Cryptography - Sage

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APPENDIXCSolutions to ExercisesIntroduction to <strong>Cryptography</strong>Classical <strong>Cryptography</strong>Substitution ciphersExercise 2.1 Determine the number of possible keys for the affine substitution ciphers. Isthis sufficient to have a secure cryptosystem?Transposition ciphersExercise 2.2 Show that for every π in S n , there exists an positive integer m, such thatπ m is the identity map, and such that m divides n!. The smallest such m is called the orderof π.Exercise 2.3 How many transpositions exist in S n ? Describe the elements of order 2 inS n and determine their number.Exercise 2.4 Show that every element of S n can be expressed as the composition of atmost n transpositions.Exercise 2.5 What is the order of a permutation with cycle lengths d 1 , . . . , d t ? How doesthis solve the previous exercise concerning the order of a permutation?Exercise 2.6 What is the block length m of an (r, s)-simple columnar transposition? Describethe permutation. Hint: it may be easier to describe the permutation if the index setis {0, . . . , m − 1}.103
Solution. The block length is the number of characters which are involved in each permutation,which equals rs for an (r, s)-simple columnar transposition. In terms of mapsof indices, the k-th position maps to (i, j) in an array where i = ((k − 1) mod s) + 1 andj = (k − 1) div s (for k in 1, . . . , rs). The transpose maps (i, j) to (j, i), which goes to thenew position π(k) = i + (j − 1)r in the ciphertext. The map k ↦→ π(k) determines the inverseof the permutation on indices. The inverse permutation is determined by exchangingthe roles of r and s.Exercise 2.7 Show that the (r, r)-simple columnar transposition has order 2. What is theorder of the cipher for (r, s) = (3, 5)? Determine the permutation in cycle notation forthis cipher. Determine the permutation in cycle notation for the (7, 36)-simple columnartransposition used in this chapter.Solution. For a (r, r)-simple columnar transposition one writes each r 2 -block into an r ×rarray, applies a transposition, and reads the columns off as rows. It has order 2 sincetwo-fold application of the cipher acts as the identity on every r 2 -block.In general, an (r, s)-simple columnar tranposition cipher does not have order 2 since thetranspose matrix does not have the same form, so r characters are read from each column,while s characters are written into each row at the next application of the cipher. Theinverse of an (r, s)-simple columnar transposition is an (s, r)-simple columnar transposition.The (3, 5)-simple columnar transposition is determined by the following map of indicesgiving the map in list notation1 ↦→ 1, 2 ↦→ 6, 3 ↦→ 11, . . .[1, 6, 11, 2, 7, 12, 3, 8, 13, 4, 9, 14, 5, 10, 15]In SAGE we can construct this sequence and determine the cycle notation for the permuationas follows:sage: (r,s) = (3,5)sage: G = SymmetricGroup(r*s)sage: S = [ i+s*j for i in range(1,s+1) for j in range(r) ]sage: S[1, 6, 11, 2, 7, 12, 3, 8, 13, 4, 9, 14, 5, 10, 15]sage: G(S)(2,6,12,14,10,4)(3,11,9,13,5,7)The equivalent construction for (r, s) = (7, 36) follows.104 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
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 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 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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