Pollard's p-1 algorithm [4]

Pollard's p − 1 algorithm is a number theoretic integer factorization algorithm, invented by John Pollard in 1974. It is a special-purpose algorithm, meaning that it is only suitable for integers with specific types of factors.

The algorithm is based on the insight that numbers of the form a^b  − 1 tend to be highly composite when b is itself composite. Since it is computationally simple to evaluate numbers of this form in modular arithmetic, the algorithm allows one to quickly check many potential factors with great efficiency. In particular, the method will find a factor p if b is divisible by p − 1, hence the name. When p − 1 is smooth (the product of only small integers) then this algorithm is well-suited to discovering the factor p.

Base concepts

Let n be a composite integer with prime factor p. By Fermat's little theorem, we know that

for a coprime to p

Let us assume that p − 1 is B-powersmooth for some reasonably sized B (more on the selection of this value later). Recall that a positive integer m is called B-smooth if all prime factors p_i  of m are such that p_i  ≤ B. m is called B-powersmooth if all prime powers
                                                   
_i dividing m are such that p_{i  i} ≤ B.

Let p₁, ..., p_L  be the primes less than B and let e₁, ..., e_L be the exponents such that

Let

As a shortcut, M = lcm{1, ..., B}. As a consequence of this, (p − 1) divides M, and also if p^e divides M this implies that p^e ≤ B. Since (p − 1) divides M we know that a^M ≡ 1 (mod p), and because p divides n this means gcd(a^M − 1, n) > 1.

Therefore if gcd(a^M − 1, n) ≠ n, then the gcd is a non-trivial factor of n.

If p − 1 is not B-power-smooth, then a^M ≢ 1 (mod p) for at least half of all a.