密码学中的计算假设

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介绍了密码学中常用的一些数学假设,及其对应的密码算法
Main Computational assumptions in Cryptography Editor Fre Vercauteren(K U. Leuven) Contributors donll Benger(Shannon Institute) Dario Catalano(UNICT), Manuel Charlemagne( Shannon Institute) David Conti(Shannon Institute), Biljana Cubaleska(RUB Hernando Fernando(Shannon Institute), Dario Fiore(UNICT Steven Galbraith(RHUL), David Galindo(Uni. Lu Jens Hermans(K U. Leuven), Vincenzo lovino(UNISA) Tibor Jager(RUB), Markulf Kohlweiss(KU. Leuven) Benoit Libert(UCL), Richard Lindner( TUD) Hans Loehr(RUB), Danny Lynch(Shannon Institute) Richard Moloney(Shannon Institute), Khaled Ouafi(EPFL Benny Pinkas(University of Haifa), Frantisek Polach(Shannon Institute) Mario Di Raimondo(UNit), Markus Ruckert TUD) Michael Schneider(TUD), Vijay Singh(Shannon Institute) Nigel Smart(UNIVBRIS), Martijn Stam(EPFL Fre Vercauteren(K U.Leuven), Jorge Villar Santos(UPC) Steve WilliamS (UNIVBRIS) 9 april 2010 Revision 1.0 The work described in this report has in part been supported by the CoInnmissioll of the European CoIl munities through the ICt program under contract ICT-2007-216G76. The information in this document is provided as is, and no warranty is given or implied that the information is fit for any particular purpose. The user thereof uses the information at its sole risk and liability Contents 1 Introduction 2 Discrete logarithm problem DLP: discrete logarithm problem 333 2. CDH: computational Diffic-Hcllman problcm 3. SDII: static Diffie-Ilellman problem 4 gap-CDH: Gap Diffie-Hellman problem 5. DDH: decision Diffie-Hellman problem 6. Strong-DDIl: strong decision Diffie-Ilellman problem 7. SDDH: skewed decision Diffie-Hellman problem 8. PDDH: parallel decision Diffie-Hellman problem 9. Square-DH: Square Dillie-Hellman problelll 10. l-DHI: -Diffie-Hellman inversion problem 11.l-DDHI: l-Decisional Diffie-Hellman inversion problem 12. REPRESENTATION: Representation problem 5566667 13. LRSW: LRSW Problem 14. Linear: Linear problem 15. D-Linear1: Decision Linear problem(version 1) 16. -SDH: l-Strong Diffie-Hellman problem 17. C-DLSE: Discrete Logarithm with Short Exponents 18. CONF: (conference-key sharing scheme 19. 3PASS: 3-Pass Message Transmission Scheme 20. LUCAS: Lucas Problern 21.XLP: x-Logarithm Problcm 22. MDIIP: Matching Diffie-llellman Problem 23 DDLP: Double Discrete Logarithm Problem 10 24. rootDLP: Root of discrctc logarithm problem 25. n-M-DDII: Multiple Decision Diffie-IIellman Problem 26. l-HENSEL-DLP: -Hensel Discrete Logarithm Problem 27. DLP(Inn(G)): Discrete Logarithm Problem over Inner Automorphism Group 28. IE: Inverse Exponent 12 29. TDH: The Twin Diffie-Hellman Assumption 30. XTR-DL: XTR discrete logarithm problem 13 31. XTR-DH: XTR Dillie-Hellian proble 13 32. XTR-DHD: XTR decision Diffie- Hellman problem 13 33. CL-DLP: discrete logarithms in class groups of imaginary quadratic orders 13 34. TV-DDH: Tzeng Variant Decision Dillie-Hellman probleIn 35. n-DHE: n-Diffic-Hcllman Exponent problem 3 Factoring 15 36. FACTORING: integer factorisation problem 15 37. SQRT: square rools nodulo a conposite 15 38. CHARACTERd: charactor problcm 15 39. MOVA: character problem 16 10. CYCLOFACTd: factorisation in Z[0 16 41. FERMAr: factorisation in Z 0 16 42. RSAP: RSA problem 17 43. Strong-RSAP: strong RSA problem 44. Difference-RSAP: Difference RSA problem 45. Partial-DL-ZN2P: Partial Discrete Logarithm problem in ZK2 18 46. DDH-ZN2P: Decision Diffie-Hellman problem over Zw2 18 47. Lift-DH-ZN2P: Lift Diffie-Hellman problem over ZK2 18 48. EPHP: Election Privacy Homomorphism problem 49. AERP: Approximate c-th root problcm 19 50. -IIENSEL-RSAP. -IIensel RSA 51. DSeRP: Decisional Small e-Residues in Z%2 52. DS2cRP: Decisional Small 2c-Rosiducs in Z%2 20 53. DSmalIRSAKP: Decisional Reciprocal RSA-Paillier in Z2 21 54. HRP: Higher Residuosity Problem 21 55. ECSQRT: Square roots in elliptic curve groups over Z /nZ 56. RFP: Root Finding Problem 57. phiA: PHI-Assumption 22 58. C-DRSA: Computational Dependent-RSA problem 23 59. D-DRSA: Decisional Dependent-RSA problem 60. E-DRSA: Extraction Dependent-RSA problem 23 61. DCR: Decisional Composite Residuosity problem 62. CRC: Composite residuosity class problem 24 63. DCRC: Decisional Composite Residuosity Class problem 64. GenBBS: generalised Blum-Blum-Shub assumption 4 Product groups 25 65. CO-CDIl: co-Computational Diffie-llellman Problem 66. XDDH: External Decision Dillie-Hellmlan problern 67. D-Linear 2: Decision Linear Problem(version 2) 26 68. FSDH: Flexible Square Diffie-Hellman Problem 26 69 KSL: Assumption 1 of Katz-Sahai-Waters 27 5 Pairings 27 70. BDHP: Bilinear Diffie-Hellman Problem 27 71. DBDH: Decision Bilinear Diffie-Hellman problem 72.l-BDlII: -Bilinear Diffie-Ilellman Inversion Problem 29 73.l-DBDHI: -Bilinear Decision Dillie-Hellman Inversion Problen 29 74.l-WBDHI: -wcak bilinear diffic-Hcllman inversion problcm 0 75.l-WDBDHI l-weak decisional bilinear diffie-Hellman Inversion Problem 30 76. KSW2: Assumption 2 of Katz-Sahai-Waters 31 77. MSEDH: Mulli-sequence of Exponents Dillie-Hellman Assunption ... 31 6 Lattices 32 6.1 Main lattice Problems 32 78. SVPP:(Approximate) Shortest vector problem 79. CVP):(Approximate)Closest vector problem 32 80. GapsvPv: Decisional shortest vector problem 81. Gapcvpp: Decisional closest vector problem 6.2 Modular lattice problems 82 SISP(n, m, 9, B: Short integer solution problem 83.I SISP(T, In 4, B): InhOmogeneous short integer solution proble 34 6.3 Misccllancous lattic problcms 84. USVPP(n, 2): Approximate unique shortest vector problem 85 SBPP(n, m): Approximate shortest basis problem 35 86. SLPP(n, y): Approximate shortest length problem 87. SIVPP(n, Y): Approximatc shortest independent vector problcm 88. heriteSVP: Ilermite shortest vector problem 89. CRP: Covering radius problem 37 6.4 Idcal lattice problems 90. Ideal-SVPP:(Approximate) Ideal shortest vector problem/Shortest polynomial problcm 7 91. Ideal-SISJ,P &: Ideal small integer solution problem 38 9,m 7 Miscellaneous problems 38 92. KEAl: Knowledge of Exponent assumption 93. MQ: Multivariable Quadratic equations 39 91. CF: Given-weight codeword finding 9 95. ConjSP: Braid group conjugacy scarch problcm 39 96. GenConjSP: Generalised braid group conjugacy search problem 40 97. ConjDecomP: Braid group conjugacy decomposition probleIll 98. Conj DP: Braid group conjugacy dccision problcm 99. DHCP: Braid group decisional Diffie-Hellman-type conjugacy problem 100. ConjSearch:(multiple simutaneous) Braid group conjugacy search prob lem 41 101. SubConjSearch: subgroup restricted Braid group conjugacy search prob m 42 102. LINPOLY: A linear algebra problem on poly Is 103. HFE-DP: Hidden Field Equations Decomposition Problem 104. IIFE-SP: Ilidden Field Equations solving problem 43 105. MKS: Multiplicative Knapsack 106. BP: Balance Problem 107. AHA: Adaptive Hardness Assumptions 45 08. SPI: Sparse Polynomia.l Interpolation 45

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