Maurer's method generates provably prime numbers, which are nearly random. The method is described in [2].
In Maurer's method, a certificate of primality for a number n is a triplet of numbers, (R, F, a), plus the prime factorization of F, where \(2RF+1 = n\), and such that (see modular arithmetic)
- 1.
\(a^{n-1} \equiv 1 {\rm mod} n\) and
- 2.
\(a^{(n-1)/q_j}-1\) is relatively prime to n for all \(1 \leq j \leq r\), where \(F = q_1^{\beta_1}, \ldots, q_r^{\beta_r}\) is the prime factorization of F.
This triplet of numbers guarantees that all prime factors of n are of the form \(mF+1\) for some positive integer m (the proof of this lemma can be found in [2] but is too complicated to be included here). In particular, if \(F \geq \sqrt{n}\) then n must be prime as the product of any two primes of the form \(mF+1\) is at least \(F^2+2F+1 > F^2 \geq n\).
Maurer's algorithm generates a prime at random by generating R and F at random with the prime factorization of Fknown, and testing to see...
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References
Bach, E. (1988). “How to generate factored random numbers.” SIAM Journal on Computing, 17 (4), 173–193.
Maurer, Ueli M. (1995). “Fast generation of prime numbers and secure public-key cryptographic parameters.” Journal of Crypology, 8 (3), 123–155.
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Liskov, M. (2005). Maurer's Method. In: van Tilborg, H.C.A. (eds) Encyclopedia of Cryptography and Security. Springer, Boston, MA . https://doi.org/10.1007/0-387-23483-7_245
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