Modular Arithmetic and Primality

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function primality(N) Input: Positive integer N Output: maybe/no // a is random positive integer between 2 and N-1 a = uniform(2 … N-1) if (modexp(a, N-1, N) == 1): return maybe a prime else: return no CS 312 - Complexity Examples - Arithmetic and RSA 30
How often does a composite number c pass the test (i.e. does a c-1 ≡ 1 mod c for a random a such that 1 < a < c) – Note always passes if we use a = 1 Fairly rare, and becomes increasingly rare with the size of the number We can prove that the probability is less than .5 (very conservative) So how do we extend the algorithm to give us more confidence CS 312 - Complexity Examples - Arithmetic and RSA 31
Page 1 and 2: Addition Multiplication Bigger Ex
Page 3 and 4: Addition of two numbers of length n
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: function primality(N) Input: Positi
Page 33 and 34: Primality testing is efficient! - O

Modular Arithmetic and Primality

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