Modular Arithmetic and Primality

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x and N are integers of length n bits, y is an integer of length m bits (i.e. m = log 2 y, n = log 2 x = log 2 N) Each multiply is n 2 How many multiplies - calling depth which is m Thus Complexity is O(n 2 m) If we assume y is of length similar to x and N then complexity is O(n 3 ) Very efficient compared to the exponential alternatives CS 312 - Complexity Examples - Arithmetic and RSA 26
function modexp (x, y, N) //Iterative version Input: Two n-bit integers x and N, an integer exponent y (arbitrarily large) Output: x y mod N if y = 0: return 1 i = y; r = 1; z = x mod N while i > 0 if i is odd: r = r z mod N z = z 2 mod N i = floor(i/2) return r – Same big-O complexity – Iteration can save overhead of calling stack (optimizing compilers) – Some feel that recursion is more elegant – Either way is reasonable CS 312 - Complexity Examples - Arithmetic and RSA 27
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: x y y binary power of x z return va
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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