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Its name is derived from the four dimensional
Gallant–Lambert–Vanstone scalar multiplication, which allows high performance calculations. The curve is defined over a two dimensional
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Both properties (four dimensional decomposition and
Mersenne prime characteristic), alongside usage of fast multiplication formulae (
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1154:{\displaystyle k=a_{1}+a_{2}\lambda _{\phi }+a_{3}\lambda _{\psi }+a_{4}\lambda _{\phi }\lambda _{\psi }{\pmod {N}}}
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2001:
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van
Oorschot, Paul C.; Wiener, Michael J. (1996). "On Diffie-Hellman Key Agreement with Short Exponents".
705:, all group operations are done in constant time, i.e. without disclosing information about key material.
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1980:
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1741:. Lecture Notes in Computer Science. Vol. 1070. Springer Berlin Heidelberg. pp. 332–343.
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Babai, L. (1 March 1986). "On Lovász' lattice reduction and the nearest lattice point problem".
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coordinates), make FourQ the currently fastest elliptic curve for the 128 bit security level.
217:
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173:
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1950:
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group operations on average. Therefore, it typically belongs to the 128 bit security level.
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Please expand the section to include this information. Further details may exist on the
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1409:. For FourQ it turns that one can guarantee an efficiently computable solution with
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1569:"FourQ: four-dimensional decompositions on a Q-curve over the Mersenne prime"
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The curve was published in 2015 by Craig
Costello and Patrick Longa from
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80:
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786:, which is usually thought as distributed uniformly at random over
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241:
137:
133:
125:
113:
1660:"Four-Dimensional Gallant–Lambert–Vanstone Scalar Multiplication"
244:
and is available for x86, x64, and ARM. It is licensed under the
2662:
2492:
2217:
2117:
338:
337:
There were some efforts to standardize usage of the curve under
1853:
1711:
Ladd, Watson; Longa, Patrick; Barnes, Richard (27 March 2017).
2559:
2362:
1474:
717:, require fast computation of scalar multiplication, i.e.
646:, a low degree efficiently computable endomorphism (see
1415:
1388:
1369:{\displaystyle P=P+\phi (P)+\psi (P)+\phi (\psi (P))}
1229:
1197:
1170:
1039:
1016:
989:
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772:
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361:
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2010:
1994:
1918:
1911:
1890:
168:
156:
143:
119:
103:
93:
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57:
35:
23:
1515:FourQ is implemented in the cryptographic library
1441:
1401:
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1153:
1022:
995:
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924:
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638:
614:
594:
536:
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480:
444:
422:
291:
874:{\displaystyle \lambda _{\psi },\lambda _{\phi }}
713:Most cryptographic primitives, and most notably
228:made by the authors of the original paper. The
1865:
341:; these efforts were withdrawn in late 2017.
212:. It is designed for key agreements schemes (
8:
817:
793:
18:
1915:
1872:
1858:
1850:
1460:, modulations can be carried efficiently.
1220:quickly by utilizing the implied equation
578:The curve is equipped with two nontrivial
423:{\displaystyle -x^{2}+y^{2}=1+dx^{2}y^{2}}
17:
1746:
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271:
1658:Longa, Patrick; Sica, Francesco (2011).
1567:Costello, Craig; Longa, Patrick (2015).
1554:
551:, all points are verified to lie in an
1739:Advances in Cryptology — EUROCRYPT '96
1693:
1682:
1592:
1581:
1845:Reference implementation by Microsoft
1608:
1606:
7:
2754:Microsoft Open Specification Promise
2463:Text Template Transformation Toolkit
1884:free and open-source software (FOSS)
1562:
1560:
1558:
481:{\displaystyle \mathbb {F} _{p^{2}}}
248:and the source code is available on
1143:
326:, New Zealand, and consequently a
14:
823:{\displaystyle \{0,\ldots ,N-1\}}
2780:
2779:
2673:TPM 2.0 Reference Implementation
2508:Windows Communication Foundation
2283:Microsoft Automatic Graph Layout
1479:
1382:technique is used to find small
841:subgroup, one can write scalars
2764:Open Source Security Foundation
2535:Windows Presentation Foundation
2278:Managed Extensibility Framework
1442:{\displaystyle a_{i}<2^{64}}
1136:
2814:Software using the MIT license
2418:Python Tools for Visual Studio
2378:Open Management Infrastructure
2288:Microsoft C++ Standard Library
1363:
1360:
1354:
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1311:
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954:
948:
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903:
897:
891:
730:
724:
1:
214:elliptic-curve Diffie–Hellman
2705:Microsoft Reciprocal License
2383:Open Neural Network Exchange
2308:Microsoft Enterprise Library
766:on the curve and an integer
2804:Elliptic curve cryptography
2483:Virtual File System for Git
2358:Neural Network Intelligence
2293:Microsoft Cognitive Toolkit
1533:Elliptic-curve cryptography
318:The paper was presented in
2830:
976:{\displaystyle \phi (P)=P}
925:{\displaystyle \psi (P)=P}
567:is specified as a 246-bit
349:The curve is defined by a
216:) and digital signatures (
41:; 9 years ago
2777:
2633:Extensible Storage Engine
2513:Windows Driver Frameworks
2423:R Tools for Visual Studio
2298:Microsoft Design Language
1898:Microsoft and open source
691:{\displaystyle 2^{122.5}}
663:The currently best known
537:{\displaystyle 2^{127}-1}
292:{\displaystyle 2^{127}-1}
232:implementation is called
69:
53:
2759:Open Letter to Hobbyists
2700:Microsoft Public License
2598:(v1.25, v2.0 & v4.0)
2540:Windows Template Library
2238:Dynamic Language Runtime
1903:Shared Source Initiative
1748:10.1007/3-540-68339-9_29
1465:extended twisted Edwards
654:Cryptographic properties
351:twisted Edwards equation
328:reference implementation
226:reference implementation
224:. It is equipped with a
220:), and offers about 128
2809:Microsoft free software
1976:Windows Package Manager
1713:"draft-ladd-cfrg-4q-01"
669:Pollard's rho algorithm
345:Mathematical properties
2749:F# Software Foundation
1692:Cite journal requires
1591:Cite journal requires
1490:is missing information
1443:
1403:
1370:
1214:
1185:
1155:
1024:
997:
977:
926:
875:
824:
780:
760:
740:
692:
667:attack is the generic
648:complex multiplication
640:
616:
596:
549:small subgroup attacks
538:
502:
482:
446:
424:
293:
2769:Outercurve Foundation
2323:Mixed Reality Toolkit
1444:
1404:
1402:{\displaystyle a_{i}}
1371:
1215:
1186:
1184:{\displaystyle a_{i}}
1156:
1025:
998:
978:
927:
876:
825:
781:
761:
741:
693:
641:
639:{\displaystyle \phi }
617:
597:
595:{\displaystyle \psi }
539:
503:
483:
447:
425:
294:
263:field defined by the
152:cryptographic library
2258:Fluent Design System
2138:.NET Micro Framework
1413:
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1195:
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987:
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790:
770:
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721:
701:In order to prevent
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515:
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456:
436:
359:
270:
1981:WorldWide Telescope
1956:Microsoft PowerToys
1820:blog.cloudflare.com
1816:"Introducing CIRCL"
1010:Hence, for a given
1007:-torsion subgroup.
833:Since we look at a
452:is a non-square in
20:
2545:Windows UI Library
2458:T2 Temporal Prover
2428:RecursiveExtractor
2090:Small Basic Online
1966:Windows Calculator
1786:10.1007/BF02579403
1618:Microsoft Research
1456:of the field is a
1439:
1399:
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1210:
1181:
1151:
1020:
993:
973:
922:
871:
820:
776:
756:
736:
688:
671:, requiring about
665:discrete logarithm
636:
612:
592:
547:In order to avoid
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498:
478:
442:
420:
309:Microsoft Research
289:
210:Microsoft Research
30:Microsoft Research
2791:
2790:
2686:
2685:
2589:Operating systems
2580:Z3 Theorem Prover
2388:Open Service Mesh
2303:Microsoft Detours
2173:ASP.NET Web Forms
2111:development tools
1758:978-3-540-61186-8
1647:. 4 October 2021.
1513:
1512:
1452:Moreover, as the
1213:{\displaystyle P}
1191:, we may compute
1164:If we find small
1023:{\displaystyle k}
996:{\displaystyle P}
779:{\displaystyle k}
759:{\displaystyle P}
739:{\displaystyle P}
615:{\displaystyle p}
501:{\displaystyle p}
445:{\displaystyle d}
330:was published on
194:
193:
2821:
2783:
2782:
2253:Entity Framework
1951:Open Live Writer
1916:
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1717:Ietf Datatracker
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983:for every point
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559:subgroup of the
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222:bits of security
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105:Operating system
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2744:.NET Foundation
2732:
2709:
2682:
2616:
2584:
2565:XDP for Windows
2503:Windows App SDK
2263:Fluid Framework
2243:eBPF on Windows
2110:
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2012:
2006:
1990:
1971:Windows Console
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1519:, published by
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604:
603:
602:related to the
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36:Initial release
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2313:Microsoft SEAL
2310:
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2128:.NET Gadgeteer
2125:
2123:.NET Framework
2120:
2114:
2112:
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2080:Project Verona
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2014:
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1926:3D Movie Maker
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1839:External links
1837:
1834:
1833:
1822:. 20 June 2019
1807:
1764:
1757:
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1703:
1694:|journal=
1650:
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1593:|journal=
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1469:
1458:Mersenne prime
1454:characteristic
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1396:
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1380:Babai rounding
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703:timing attacks
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2018:
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2015:
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1774:Combinatorica
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1488:This section
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1033:
1032:
1031:
1030:we may write
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1008:
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990:
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683:
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658:
653:
651:
649:
633:
625:
624:Frobenius map
609:
589:
581:
580:endomorphisms
576:
574:
571:dividing the
570:
566:
562:
558:
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550:
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208:developed by
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2468:TLA+ Toolbox
2393:Open XML SDK
2213:CLR Profiler
2158:ASP.NET Core
2153:ASP.NET AJAX
2100:Visual Basic
1946:File Manager
1919:Applications
1824:. Retrieved
1819:
1810:
1777:
1773:
1767:
1738:
1732:
1720:. Retrieved
1716:
1706:
1685:cite journal
1673:. Retrieved
1653:
1642:
1639:"References"
1633:
1621:. Retrieved
1617:
1584:cite journal
1572:. Retrieved
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712:
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336:
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317:
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254:
236:and runs on
233:
201:
198:cryptography
195:
25:Developer(s)
15:
2612:Azure Linux
2343:MonoDevelop
2163:ASP.NET MVC
2109:Frameworks,
2011:Programming
1995:Video games
1986:XML Notepad
1941:Family.Show
1780:(1): 1–13.
1492:about uses.
322:in 2015 in
246:MIT License
230:open source
163:MIT License
2798:Categories
2678:WikiBhasha
2653:Project Mu
2628:ChronoZoom
2602:Barrelfish
2438:Sandcastle
2403:Playwright
2208:ChakraCore
2183:Babylon.js
2095:TypeScript
2075:PowerShell
2050:IronPython
2002:Allegiance
1614:"FourQlib"
1549:References
1538:Curve25519
1521:Cloudflare
881:such that
709:Efficiency
176:.microsoft
110:Windows 10
94:Written in
85:/microsoft
75:Repository
2658:ReactiveX
2575:xUnit.net
2488:Voldemort
2453:SVNBridge
2268:Infer.NET
2223:DeepSpeed
2198:C++/WinRT
2188:BitFunnel
2133:.NET MAUI
2013:languages
1881:Microsoft
1794:1439-6912
1669:1106.5149
1504:July 2019
1496:talk page
1352:ψ
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332:Microsoft
320:Asiacrypt
284:−
257:extension
186:/fourqlib
182:/research
87:/FourQlib
2785:Category
2723:CodePlex
2692:Licenses
2638:FlexWiki
2448:StyleCop
2408:ProcDump
2333:mod_mono
2318:mimalloc
2273:LightGBM
2248:Electron
2178:Avalonia
2070:Power Fx
2055:IronRuby
2045:GW-BASIC
1961:Terminal
1912:Software
1891:Overview
1543:Curve448
1527:See also
659:Security
563:, where
488:, where
324:Auckland
234:FourQlib
184:/project
121:Platform
64:v3.1
2737:Related
2555:WinObjC
2473:U-Prove
2443:SignalR
2413:ProcMon
2398:Orleans
2373:OneFuzz
2348:MSBuild
2228:DiskSpd
2148:ASP.NET
1826:28 July
1802:7914792
1003:in the
626:, and
622:-power
557:torsion
508:is the
303:History
259:of the
238:Windows
218:Schnorr
169:Website
158:License
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2728:GitHub
2715:Forges
2648:Gollum
2596:MS-DOS
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2353:MsQuic
2328:ML.NET
2193:Blazor
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2020:Bosque
1800:
1792:
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1722:23 May
1675:23 May
1644:GitHub
1623:23 May
1574:23 May
839:cyclic
837:order
313:ePrint
250:GitHub
204:is an
180:/en-us
130:x86-64
81:github
2668:TLAPS
2643:FourQ
2621:Other
2607:SONiC
2550:WinJS
2478:vcpkg
2368:NuGet
2233:Dryad
2030:Dafny
1798:S2CID
1664:arXiv
1517:CIRCL
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569:prime
261:prime
242:Linux
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126:IA-32
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2118:.NET
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1931:Atom
1828:2019
1790:ISSN
1753:ISBN
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1471:Uses
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