Number Theoretic Bit Generators#

RandomSequenceRSA(b, t): RngIntElt, RngIntElt -> SeqEnum#: Generates a sequence of \(t\) bits using the RSA pseudo-random bit generator with an RSA modulus of approximately \(b\) bits in length. The modulus \(n\) is computed by finding (pseudo-)random primes with the RandomPrime function. If gcd\((\phi(n),\ 3)\) is 1, then the exponent 3 will be used. Otherwise, a (pseudo-)random exponent \(e\) is chosen so that gcd\(( \phi(n),\ e)=1\). The seed is also chosen as a (pseudo-)random number modulo \(n\). Bits are represented as elements of \({\bf F}_{2}\).

Example: Rsa Stats (ex-f17b7a)#

The code below counts the number of \(1\)’s that appear in a sequence of \(1000\) bits generated from a 100-bit RSA modulus.

> Z := Integers();
> &+[ Z | b : b in RandomSequenceRSA(100, 1000) ];
497

RandomSequenceRSA(n, e, s, t): RngIntElt, RngIntElt, RngIntElt, RngIntElt -> SeqEnum#: Generates a sequence of \(t\) bits using the RSA pseudo-random bit generator with modulus \(n\), exponent \(e\), and seed value \(s\). Bits are represented as elements from \({\bf F}_{2}\). The integer \(n\) must be larger than 1.

RSAModulus(b): RngIntElt -> RngIntElt, RngIntElt#: Returns an RSA Modulus \(n\) of \(b\) bits in length, and an exponent \(e\) such that Gcd(EulerPhi(n),e)=1. The resulting values can be used to generate random bits with the function RandomSequenceRSA. The argument \(b\) must be at least 16. Warning: RSA Moduli generated by Magma should not be used for real world cryptographic applications. Such applications require a “true random” source to seed the random number generator. Magma’s method of seeding may not be sufficiently random to meet the requirements of cryptographic standards.

RSAModulus(b, e): RngIntElt, RngIntElt -> RngIntElt#: Returns an RSA Modulus \(n\) of \(b\) bits in length such that Gcd(EulerPhi(n),e)=1. The resulting value can be used with \(e\) for the exponent to generate random bits with the function RandomSequenceRSA. The argument \(b\) must be at least 16. The argument \(e\) must be odd and must also be in the range \(1 < e < 2^b\). Warning: RSA Moduli generated by Magma should not be used for real world cryptographic applications. Such applications require a “true random” source to seed the random number generator. Magma’s method of seeding may not be sufficiently random to meet the requirements of cryptographic standards.

RandomSequenceBlumBlumShub(b, t): RngIntElt, RngIntElt -> SeqEnum#
BlumBlumShub(b, t): RngIntElt, RngIntElt -> SeqEnum#: Generates a sequence of \(t\) bits using the Blum-Blum-Shub pseudo-random bit generator with a Blum-Blum-Shub modulus of approximately \(b\) bits in length. The modulus \(n\) is computed within Magma by finding (pseudo-)random primes with the RandomPrime function (the condition being that the primes are congruent to \(3\) mod \(4\)). The seed is chosen as a (pseudo-)random number modulo \(n\). Bits are represented as elements from \({\bf F}_{2}\). \(b\) must be at least \(16\).

RandomSequenceBlumBlumShub(n, s, t): RngIntElt, RngIntElt, RngIntElt -> SeqEnum#
BlumBlumShub(n, s, t): RngIntElt, RngIntElt, RngIntElt -> SeqEnum#: Generates a sequence of \(t\) bits using the Blum-Blum-Shub pseudo-random bit generator with modulus \(n\) and seed value \(s\). Bits are represented as elements from \({\bf F}_{2}\). The argument \(n\) must be larger than 1 and gcd(\(s\), \(n\)) must be 1.

BBSModulus(b): RngIntElt -> RngIntElt#
BlumBlumShubModulus(b): RngIntElt -> RngIntElt#: Returns a Blum-Blum-Shub Modulus \(b\) bits in length. The resulting value can be used to generate random bits with the function RandomSequenceBlumBlumShub. The argument \(b\) must be at least 16. Warning: Blum-Blum-Shub Moduli generated by Magma should not be used for real world cryptographic applications. Such applications require a “true random” source to seed the random number generator. Magma’s method of seeding may not be sufficiently random to meet the requirements of cryptographic standards.