:
1548:
1543:
1539:
1534:
1530:
1526:
1521:
1517:
1513:
1510:
1503:
1499:
1495:
1490:
1486:
1481:
1477:
1474:
1471:
1469:
1467:
1464:
1457:
1453:
1448:
1444:
1439:
1435:
1431:
1424:
1420:
1415:
1411:
1408:
1401:
1397:
1392:
1388:
1383:
1379:
1375:
1368:
1364:
1359:
1355:
1352:
1349:
1347:
1345:
1340:
1336:
1332:
1327:
1322:
1319:
1314:
1310:
1306:
1301:
1296:
1295:
1272:
1269:
1266:
1263:
1260:
1257:
1254:
1251:
1248:
1245:
1242:
1222:
1217:
1213:
1209:
1206:
1203:
1198:
1194:
1190:
1187:
1184:
1181:
1178:
1173:
1151:
1131:
1109:
1105:
1101:
1098:
1078:
1075:
1072:
1069:
1066:
1063:
1060:
1057:
1054:
1034:
1014:
994:
974:
973:
958:
953:
949:
945:
940:
936:
932:
927:
922:
919:
917:
915:
912:
905:
898:
894:
888:
884:
878:
874:
870:
867:
864:
862:
860:
857:
850:
843:
838:
834:
828:
823:
819:
815:
812:
810:
808:
803:
799:
795:
790:
785:
782:
777:
773:
769:
764:
759:
758:
735:
728:
721:
717:
713:
710:
707:
704:
699:
677:
657:
637:
605:
585:
582:
579:
574:
560:
557:
539:
536:
515:
512:
492:
489:
486:
482:
479:
476:
473:
470:
467:
464:
460:
457:
437:
417:
375:
374:
367:
365:
360:
353:
333:
330:
326:David Naccache
321:Cohen's method
307:ideal lattices
292:ideal lattices
285:bootstrappable
265:
262:
261:
260:
253:
250:
247:
241:
235:
225:
219:
207:
204:
198:
195:
186:
185:
179:
173:
167:
113:
110:
68:
67:
62:
58:
57:
42:
38:
37:
26:
24:
14:
13:
10:
9:
6:
4:
3:
2:
5956:
5945:
5942:
5940:
5937:
5935:
5932:
5930:
5927:
5926:
5924:
5915:
5911:
5907:
5896:
5892:
5888:
5886:
5883:
5881:
5878:
5876:
5873:
5872:
5868:
5860:
5854:
5851:
5839:
5833:
5830:
5818:
5812:
5809:
5799:
5798:
5790:
5787:
5775:
5774:
5769:
5762:
5759:
5747:
5746:
5741:
5734:
5731:
5719:
5718:
5713:
5706:
5703:
5698:
5692:
5677:
5676:
5671:
5664:
5661:
5649:
5648:
5643:
5636:
5633:
5628:
5627:
5622:
5615:
5612:
5600:
5599:
5594:
5587:
5584:
5572:
5571:
5566:
5559:
5556:
5544:
5540:
5536:
5530:
5526:
5522:
5518:
5511:
5504:
5501:
5496:
5495:
5490:
5483:
5480:
5476:
5470:
5467:
5463:
5459:
5453:
5450:
5438:
5437:
5432:
5425:
5422:
5410:
5409:
5404:
5397:
5394:
5382:
5381:
5376:
5369:
5366:
5354:
5353:
5348:
5341:
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4707:
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4025:
4022:
4019:
4016:
4013:
4010:
4007:
4004:
4001:
4000:
3997:
3994:
3991:
3988:
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3827:
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3811:
3808:
3805:
3802:
3799:
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3790:
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3785:
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3700:
3697:
3694:
3691:
3690:
3687:
3684:
3681:
3678:
3675:
3672:
3669:
3667:CryptoExperts
3666:
3663:
3662:
3659:
3656:
3653:
3650:
3647:
3644:
3641:
3638:
3635:
3634:
3631:
3628:
3625:
3622:
3619:
3616:
3613:
3610:
3607:
3606:
3603:
3600:
3597:
3594:
3591:
3588:
3585:
3582:
3580:
3577:
3576:
3572:
3568:
3565:
3562:
3559:
3556:
3553:
3550:
3547:
3543:
3539:
3535:
3532:
3530:
3527:
3526:
3522:
3519:Successor to
3518:
3515:
3512:
3509:
3506:
3503:
3500:
3497:
3493:
3489:
3483:
3478:
3474:
3471:
3469:
3466:
3465:
3462:
3459:
3456:
3453:
3450:
3447:
3444:
3442:
3439:
3437:
3434:
3433:
3429:
3426:
3423:
3420:
3417:
3414:
3411:
3409:
3406:
3404:
3401:
3400:
3396:
3393:
3390:
3387:
3384:
3381:
3378:
3375:
3372:
3371:
3365:
3363:
3357:
3348:
3345:
3337:
3326:
3323:
3319:
3316:
3312:
3309:
3305:
3302:
3298:
3295: –
3294:
3290:
3289:Find sources:
3283:
3277:
3276:
3272:
3267:This section
3265:
3261:
3256:
3255:
3249:
3247:
3245:
3238:
3232:
3227:
3224:
3221:
3218:
3215:
3213:
3210:
3208:
3205:
3203:
3200:
3199:
3198:
3197:
3175:
3167:
3163:
3159:
3154:
3150:
3136:
3134:
3124:
3120:
3105:
3095:
3091:
3085:
3081:
3070:
3066:
3062:
3057:
3053:
3048:
3044:
3042:
3032:
3028:
3010:
3005:
3001:
2993:
2989:
2984:
2972:
2967:
2963:
2955:
2951:
2946:
2939:
2937:
2927:
2923:
2909:
2901:
2897:
2875:
2874:
2873:
2856:
2853:
2850:
2847:
2844:
2841:
2838:
2832:
2829:
2807:
2803:
2788:
2784:
2778:
2774:
2770:
2764:
2734:
2714:
2707:and the base
2694:
2686:
2681:
2680:
2658:
2652:
2638:
2634:
2630:
2625:
2621:
2607:
2605:
2597:
2583:
2573:
2569:
2563:
2559:
2548:
2544:
2540:
2535:
2531:
2526:
2522:
2520:
2512:
2495:
2490:
2486:
2478:
2474:
2469:
2457:
2452:
2448:
2440:
2436:
2431:
2424:
2422:
2412:
2408:
2394:
2386:
2382:
2360:
2359:
2358:
2341:
2338:
2335:
2332:
2329:
2326:
2323:
2317:
2314:
2294:
2280:
2276:
2270:
2266:
2262:
2256:
2226:
2206:
2186:
2179:and the base
2166:
2158:
2153:
2152:
2148:
2146:
2130:
2103:
2095:
2091:
2087:
2082:
2078:
2064:
2062:
2054:
2040:
2030:
2026:
2020:
2016:
2005:
2001:
1997:
1992:
1988:
1983:
1979:
1977:
1969:
1955:
1950:
1946:
1938:
1934:
1929:
1923:
1918:
1914:
1906:
1902:
1897:
1893:
1891:
1881:
1877:
1863:
1855:
1851:
1829:
1828:
1827:
1810:
1807:
1804:
1801:
1798:
1795:
1792:
1786:
1783:
1763:
1749:
1745:
1739:
1735:
1731:
1725:
1695:
1675:
1655:
1647:
1642:
1641:
1619:
1611:
1607:
1603:
1598:
1594:
1580:
1578:
1563:
1559:
1555:
1550:
1546:
1541:
1532:
1528:
1524:
1519:
1515:
1508:
1501:
1497:
1493:
1488:
1484:
1479:
1472:
1470:
1455:
1451:
1446:
1442:
1437:
1433:
1429:
1422:
1418:
1413:
1399:
1395:
1390:
1386:
1381:
1377:
1373:
1366:
1362:
1357:
1350:
1348:
1338:
1334:
1320:
1312:
1308:
1286:
1285:
1284:
1267:
1264:
1261:
1258:
1255:
1252:
1249:
1243:
1240:
1215:
1211:
1207:
1204:
1201:
1196:
1192:
1185:
1179:
1149:
1129:
1107:
1103:
1099:
1096:
1073:
1070:
1067:
1064:
1061:
1058:
1055:
1032:
1012:
992:
984:
979:
978:
951:
947:
943:
938:
934:
920:
918:
910:
896:
886:
882:
876:
872:
865:
863:
855:
841:
836:
832:
826:
821:
817:
813:
811:
801:
797:
783:
775:
771:
749:
748:
747:
733:
719:
715:
711:
705:
675:
655:
635:
627:
622:
621:
617:
603:
580:
558:
556:
552:
550:
545:
537:
535:
531:
529:
525:
521:
513:
511:
508:
506:
487:
458:
455:
435:
415:
407:
403:
399:
394:
390:
388:
384:
383:overstretched
380:
372:
368:
366:cryptosystem;
363:
361:
358:
354:
351:
350:
349:
347:
343:
339:
331:
329:
327:
322:
318:
316:
312:
308:
304:
300:
295:
293:
288:
286:
281:
278:
274:
270:
263:
258:
254:
251:
248:
245:
242:
239:
236:
233:
229:
226:
223:
220:
217:
214:
213:
212:
205:
203:
196:
194:
192:
183:
180:
177:
174:
171:
168:
165:
162:
161:
160:
158:
154:
151:
148:homomorphic,
147:
144:homomorphic,
143:
137:
135:
131:
127:
123:
119:
111:
109:
106:
102:
96:
94:
89:
87:
83:
78:
75:is a form of
74:
66:
63:
59:
55:
51:
47:
43:
39:
34:
19:
5899:. Retrieved
5894:
5853:
5842:. Retrieved
5832:
5821:. Retrieved
5811:
5801:, retrieved
5796:
5789:
5777:. Retrieved
5771:
5761:
5749:. Retrieved
5743:
5733:
5721:. Retrieved
5715:
5705:
5679:. Retrieved
5673:
5663:
5651:. Retrieved
5645:
5635:
5624:
5614:
5602:. Retrieved
5596:
5586:
5574:. Retrieved
5568:
5558:
5546:. Retrieved
5516:
5503:
5492:
5482:
5469:
5461:
5452:
5442:13 September
5440:. Retrieved
5434:
5424:
5412:. Retrieved
5406:
5396:
5384:. Retrieved
5378:
5368:
5356:. Retrieved
5350:
5340:
5328:. Retrieved
5322:
5312:
5300:. Retrieved
5291:
5279:. Retrieved
5273:
5263:
5251:. Retrieved
5245:
5235:
5227:
5218:
5207:. Retrieved
5205:. p. 96
5192:
5175:
5162:
5129:
5125:
5112:
5100:. Retrieved
5091:
5082:
5069:
5060:
5047:
5039:
5030:
5003:
4986:
4977:
4965:. Retrieved
4938:. Retrieved
4932:
4907:
4888:
4879:
4871:
4862:
4830:(1): 57–81.
4827:
4823:
4813:
4805:
4796:
4788:
4779:
4771:
4762:
4743:
4739:
4729:
4721:
4701:
4692:
4684:
4663:
4642:
4633:
4625:
4616:
4609:
4588:
4566:
4523:
4516:
4507:
4497:
4489:the original
4475:
4453:
4444:
4434:
4425:
4415:
4397:
4389:
4380:
4372:
4367:
4359:
4354:
4319:
4313:
4305:
4300:
4285:
4276:
4266:
4229:
4225:
4215:
4204:. Retrieved
4200:
4190:
4092:
3911:
3882:Description
3822:Liberate.FHE
3815:
3397:Description
3358:
3355:
3340:
3331:
3321:
3314:
3307:
3300:
3288:
3268:
3236:
3195:
3194:
2682:
2678:
2677:
2154:
2150:
2149:
2145:exclusive-or
2122:
1643:
1639:
1638:
980:
976:
975:
688:is given by
623:
620:Unpadded RSA
619:
618:
562:
553:
541:
532:
528:Brent Waters
520:Craig Gentry
517:
509:
398:Craig Gentry
395:
391:
382:
376:
342:Craig Gentry
335:
319:
299:Craig Gentry
296:
289:
284:
282:
276:
269:Craig Gentry
267:
232:exclusive or
209:
200:
187:
181:
175:
169:
163:
156:
152:
149:
145:
141:
138:
134:homomorphism
129:
115:
97:
90:
72:
71:
41:Derived from
5548:17 November
5347:"FV-NFLlib"
5281:20 February
5253:31 December
5146:2117/103661
5040:CT-RSA 2022
4967:31 December
4940:31 December
4908:CRYPTO 2014
4872:CRYPTO 2013
4806:CRYPTO 2012
4722:CRYPTO 2016
4685:CRYPTO 2012
4481:Cohen, Bram
3780:distributed
3548:and others.
3498:and others.
3480: [
406:Nigel Smart
402:Shai Halevi
303:Shai Halevi
234:operations)
130:Homomorphic
112:Description
86:computation
5923:Categories
5901:2018-05-08
5844:2022-05-12
5823:2022-05-12
5803:2024-07-18
5723:3 February
5681:1 November
5593:"Concrete"
5429:EPFL-LDS.
5386:1 November
5373:NuCypher.
5358:1 November
5209:2010-03-17
5042:(Springer)
4989:(Springer)
4910:(Springer)
4874:(Springer)
4808:(Springer)
4791:(SpringeR)
4774:(Springer)
4724:(Springer)
4704:(Springer)
4702:IMACC 2013
4687:(Springer)
4206:2023-08-18
4183:References
4061:TwC Group
3304:newspapers
3271:references
524:Amit Sahai
315:Bram Cohen
132:refers to
122:secret key
118:encryption
77:encryption
5543:231732347
5489:"TFHE-rs"
5302:1 January
5275:Microsoft
4889:ITCS 2014
4832:CiteSeerX
4643:STOC 2012
4626:FOCS 2011
4610:ITCS 2012
4099:Microsoft
4033:TwC Group
4005:TwC Group
3977:TwC Group
3861:Developer
3724:TwC Group
3664:FV-NFLlib
3583:CryptoLab
3441:Microsoft
3376:Developer
3334:July 2022
2910:⋅
2854:−
2845:…
2833:∈
2395:⋅
2339:−
2330:…
2318:∈
2131:⊕
2088:⊕
1864:⋅
1808:−
1799:…
1787:∈
1604:⋅
1525:⋅
1443:⋅
1387:⋅
1321:⋅
1265:−
1256:…
1244:∈
1208:⋅
1005:of order
944:⋅
784:⋅
518:In 2013,
459:⋅
317:in 1998.
191:malleable
142:partially
5691:cite web
5563:Desilo.
5102:10 March
4854:11202438
4789:PKC 2012
4589:PKC 2014
4258:35531323
4121:See also
3887:Concrete
3876:PALISADE
3721:REDcuFHE
3695:NuCypher
3529:PALISADE
3521:PALISADE
2679:Paillier
1089:, where
311:Naccache
271:, using
146:somewhat
52:or even
5779:25 June
5751:29 July
5653:27 July
5576:7 March
5414:7 March
4392:, 2009.
4375:, 2007.
4362:, 2005.
4346:1976588
4308:, 1978.
4249:9062639
3879:Lattigo
3791:TFHE-rs
3750:Lattigo
3571:OpenFHE
3468:OpenFHE
3318:scholar
2683:In the
2155:In the
2151:Benaloh
1644:In the
981:In the
977:ElGamal
624:If the
206:Pre-FHE
197:History
150:leveled
82:storage
36:General
5914:GitHub
5773:GitHub
5745:GitHub
5717:GitHub
5675:GitHub
5647:GitHub
5604:20 May
5598:GitHub
5591:Zama.
5570:GitHub
5541:
5531:
5494:GitHub
5436:GitHub
5408:GitHub
5380:GitHub
5352:GitHub
5330:15 May
5324:GitHub
5247:GitHub
5152:
5018:
4934:GitHub
4852:
4834:
4628:(IEEE)
4541:
4344:
4334:
4256:
4246:
4201:Forbes
4105:, the
4030:Juliet
3825:Desilo
3320:
3313:
3306:
3299:
3291:
2123:where
1122:, and
526:, and
404:, and
5539:S2CID
5513:(PDF)
5460:. In
5226:. In
5172:(PDF)
5154:62063
5150:S2CID
5122:(PDF)
5079:(PDF)
5057:(PDF)
5038:, In
4985:. In
4906:. In
4887:. In
4870:. In
4850:S2CID
4804:. In
4787:. In
4770:. In
4720:, In
4700:. In
4683:, In
4645:(ACM)
4641:. In
4624:. In
4608:, In
4467:(PDF)
4407:(PDF)
4388:. In
4342:S2CID
4103:Intel
4058:PEEV
3946:SHEEP
3870:HElib
3692:NuFHE
3579:HEAAN
3488:Intel
3484:]
3403:HElib
3325:JSTOR
3311:books
157:fully
153:fully
5781:2024
5753:2024
5725:2023
5697:link
5683:2019
5655:2019
5626:Pypi
5606:2022
5578:2024
5550:2022
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