Pull-up resistor

2013-10-24T21:37:17Z

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'''Pollard's rho algorithm for logarithms''' is an algorithm introduced by [[John Pollard (mathematician)|John Pollard]] in 1978 for solving the [[discrete logarithm]] problem analogous to [[Pollard's rho algorithm]] for solving 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 [[Group (mathematics)|group]] <math>G</math> 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</math> as a solution of the equation <math>(B-b)\gamma = (a-A) \pmod{n}</math>.

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 disjoint subsets of approximately equal size: <math>S_0</math>, <math>S_1</math>, and <math>S_2</math>. 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>p</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 a map

<math>
f(x) = \left\{\begin{matrix}
\beta x & x\in S_0\\
x^2 & x\in S_1\\
\alpha x & x\in S_2
\end{matrix}\right.
</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>
g(x,n) = \left\{\begin{matrix}
n & x\in S_0\\
2n \ (\bmod \ p) & x\in S_1\\
n+1 \ (\bmod \ p) & x\in S_2
\end{matrix}\right.
</math>

<math>
h(x,n) = \left\{\begin{matrix}
n+1 \ (\bmod \ p) & x\in S_0\\
2n \ (\bmod \ p) & x\in S_1\\
n & x\in S_2
\end{matrix}\right.
</math>

:'''Inputs''' ''a'' a generator of ''G'', ''b'' an element of ''G''
:'''Output''' An integer ''x'' such that ''ax = b'', or failure
:# Initialise ''a0'' ← 0
:#::''b0'' ← 0
:#::''x0'' ← 1 ∈ ''G''
:#::''i'' ← 1
:# ''xi'' ← ''f(xi-1)'', ''ai'' ← ''g(xi-1,ai-1)'', ''bi'' ← ''h(xi-1,bi-1)''
:#''x2i'' ← ''f(f(x2i-2))'', ''a2i'' ← ''g(f(x2i-2),g(x2i-2,a2i-2))'', ''b2i'' ← ''h(f(x2i-2),h(x2i-2,b2i-2))''
:# If ''xi'' = ''x2i'' then
:## ''r'' ← ''bi'' - ''b2i''
:## If r = 0 return failure
:## x ← ''r -1 ''(''a2i'' - ''ai'') mod ''p''
:## return x
:# If ''xi'' ≠ ''x2i'' then ''i'' ← ''i+1'', and go to step 2.

==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:

<source lang="c">
#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;
}
</source>

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, 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 O(<math>\sqrt{p}</math>) where p is n's largest prime factor.

==References==
*{{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= |jstor=2006496 }}
*{{cite book |first=Alfred J. |last=Menezes |first2=Paul C. |last2=van Oorschot |first3=Scott A. |last3=Vanstone |chapterurl=http://www.cacr.math.uwaterloo.ca/hac/about/chap3.pdf |title=Handbook of Applied Cryptography |chapter=Chapter 3 |year=2001 |location= |publisher= |isbn= }}

{{Number-theoretic algorithms}}

[[Category:Logarithms]]
[[Category:Number theoretic algorithms]]

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