Prime Numbers

9.2 Enhancements to modular arithmetic 455
although the techniques we shall describe apply equally well to general moduli
2 q ±1, any q. That is, whether or not the modulus N has additional properties
of primality or special structure is of no consequence for the mod algorithm
of this section. A relevant result is the following:
Theorem 9.2.12 (Special-form modular arithmetic). For N =2 q +c, c an
integer, q a positive integer, and for any integer x,
x ≡ (x mod 2 q ) − c⌊x/2 q ⌋ (mod N). (9.15)
Furthermore, in the Mersenne case c = −1, multiplication by 2 k modulo N is
equivalent to left-circular shift by k bits (so if k>,
left-shift