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: 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 Why not Thus, addition is Θ(n) CS 312 - Complexity Examples - Arithmetic and RSA 6
Isn’t addition in a computer just 1 time step Can be, If the size of the numbers is fixed – 32/64/128 bit computer word - 1 step if we assume numbers within the word size because of hardware parallelism. – Would if bigger (double int), but still fixed. Then still c·(1). – O(n) when n can be arbitrarily large. For most applications, the problem size parameter(s) will not be tied to the specific data elements (numbers, characters, etc.) but to the number of data elements. What is the complexity of adding two arrays each containing m fixed size numbers - common What is the complexity of adding two arrays each containing m numbers of arbitrary size n - less common Typically work with fixed size data elements, and our concern is number of elements, rather than varying size of the elements CS 312 - Complexity Examples - Arithmetic and RSA 7
Page 1 and 2: Addition Multiplication Bigger Ex
Page 3 and 4: Addition of two numbers of length n
Page 5: Addition of two numbers of length n
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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