Prime Numbers
Prime Numbers 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
456 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC Algorithm 9.2.13 (Fast mod operation for special-form moduli). Assume modulus N =2 q + c, with B(|c|) 0. The method is generally more efficient for smaller |c|. 1. [Perform reduction] while(B(x) >q) { y = x>>q; // Right-shift does ⌊x/2 q ⌋. x = x − (y
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456 Chapter 9 FAST ALGORITHMS FOR LARGE-INTEGER ARITHMETIC<br />
Algorithm 9.2.13 (Fast mod operation for special-form moduli). Assume<br />
modulus N =2 q + c, with B(|c|) 0. The method is generally more efficient for smaller |c|.<br />
1. [Perform reduction]<br />
while(B(x) >q) {<br />
y = x>>q; // Right-shift does ⌊x/2 q ⌋.<br />
x = x − (y
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