By Firoz Kaderali
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Additional resources for Foundations and Applications of Cryptology [Lecture notes]
We look at how to find the multiplicative inverse in the set Z∗n . e. we must determine an element x ∈ Z∗n so that ax ≡ 1 mod n. e. there is an integer k so that ax = 1 + nk. This can be written as: ax − nk = 1. 3-1 Since the integers a, x ∈ Z∗n , we have gcd(a, x) = 1 and Eq. 3-2 and, if we put y = (−k), we obtain: ax + ny = gcd(a, n). 3-1 says that the equation ax + by = gcd(a, b) will always be solvable and its solution will be determined by the two integers x and y. We can now use the Extended Euclidean algorithm to find such integers x and y.
It is based on the following simple theorem. 3-3: Let a, b ∈ Z. Then, 1. if b = 0, then gcd(a, b) = |b|, 2. if b = 0, then gcd(a, b) = gcd(|b|, a mod |b|) . The previous theorem enables the computation of the greatest common divisor gcd(a, b) as follows: We suppose r0 = a, r1 = b and a > b > 0, then we introduce the notation ri+1 = ri−1 mod ri for each integer i ≥ 1 and ri = 0. Then, we compute ri+1 = ri−1 mod ri for i = 1, 2, 3 . . until we obtain for a fixed i0 ≥ 1: ri0 +1 = 0. Then, the greatest common divisor is ri0 .
If f (n) = O(h(n)) and g(n) = O(l(n)), then (f · g)(n) = O(h(n)l(n)). 8 Complexity classes We define a polynomial time algorithm as an algorithm having the complexity O(nk ), where k is a constant. Algorithms whose complexities are O(tf (n) ), with t a constant and f (n) a polynomial of n are called exponential time algorithms. A subexponential time algorithm is an algorithm with the complexity eo(n) . There is a given way to classify the running time of algorithms between polynomial time and exponential time.
Foundations and Applications of Cryptology [Lecture notes] by Firoz Kaderali