LittleNotesOnZnZnMatrixInverse.pdf - A little note on...

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A little note on Cryptographic Background Mathematics Sk Md Mizanur Rahman, Ph.D. Professor, Cybersecurity Centennial College, Toronto, Canada
1. Z n and Z n * sets: In cryptography uses an integer as the encryption key, the inverse of that integer is used as the decryption key. If the encryption / decryption operation is addition, then Z n is the set of possible keys, as each element of Z n has an inverse. For multiplication, it is not true. Not all elements of Z n has an inverse. Z n * is a new set, a subset of Z n , which includes those members of Z n which has a unique multiplicative inverse. 2. Additive and Multiplicative inverse We need to use Z n when additive inverses are needed; we need to use Z n * when multiplicative inverses are needed Additive Inverse: An additive inverse is relative to an addition operation. In modular arithmetic , each integer has an additive inverse. The sum of an integer and its additive inverse is congruent to 0 modulo n . Example 1: Find all additive inverse pairs in Z 10 . We Know, Z 10 = {0, 1, 2, 3, 4, 5, 6, 7, 8, 9}
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3 | P a g e o f 1 2 In Z n , two numbers a and b are the multiplicative inverse of each other if Thus, if gcd (n, a) = 1 , then “a” has a multiplicative in Z n, which is “b” . And, the extended Euclidean algorithm finds the multiplicative inverses of a (i.e., a -1 = b ) in Z n when n and a are given. Extended Euclidean Algorithm for finding multiplicative inverse of “a” in Z n Answer : r 1 n r 2 ← a t 1 ← 0 t 2 ← 1 While (r 2 >0) { q ← r 1 /r 2 r← r 1 q x r 2 r 1 ← r 2 r 2 ← r t ← t 1 - q x t 2 t 1 ← t 2 t 2 ← t } If (r 1 =1) then a -1 ← t 1 [ In fact r 1 is the gcd of (n,a)]
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