Download Cryptography and Group Theory

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.
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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.
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