Modular Arithmetic and Primality

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Factoring: Given a number N, express it as a product of its prime numbers Primality: Given a number N, determine whether it is prime Which one is harder This gulf will be a key to modern encryption algorithms (RSA, etc), which makes secure communication (internet, etc.) currently reasonable. CS 312 - Complexity Examples - Arithmetic and RSA 2
Addition of two numbers of length n – Pad smaller number with leading 0’s if necessary CS 312 - Complexity Examples - Arithmetic and RSA 3
Page 1: Addition Multiplication Bigger Ex
Page 5 and 6: Addition of two numbers of length n
Page 7 and 8: Isn’t addition in a computer just
Page 9 and 10: x y 15 11 At each step double x and
Page 11 and 12: ⎧⎧ 2(x⋅ ⎣⎣y/2⎦⎦) if y
Page 13 and 14: x y y binary # of x’s 15 8 0 1 30
Page 15 and 16: function multiply(x,y) Input: Two n
Page 17 and 18: Is multiplication O(n 2 ) - Could w
Page 19 and 20: Assume we start with numbers Mod N
Page 21 and 22: There is a faster algorithm - Rathe
Page 23 and 24: ⎧⎧ (x ⎣⎣y/2⎦⎦ ) 2 if y
Page 25 and 26: x y y binary power of x z return va
Page 27 and 28: function modexp (x, y, N) //Iterati
Page 29 and 30: function primality(N) Input: Positi
Page 31 and 32: How often does a composite number c
Page 33 and 34: Primality testing is efficient! - O

Modular Arithmetic and Primality

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