Pollard's rho algorithm for logarithms: Difference between revisions

Revision as of 02:52, 8 October 2005

Revision as of 15:45, 1 January 2006

Line 1:

'''Pollard's rho algorithm for logarithms''' is an algorithm for solving the [[discrete logarithm]] problem analogous to [[Pollard's rho algorithm]] for solving the [[Integer factorization]] problem.

The algorithm computes <math>\gamma</math> such that <math>\alpha ^ \gamma = \beta</math>, where <math>\beta</math> is in a [[group]] <math>G</math> [[generator|generated]] by <math>\alpha</math>. The algorithm computes integers <math>a</math>, <math>b</math>, <math>A</math>, and <math>B</math> such that <math>\alpha^a \beta^b = \alpha^A \beta^B</math>. Assuming, for simplicity, that the underlying group is cyclic of order <math>n</math>, we can calculate <math>\gamma = \frac{a-A}{B-b} \pmod{n}</math>.

{{comp-stub}}

To find the needed <math>a</math>, <math>b</math>, <math>A</math>, and <math>B</math> the algorithm uses [[Floyd's cycle-finding algorithm]] to find a cycle in the sequence <math>x_i = \alpha^{a_i} \beta^{b_i}</math>, where the function <math>f: x_i \mapsto x_{i+1}</math> is assumed to be random-looking and thus is likely to enter into a loop after approximately <math>\sqrt{\frac{\pi n}{2}}</math> steps. One way to define such a function is to use the following rules: Divide <math>G</math> into three subsets (not necessarily [[subgroup]]s) of approximately equal size: <math>G_0</math>, <math>G_1</math>, and <math>G_2</math>. If <math>x_i</math> is in <math>G_0</math> then double both <math>a</math> and <math>b</math>; if <math>x_i \in G_1</math> then increment <math>a</math>, if <math>x_i \in G_2</math> then increment <math>b</math>.

==Example==

Consider, for example, the group generated by 2 modulo <math>N=1019</math> (the order of the group is <math>n=509</math>). The algorithm is implemented by the following [[C plus plus|C++]] program:

#include <stdio.h>

const int n = 509, N = 2*n + 1; // N = 1019 -- prime

const int alpha = 2; // generator

const int beta = 5; // 2^{10} = 1024 = 5 (N)

void new_xab(int& x, int& a, int& b){

switch(x%3){

case 0: x = x*x % N; a = a*2 % n; b = b*2 % n; break;

case 1: x = x*alpha % N; a = a+1 % n; break;

case 2: x = x*beta % N; b = b+1 % n; break;

}

}

int main(){

int x=1, a=0, b=0;

int X=x, A=a, B=b;

for(int i = 1; i < n; ++i){

new_xab(x, a, b);

new_xab(X, A, B); new_xab(X, A, B);

printf("%3d %4d %3d %3d %4d %3d %3d\n", i, x, a, b, X, A, B);

if(x == X) break;

}

return 0;

}

The results are as follows (edited):

i x a b X A B

------------------------------

1 2 1 0 10 1 1

2 10 1 1 100 2 2

3 20 2 1 1000 3 3

4 100 2 2 425 8 6

5 200 3 2 436 16 14

6 1000 3 3 284 17 15

7 981 4 3 986 17 17

8 425 8 6 194 17 19

..............................

48 224 171 376 86 299 412

49 101 171 377 860 300 413

50 505 171 378 101 300 415

51 1010 172 378 1010 301 416

That is <math>2^{172} 5^{378} = 1010 = 2^{301} 5^{416} \pmod{1019}</math> and so <math>\gamma = \frac{172-301}{416-378} = 10 \pmod{509}</math>, as expected.

==References==

* J. Pollard, ''Monte Carlo methods for index computation mod p'', Mathematics of Computation, Volume 32, 1978.

* Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, [http://www.cacr.math.uwaterloo.ca/hac/about/chap3.pdf Handbook of Applied Cryptography, Chapter 3], 2001.