Cryptography - Sage

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Then verify that c n 5= a 129n 5and c n 6= a 127n 6. Find similar relations forn 1 = (p − 1) div 2 n 3 = (p − 1) div 5,n 2 = (p − 1) div 3 n 4 = (p − 1) div 37.and use this information to find the discrete logarithm of c with respect to a.Solution.We set up the problem in SAGE in the following way.sage: p = 2^131+1883sage: fac = (p-1).factor(); fac2 * 3 * 5 * 37 * 634466267339108669 * 3865430919824322067sage: primes = [ f[0] for f in (p-1).factor() ]sage: (p1, p2, p3, p4, p5, p6) = primessage: expons = ( (p-1) // r for r in primes )sage: (n1, n2, n3, n4, n5, n6) = exponsIn this way, p 1 , p 2 , p 3 , p 4 , p 5 , and p 6 are assigned to be the prime divisors of p − 1 and n 1 ,n 2 , n 3 , n 4 , n 5 , and n 6 their cofactors.Raising both generator a and its power c to the large exponents n 1 , n 2 , n 3 , and n 4reduces the solution to the discrete logarithm to one modulo p 1 = 2, p 2 = 3, p 3 = 5, andp 4 = 37, which can be easily solved by enumerating all possibilities.sage: FF = FiniteField(p)sage: a = FF(109)sage: c = FF(1014452131230551128319928312434869768346)sage: [ a^(n1*i) for i in range(2) ].index(c^n1)1sage: [ a^(n2*i) for i in range(3) ].index(c^n2)2sage: [ a^(n3*i) for i in range(5) ].index(c^n3)4sage: [ a^(n4*i) for i in range(37) ].index(c^n4)29sage: crt([1,2,4,29],[2,3,5,37])29For the larger primesp 5 = 634466267339108669 and p 6 = 3865430919824322067,we verify the given discrete logarithms.131
sage: a5 := a^n5sage: c5 := c^n5sage: a5^1291106532280219457618983939634726858708298sage: c51106532280219457618983939634726858708298sage: a6 = a^n6sage: c6 = c^n6sage: a6^127809579285918008980133272648385832028198sage: c6809579285918008980133272648385832028198The discrete logarithm x can be recovered from the discrete logarithms x i = log ai(c i ) wherea i = a n iand c i = c n iby using the function crt to find the Chinese remainder lifting.sage: x = CRT([29,129,127],[2*3*5*37,p5,p6]);sage: x1075217203476555175652504438224037579sage: a^x eq cTrueDigital SignaturesSecret SharingExercise 10.1 Verify the correctness of the formula for the secret secret in Shamir’s secretsharing scheme by substituting into the formula of Lagrange’s interpolation theorem.132 Appendix C. Solutions to Exercises
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Author (David R. Kohel) /Title (Cry
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CONTENTS1 Introduction to Cryptogra
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PrefaceWhen embarking on a project
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information. We introduce here some
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ut strings in A ∗ map injectively
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CHAPTERTWOClassical Cryptography2.1
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LV MJ CW XP QO IG EZ NB YH UA DS RK
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As a special case, consider 2-chara
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Note that if d k = 1, then we omit
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ExercisesSubstitution ciphersExerci
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Ciphertext-only AttackThe cryptanal
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of size n, suppose that p i is the
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Note that ZKZ and KZA are substring
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Checking possible keys, the partial
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sage: X = pt.frequency_distribution
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CHAPTERFOURInformation TheoryInform
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For each of these we can extend our
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in terms of the cryptosystem), then
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CHAPTERFIVEBlock CiphersData Encryp
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Deciphering. Suppose we begin with
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The Advanced Encryption Standard al
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1. Malicious substitution of a ciph
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locks M j−1 , . . . , M 1 as well
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where X = K ⊕ M = (X 1 , X 2 , X
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6.2 Properties of Stream CiphersSyn
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Exercise. Verify that the equality
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n 2 n − 11 12 33 74 155 316 637 1
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Exercise 6.6 In the previous exerci
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Exercise 6.9 Compute the first 8 te
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which holds since −4 = 17 + (−1
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must therefore have a divisor of de
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Shrinking Generator cryptosystemLet
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CHAPTEREIGHTPublic Key Cryptography
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Initial setup:1. Alice and Bob publ
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We apply this rule in the RSA algor
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the discrete logarithm problem (DLP
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Man in the Middle AttackThe man-in-
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Exercise 8.6 Fermat’s little theo
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Page 88 and 89: CHAPTERTENSecret SharingA secret sh
Page 90: using any t shares (x 1 , y 1 ), .
Page 93 and 94: sage-------------------------------
Page 95 and 96: sage: x.is_unit?Type:builtin_functi
Page 97 and 98: Python (hence SAGE) has useful data
Page 99 and 100: sage: n = 12sage: for i in range(n)
Page 101 and 102: sage: I = [55+i for i in range(3)]
Page 103 and 104: sage: I = [7, 4, 11, 11, 14, 22, 14
Page 105 and 106: ExercisesRead over the above SAGE t
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Page 109 and 110: Solution. The block length is the n
Page 111 and 112: Solution.below.The coincidence inde
Page 113 and 114: analysis of the each of the decimat
Page 115 and 116: arbitrary permutation of the alphab
Page 117 and 118: In order to understand naturally oc
Page 119 and 120: We do this by first verifying the e
Page 121 and 122: Solution.None provided.Linear feedb
Page 123 and 124: Multiplying each through by the con
Page 125 and 126: Solution. The linear complexity of
Page 127 and 128: If a, b, and c are as above, then f
Page 129 and 130: Exercise 8.5 Use SAGE to find a lar
Page 131 and 132: Solution. Now we can verify that e
Page 133 and 134: which has no common factors with p
Page 135: sage: p = 2^32+61sage: m = (p-1).qu
Page 139 and 140: The application of this function E
Page 141 and 142: 5. (∗) How many elements a of G h
Page 143: 1. The value f(0) of the polynomial
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Cryptography - Sage

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