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Cryptography and Group Theory Motivation: For much of cryptography, the concept of a group is an important underlying thread. Preview: 1) Symmetric Cryptosystems: many of these are based on the action of either: (i) the symmetric group Sn (block ciphers) or (ii) the linear group GLn(Fp) (linear ciphers). 2) Public Key Cryptosystems: many of these are based on the following idea: Given: a group G which has been represented in some way in the computer such that the basic operation of multiplication is fast. Then: (i) To encode a message: given g ∈ G and n ∈ N, compute g n. (fast!) (ii) To decode a message: recover g or n from g n. (expected to be difficult without extra information) Examples: 1) RSA: g = message, n is known; 2) DL (Discrete log): g is known, n = message. 1 Definition: A group is a set G together with a binary operation ∗ such that: (i) (Associative Law) We have x ∗ (y ∗ z) = (x ∗ y) ∗ z, for all x, y, z ∈ G. (ii) (Identity Law) There is an e ∈ G such that e ∗ x = x ∗ e = x, for all x ∈ G. (ii) (Inverse Law) For each x ∈ G there is an x0 ∈ G such that x ∗ x0 = x0 ∗ x = e. Notes: 1) One usually writes · in place of ∗, and 1 in place of e. Moreover, we write x0 = x−1 in (iii). 2) If G is abelian, i.e. if x∗y = y∗x, for all x, y ∈ G, then one usually writes + for ∗, and 0 for e, etc. Examples: 1) (G, ∗) = (Z, +) integers wrt. addition; 2) (Z/mZ, +) integers modulo m wrt. addition; 3) the group of units R× = {x ∈ R : x is invertible} of a ring R; in particular: (Z/mZ)× and GLn(K) = Mn(K)× 4) The symmetric group Sn of degree n is the set of all bijections f : {1, . . . , n} → {1, . . . , n} wrt. function composition. 2