文档介绍：Chapter 2
Key Exchange Protocols
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Mathematical Preliminaries
◆ Groups
Throughout, g, denotes a cyclic group of finite order n, written
multiplicatively.
Usually, but not necessarily the group order n is prime. Recall that
any group of prime order is cyclic, and that any finite cyclic group is
abellan
Also, Gn=<g>, where g is any generator(. an element of order n)
of Gni thus, the elements of g, are enumerated as 1,g, gg
g-l. Often we write G as a shorthand for G
The discrete /og of an element ye g is defined as the least
nonnegative integer x satisfying y=gx. We write x=log, y Fory e
G, we let ord(y) denote its order.
Note that ord(y)In
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Example Take G=p, for a prime p. Then G is a cyclic group of
order n=p-1. Clearly, n is not prime(amles s p=3, of course)
More generally. one may taheG-Fa the multiplicative group of a finite
field of order q=p, for some positive integer r. Then g is a cyclic group
of ordern=g-1.
Example 1. 4 Tahe G-g), where g denotes an element of order p' in 2
with p-1, p, p prime. Then G is a cyclic group of order n=p. Cleard
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th
More generally, one may tahe G=(g), where g denotes an element of
Example 1. 5 Consider E(FQ). the finite group of points of an elliptic curve
over Fo, which is usually written additively, with the point at infinity O
as identity element. Then E(Fo is abelian, but not necessarily cyclic. Io
general. E(FQ) is isomorphic to Zr, x Enz, where n1 n2 and m1 q-1. So,
E(FQ) is cyclic if and only f m1=1. Hence, one may take G=E(FQ) f
m1=1: otherwise, one may tahe a cyclic subgroup ofE(Fg for G
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Mathematical Preliminaries
◆ Probability
Throughout, we will use basic notions from probability theory, such
as sample space, events, probability distributions, and random
variables. We will only be concerned with discrete random variables
Specifically, we use the following notion of statistical distance
Definition 1. 6 The statistical distance