Pollard's rho algorithm for logarithms: Difference between revisions

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{{Short description|Mathematical algorithm}}

'''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.

'''Pollard's rho algorithm for logarithms''' is an algorithm introduced by [[John Pollard (mathematician)|John Pollard]] in 1978 to solve the [[discrete logarithm]] problem, analogous to [[Pollard's rho algorithm]] to solve the [[integer factorization]] problem.

The goal is to compute <math>\gamma</math> such that <math>\alpha ^ \gamma = \beta</math>, where <math>\beta</math> belongs to a [[cyclic group]] <math>G</math> generated by <math>\alpha</math>. The algorithm computes [[integer]]s <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>. If the underlying [[group (mathematics)|group]] is cyclic of [[order of a group|order]] <math>n</math>, by substituting <math> \beta </math> as <math> {\alpha}^{\gamma} </math> and noting that two powers are equal [[if and only if]] the exponents are equivalent modulo the order of the base, in this case modulo <math>n</math>, we get that <math>\gamma</math> is one of the solutions of the equation <math>(B-b) \gamma = (a-A) \pmod n</math>. Solutions to this equation are easily obtained using the [[extended Euclidean algorithm]].

{{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 (mathematics)|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 of approximate length <math>\sqrt{\frac{\pi n}{8}}</math> after <math>\sqrt{\frac{\pi n}{8}}</math> steps. One way to define such a function is to use the following rules: [[Partition of a set|Partition]] <math>G</math> into three [[disjoint subset]]s <math>S_0</math>, <math>S_1</math>, and <math>S_2</math> of approximately equal size using a [[hash function]]. If <math>x_i</math> is in <math>S_0</math> then double both <math>a</math> and <math>b</math>; if <math>x_i \in S_1</math> then increment <math>a</math>, if <math>x_i \in S_2</math> then increment <math>b</math>.

==Algorithm==

Let <math>G</math> be a [[cyclic group]] of order <math>n</math>, and given <math>\alpha, \beta\in G</math>, and a partition <math>G = S_0\cup S_1\cup S_2</math>, let <math>f:G\to G</math> be the map

:<math>

f(x) = \begin{cases}

\beta x & x\in S_0\\

x^2 & x\in S_1\\

\alpha x & x\in S_2

\end{cases}

</math>

and define maps <math>g:G\times\mathbb{Z}\to\mathbb{Z}</math> and <math>h:G\times\mathbb{Z}\to\mathbb{Z}</math> by

:<math>\begin{align}

g(x,k) &= \begin{cases}

k & x\in S_0\\

2k \pmod {n} & x\in S_1\\

k+1 \pmod {n} & x\in S_2

\end{cases}

\\

h(x,k) &= \begin{cases}

k+1 \pmod {n} & x\in S_0\\

2k \pmod {n} & x\in S_1\\

k & x\in S_2

\end{cases}

\end{align}</math>

'''input:''' ''a'': a generator of ''G''

''b'': an element of ''G''

'''output:''' An integer ''x'' such that ''a^x'' = ''b'', or failure

Initialise ''i'' &larr; 0, ''a''₀ &larr; 0, ''b''₀ &larr; 0, ''x''₀ &larr; 1 &isin; ''G''

'''loop'''

''i'' &larr; ''i'' + 1

''x_i'' &larr; ''f''(''x''_''i''−1),

''a_i'' &larr; ''g''(''x''_''i''−1, ''a''_''i''−1),

''b_i'' &larr; ''h''(''x''_''i''−1, ''b''_''i''−1)

''x''_2''i''−1 &larr; ''f''(''x''_2''i''−2),

''a''_2''i''−1 &larr; ''g''(''x''_2''i''−2, ''a''_2''i''−2),

''b''_2''i''−1 &larr; ''h''(''x''_2''i''−2, ''b''_2''i''−2)

''x''_2''i'' &larr; ''f''(''x''_2''i''−1),

''a''_2''i'' &larr; ''g''(''x''_2''i''−1, ''a''_2''i''−1),

''b''_2''i'' &larr; ''h''(''x''_2''i''−1, ''b''_2''i''−1)

'''while''' ''x_i'' &ne; ''x''_2''i''

''r'' &larr; ''b_i'' − ''b''_2''i''

'''if''' r = 0 '''return failure'''

'''return''' ''r''^{<span class="nowrap" style="padding-left:0.1em">−1</span>}(''a''_2''i'' − ''a_i'') mod ''n''

==Example==

Consider, for example, the group generated by 2 modulo <math>N=1019</math> (the order of the group is <math>n=1018</math>, 2 generates the group of units modulo 1019). The algorithm is implemented by the following [[C++]] program:

<syntaxhighlight lang="cpp">

#include <stdio.h>

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(void) {

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;

}

</syntaxhighlight>

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 <math>2^{681} 5^{378} = 1010 = 2^{301} 5^{416} \pmod{1019}</math> and so <math>(416-378)\gamma = 681-301 \pmod{1018}</math>, for which <math>\gamma_1=10</math> is a solution as expected. As <math>n=1018</math> is not [[prime number|prime]], there is another solution <math>\gamma_2=519</math>, for which <math>2^{519} = 1014 = -5\pmod{1019}</math> holds.

==Complexity==

The running time is approximately <math>\mathcal{O}(\sqrt{n})</math>. If used together with the [[Pohlig–Hellman algorithm]], the running time of the combined algorithm is <math>\mathcal{O}(\sqrt{p})</math>, where <math>p</math> is the largest prime [[divisor|factor]] of <math>n</math>.

==References==

{{Reflist}}

*{{cite journal |first=J. M. |last=Pollard |title=Monte Carlo methods for index computation (mod ''p'') |journal=[[Mathematics of Computation]] |volume=32 |year=1978 |issue=143 |pages=918–924 |doi= 10.2307/2006496 |jstor=2006496 }}

*{{cite book |first1=Alfred J. |last1=Menezes |first2=Paul C. |last2=van Oorschot |first3=Scott A. |last3=Vanstone |chapter-url=https://cacr.uwaterloo.ca/hac/about/chap3.pdf |title=Handbook of Applied Cryptography |chapter=Chapter 3 |year=2001 }}

{{Number-theoretic algorithms}}

[[Category:Logarithms]]

[[Category:Number theoretic algorithms]]