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Cryptanalysis of the Affine Cipher

Cryptanalysis of the Affine Cipher. Ciphertext-only attack: Brute-force (try possible keys) Frequency analysis Any other ideas ? Suppose we know two symbols and what they map to. Example: 0  3 7  10. Cryptanalysis of the Affine Cipher. More challenging examples: 4  17

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Cryptanalysis of the Affine Cipher

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  1. Cryptanalysis of the Affine Cipher • Ciphertext-only attack: • Brute-force (try possible keys) • Frequency analysis • Any other ideas ? • Suppose we know two symbols and what they map to. • Example: 0  3 • 7  10

  2. Cryptanalysis of the Affine Cipher More challenging examples: 4  17 19  3 4  17 19  10

  3. Cryptanalysis of the Affine Cipher • Remarks: • For which attacks do we have two pairs of symbols and their maps ? • Can we use this idea for the ciphertext-only attack ? • What if we have only one pair of a symbol and its map ? • Read Section 3.3 to learn more about congruences.

  4. Some More Number Theory Recall that there are 12 elements a2Z26 such that gcd(a,26)=1. In general, for m>0, the number of elements of Zm that are relatively prime to m is denoted by Á(m) and it is usually referred to as the Euler phi function. Examples: Á(26) = Á(p) = if p is a prime

  5. Some More Number Theory For non-prime numbers: Suppose m = ¦i=1,…,n pie where the pi’s are distinct primes and ei>0 for i2{1,2,…,n}. Then, Á(m) = ¦i=1,…,n (pie – pie -1) How many keys do we have for the affine cipher over Zm ?

  6. Some More Number Theory • Computing a-1 (mod m): • Option 1: brute-force • advantages / disadvantages ? • Option 2: ??? • First, some math analysis: For which a 2 Zm the inverse a-1 does not exist in Zm ?

  7. Some More Number Theory Computing gcd(a,b):

  8. Euclidean Algorithm Def EuclideanAlgorithm (a,b): // a,b>0, integers r0 = a, r1 = b m = 0 while rm+1 0: m++ qm = b rm-1/rmc rm+1 = rm-1 – qmrm return rm

  9. Extended Euclidean Algorithm Given integers a,b>0, it computes r,s,t such that: r = gcd(a,b) sa + tb = r How is it helpful for computation of a-1 ?

  10. Extended Euclidean Algorithm Def ExtendedEuclideanAlgorithm (a,b): // a,b>0, integers r0 = a, r1 = b, s0 = 1, s1 = 0, t0 = 0, t1 = 1 m = 0 while rm+1 0: m++ qm = b rm-1/rmc rm+1 = rm-1 – qmrm tm+1 = tm-1 – qmtm sm+1 = sm-1 – qmsm return rm, sm, tm

  11. Extended Euclidean Algorithm • Remarks: • By induction on j we can show that for j2{0,1,…,m}: • rj = sja + tjb • Hence, the algorithm is correct. • We can save space by using many fewer variables. • Running time ?

  12. Solving ax ´ c (mod m) • useful for cryptanalysis of e.g. the affine cipher • Possibilities: • if gcd(a,m)=1, then: • if gcd(a,m)=d>1 and d does not divide c, then: • if gcd(a,m)=d>1 and d divides c, then:

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