In mathematics, a Solinas prime, or generalized Mersenne prime, is a prime number that has the form , where is a low-degree polynomial with small integer coefficients.[1][2] These primes allow fast modular reduction algorithms and are widely used in cryptography. They are named after Jerome Solinas.

This class of numbers encompasses a few other categories of prime numbers:

Modular Reduction Algorithm

Let be a monic polynomial of degree with coefficients in and suppose that is a Solinas prime. Given a number with up to bits, we want to find a number congruent to mod with only as many bits as – that is, with at most bits.

First, represent in base :

Next, generate a -by- matrix by stepping times the linear-feedback shift register defined over by the polynomial : starting with the -integer register , shift right one position, injecting on the left and adding (component-wise) the output value times the vector at each step (see [1] for details). Let be the integer in the th register on the th step and note that the first row of is given by . Then if we denote by the integer vector given by:

,

it can be easily checked that:

.

Thus represents an -bit integer congruent to .

For judicious choices of (again, see [1]), this algorithm involves only a relatively small number of additions and subtractions (and no divisions!), so it can be much more efficient than the naive modular reduction algorithm ().

Examples of Solinas primes

Four of the recommended primes in NIST's document "Recommended Elliptic Curves for Federal Government Use" are Solinas primes:

Curve448 uses the Solinas prime

See also

References

  1. ^ Solinas, Jerome A. (1999). Generalized Mersenne Numbers (PDF) (Technical report). Center for Applied Cryptographic Research, University of Waterloo. CORR-99-39.
  2. ^ Solinas, Jerome A. (2011). "Generalized Mersenne Prime". In Tilborg, Henk C. A. van; Jajodia, Sushil (eds.). Encyclopedia of Cryptography and Security. Springer US. pp. 509–510. doi:10.1007/978-1-4419-5906-5_32. ISBN 978-1-4419-5905-8.
  3. ^ US patent 5159632, Richard E. Crandall, "Method and apparatus for public key exchange in a cryptographic system", issued 1992-10-27, assigned to NeXT Computer, Inc.