Algebraic-group factorisation algorithms are algorithms for factoring an integer N by working in an algebraic group defined modulo N whose group structure is the direct sum of the 'reduced groups' obtained by performing the equations defining the group arithmetic modulo the unknown prime factors p1, p2, ... By the Chinese remainder theorem, arithmetic modulo N corresponds to arithmetic in all the reduced groups simultaneously. The aim is to find an element which is not the identity of the group modulo N, but is the identity modulo one of the factors, so a method for recognising such one-sided identities is required. In general, one finds them by performing operations that move elements around and leave the identities in the reduced groups unchanged. Once the algorithm finds a one-sided identity all future terms will also be one-sided identities, so checking periodically suffices. Computation proceeds by picking an arbitrary element x of the group modulo N and computing a large and smooth multiple Ax of it; if the order of at least one but not all of the reduced groups is a divisor of A, this yields a factorisation. It need not be a prime factorisation, as the element might be an identity in more than one of the reduced groups. Generally, A is taken as a product of the primes below some limit K, and Ax is computed by successive multiplication of x by these primes; after each multiplication, or every few multiplications, the check is made for a one-sided identity. The two-step procedure It is often possible to multiply a group element by several small integers more quickly than by their product, generally by difference-based methods; one calculates differences between consecutive primes and adds consecutively by the . This means that a two-step procedure becomes sensible, first computing Ax by multiplying x by all the primes below a limit B1, and then examining p Ax for all the primes between B1 and a larger limit B2. Methods corresponding to particular algebraic groups If the algebraic group is the multiplicative group mod N, the one-sided identities are recognised by computing greatest common divisors with N, and the result is the p − 1 method. If the algebraic group is the multiplicative group of a quadratic extension of N, the result is the p + 1 method; the calculation involves pairs of numbers modulo N. It is not possible to tell whether is