Core Ideas
The rho algorithm is based on Floyd's cycle-finding algorithm and on the observation that (as in the birthday problem) two numbers x and y are congruent modulo p with probability 0.5 after numbers have been randomly chosen. If p is a factor of n, the integer we are aiming to factor, then since p divides both and .
The rho algorithm therefore uses a function modulo n as a generator of a pseudo-random sequence. It runs one sequence twice as "fast" as the other; i.e. for every iteration made by one copy of the sequence, the other copy makes two iterations. Let x be the current state of one sequence and y be the current state of the other. The GCD of |x − y| and n is taken at each step. If this GCD ever comes to n, then the algorithm terminates with failure, since this means x = y and therefore, by Floyd's cycle-finding algorithm, the sequence has cycled and continuing any further would only be repeating previous work.
Read more about this topic: Pollard's Rho Algorithm
Famous quotes containing the words core and/or ideas:
“The threadbare trees, so poor and thin,
They are no wealthier than I;
But with as brave a core within
They rear their boughs to the October sky.
Poor knights they are which bravely wait
The charge of Winter’s cavalry,
Keeping a simple Roman state,
Discumbered of their Persian luxury.”
—Henry David Thoreau (1817–1862)
“A few ideas seem to be agreed upon. Help none but those who help themselves. Educate only at schools which provide in some form for industrial education. These two points should be insisted upon. Let the normal instruction be that men must earn their own living, and that by the labor of their hands as far as may be. This is the gospel of salvation for the colored man. Let the labor not be servile, but in manly occupations like that of the carpenter, the farmer, and the blacksmith.”
—Rutherford Birchard Hayes (1822–1893)