Random Numbers

Pseudo-random integers in Magma are generated using the Monster random number generator of G. Marsaglia [Mar00] (see also [MZ91] and [Mar92]). The period of the Monster generator is 2²⁹⁴³⁰ - 2²⁷³⁸² (approximately 10⁸⁸⁵⁹), and the generator passes all of the stringent tests in Marsaglia's Diehard test suite [Mar95]. Throughout the following text, the word `random' is used to mean `pseudo-random'.

Random(a, b) : RngIntElt, RngIntElt -> RngIntElt

A random integer lying in the interval [a, b], where a≤b.

Random(b) : RngIntElt -> RngIntElt

A random integer lying in the interval [0, b], where b is a non-negative integer. Because of the good properties of the underlying Monster generator, calling Random(1) is a good safe way of producing a sequence of random bits.

RandomBits(n) : RngIntElt -> RngIntElt

A random integer m such that 0 ≤m < 2ⁿ, where n is a small non-negative integer. Thus, m has n random bits with a probability of 1/2 for each bit. The function always returns 0 when n=0.

RandomPrime(n: parameter) : RngIntElt -> RngIntElt

    Proof: BoolElt                      Default: true

A random prime integer m such that 0 < m < 2ⁿ, where n is a small non-negative integer. The function always returns 0 for n=0 or n=1. A rigorous method will be used to check primality, unless m > 3317044064679887385961981 and the optional parameter Proof is set to Proof := false, in which case the result indicates that m is a probable prime (of order 20).

RandomPrime(n, a, b, x: parameter) :RngIntElt, RngIntElt, RngIntElt -> BoolElt, RngIntElt

    Proof: BoolElt                      Default: true

Tries up to x iterations to find a random prime integer m congruent to a modulo b such that 0 < m < 2ⁿ. If successful, the function returns true and the integer m, otherwise false. The integer n must be larger than 0, a must lie between 0 and b - 1 and b must be larger than 0. A rigorous method will be used to establish primality, unless m > 3317044064679887385961981 and the optional parameter Proof is set to Proof := false, in which case the result indicates that m is a probable prime (of order 20).

RandomConsecutiveBits(n, a, b) : RngIntElt, RngIntElt -> RngIntElt

A integer m such that 0 ≤m < 2ⁿ, and the binary expansion of n consists of consecutive strings of zeros or ones each of random length in the range [a ... b].