Pollard's rho algorithm for logarithms

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 goal is to compute $$gamma such that $$alpha ^ gamma = eta(mod N), where $$eta belongs to the group $$G generated by $$alpha. The algorithm computes integers $$a, $$b, $$A, and $$B such that $$alpha^a eta^b = alpha^A eta^B(mod N). Assuming, for simplicity, that the underlying group is cyclic of order $$N and that $$n = phi(N), we can calculate $$gamma as a solution of the equation $$(B-b)gamma = (a-A) pmod{n}.

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

Algorithm

Let $$G be a cyclic group of prime order $$p, and given $$a,bin G, and a partition $$G = G_0cup G_1cup G_2, let $$f:G o G be a map

$$f(x) = left{egin{matrix}bcdot x & xin G_0\x^2 & xin G_1\acdot x & xin G_2end{matrix} ight.

and define maps $$g:G imesmathbb{Z} omathbb{Z} and $$h:G imesmathbb{Z} omathbb{Z} by

$$g(x,n) = left{egin{matrix}n & xin G_0\2n (mod p) & xin G_1\n+1 (mod p) & xin G_2end{matrix} ight.

$$h(x,n) = left{egin{matrix}n+1 (mod p) & xin G_0\2n (mod p) & xin G_1\n & xin G_2end{matrix} ight.

:Inputs "a" a generator of "G", "b" an element of "G":Output An integer "x" such that "a^x = b", or failure:# Initialise "a₀" ← 0:#::"b₀" ← 0:#::"x₀" ← 1 ∈ "G":#::"i" ← 1:# "x_i" ← "f(x_i-1)", "a_i" ← "g(x_i-1,a_i-1)", "b_i" ← "h(x_i-1,b_i-1)":#"x_2i" ← "f(f(x_2i-2))", "a_2i" ← "g(f(x_2i-2),g(x_2i-2,a_2i-2))", "b_2i" ← "h(f(x_2i-2),h(x_2i-2,b_2i-2))":# If "x_i" = "x_2i" then:## "r" ← "b_i" - "b_2i":## If r = 0 return failure:## x ← "r^-1"("a_2i" - "a_i") mod "p":## return x:# If "x_i" ≠ "x_2i" then "i" ← "i+1", and go to step 2.

Example

Consider, for example, the group generated by 2 modulo $$N=1019 (the order of the group is $$n=1018, 2generates the group of units modulo 1019). The algorithm is implemented by the following C program:

#include const int n = 1018, N = 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 ", 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 .............................. 48 224 680 376 86 299 412 49 101 680 377 860 300 413 50 505 680 378 101 300 415 51 1010 681 378 1010 301 416

That is $$2^{681} 5^{378} = 1010 = 2^{301} 5^{416} pmod{1019} and so $$(416-378)gamma = 681-301 pmod{1018}, for which $$gamma_1=10 is a solution as expected. As $$n=1018 is not prime, there is another solution $$gamma_2=519, for which $$2^{519} = 1014 = -5pmod{1019} holds.

Complexity

The running time is approximately O($$sqrt{n}) for a number "n".

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.