Modular Arithmetic and Primality

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Addition of two numbers of length n – Pad smaller number with leading 0’s if necessary – At each step 3 single digit numbers (including carry) are added to create a 2 digit number (could have a leading 0) – Time needed: CS 312 - Complexity Examples - Arithmetic and RSA 4
Addition of two numbers of length n – Pad smaller number with leading 0’s if necessary – At each step 3 single digit numbers (including carry) are added to create a 2 digit number (could have a leading 0) – Time needed: c 0 (overhead) + c 1· n (which is linear) – Complexity O(n), where n is the size (in bits) of the numbers – Base an issue - Any number N can represented by log b (N+1) symbols Difference between decimal and binary (3.32) Constant factor between any two bases - Thus we’ll usually just use binary examples – Can we go faster CS 312 - Complexity Examples - Arithmetic and RSA 5
Page 1 and 2: Addition Multiplication Bigger Ex
Page 3: 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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