Modular Arithmetic and Primality

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Note that we can use the same style trick as in multiplication a la Francais ⎧⎧ (x ⎣⎣y/2⎦⎦ ) 2 if y is even x y = ⎨⎨ ⎩⎩ x⋅ (x ⎣⎣y/2⎦⎦ ) 2 if y is odd € CS 312 - Complexity Examples - Arithmetic and RSA 22
⎧⎧ (x ⎣⎣y/2⎦⎦ ) 2 if y is even x y = ⎨⎨ ⎩⎩ x⋅ (x ⎣⎣y/2⎦⎦ ) 2 if y is odd function modexp (x, y, N) Input: € Two n-bit integers x and N, an integer exponent y (arbitrarily large) Output: x y mod N if y = 0: return 1 z = modexp(x, floor(y/2), N) if y is even: return z 2 mod N else: return x · z 2 mod N CS 312 - Complexity Examples - Arithmetic and RSA 23
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: There is a faster algorithm - Rathe
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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