Modular Arithmetic and Primality

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x y y binary # of x’s 15 11 1 1 30 5 1 2 60 2 0 4 120 1 1 8 165 11 in binary is 1011 - We show it from bottom to top in binary column Makes sense since each position in binary is a doubling of the previous position Interesting mix of binary and decimal CS 312 - Complexity Examples - Arithmetic and RSA 10
⎧⎧ 2(x⋅ ⎣⎣y/2⎦⎦) if y is even x⋅ y = ⎨⎨ ⎩⎩ x + 2(x⋅ ⎣⎣y/2⎦⎦) if y is odd € Obvious equality, yet it gives us a divide and conquer opportunity since we halve the size at each step Prep us for Modular exponentiation which uses the same basic approach
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: x y 15 11 At each step double x and
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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