Cryptography - Sage

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at the command line:sage: x = 2sage: y = 2/1sage: z = ’2’sage: x2sage: y2sage: z’2’The operator == tests mathematical equality, which may involve an evaluation of one objectin the parent of the other to carry out the comparison.> x == yTrue> z == yFalseIn the first line above, the integer x is interpretted as a rational number and found to beequal to y (and vice versa).Booleans and boolean operatorsThe boolean truth values True and False have their own types in SAGE (in fact inPython), which take the special binary boolean operators and, or, and unary operatornot.sage: a = Truesage: type(a)sage: b = Falsesage: a and bFalsesage: a or bTruesage: not aFalseLists, tuples, sets, and dictionaries91
Python (hence SAGE) has useful datastructures called lists, tuples, and dictionaries whichcan be used to collect objects in SAGE.sage: type([])sage: type(())sage: type({})The list and tuple types collects sequences of data which is indexed like strings:sage: s = [ 16, 9, 4, 1, 0, 1, 4, 9, 16 ]sage: t = ( 16, 9, 4, 1, 0, 1, 4, 9, 16 )sage: [ s[i] == t[i] for i in range(9) ][True, True, True, True, True, True, True, True, True]Note that all strings, lists, and tuples are indexed from 0 and range(n) returns the list ofelements from 0 to 9:sage: range(9)[0, 1, 2, 3, 4, 5, 6, 7, 8]The main distinction is that tuples are immutable:sage: t[0] = 4---------------------------------------------------------------------------Traceback (most recent call last)...: ’tuple’ object does not support item assignmentwhile list elements can be reassigned:sage: s[0] = 4sage: s[8] = 4sage: s[4, 9, 4, 1, 0, 1, 4, 9, 4]92 Appendix A. SAGE Constructions
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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
Page 46 and 47: The Advanced Encryption Standard al
Page 48 and 49: 1. Malicious substitution of a ciph
Page 50 and 51: locks M j−1 , . . . , M 1 as well
Page 52: where X = K ⊕ M = (X 1 , X 2 , X
Page 55 and 56: 6.2 Properties of Stream CiphersSyn
Page 57 and 58: Exercise. Verify that the equality
Page 59 and 60: n 2 n − 11 12 33 74 155 316 637 1
Page 61 and 62: Exercise 6.6 In the previous exerci
Page 63 and 64: Exercise 6.9 Compute the first 8 te
Page 65 and 66: which holds since −4 = 17 + (−1
Page 67 and 68: must therefore have a divisor of de
Page 69 and 70: Shrinking Generator cryptosystemLet
Page 72 and 73: CHAPTEREIGHTPublic Key Cryptography
Page 74 and 75: Initial setup:1. Alice and Bob publ
Page 76 and 77: We apply this rule in the RSA algor
Page 78 and 79: the discrete logarithm problem (DLP
Page 80 and 81: Man in the Middle AttackThe man-in-
Page 82: Exercise 8.6 Fermat’s little theo
Page 85 and 86: k < p − 1 with GCD(k, p − 1) =
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: sage: x.is_unit?Type:builtin_functi
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 and 136: sage: p = 2^32+61sage: m = (p-1).qu
Page 137 and 138: sage: a5 := a^n5sage: c5 := c^n5sag
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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