Pseudo-Mersenne Prime

Definition

Pseudo-Mersenne Prime is a prime of the form

$$p = {2}^{m} - k,$$

where k is an integer for which

$$0 < \left \vert k\right \vert <{ \mathrm{2}}^{\lfloor m/2\rfloor }$$

If k = 1, then p is a Mersenne prime (and m must necessarily be a prime). If $k = -1$, then p is called a Fermat prime (and m must necessarily be a power of two).

Applications

Pseudo-Mersenne primes are useful in public-key cryptography because they admit fast modular reduction (modular arithmetic) similar to Mersenne primes. If n is a positive integer less than p ², then n can be written as

$$ n = u \cdot {2}^{2m} + a \cdot {2}^{m} + b, $$

where u = 0 or 1 and a and b are nonnegative integers less than 2^m. (It is only rarely true that u = 1, and never true if k > 0.) Then

$$ n \equiv u \cdot {k}^{2} + a \cdot k + b(\ {\rm mod \ }\,\,\ p).$$

Repeating this substitution a few times will yield n modulo p. This method of modular reduction requires a small number of additions and subtractions rather than the...