Prime Numbers

98 Chapter 2 NUMBER-THEORETICAL TOOLS
and special relations
−1
=(−1)
m
(m−1)/2 , (2.9)
2
=(−1)
m
(m2−1)/8 . (2.10)
Furthermore, we have the law of quadratic reciprocity for coprime m, n:
m n
n m
=(−1) (m−1)(n−1)/4 . (2.11)
Already (2.6) shows that when |a| < p, the Legendre symbol
a
p can be
computed in O ln 3 p bit operations using naive arithmetic and Algorithm
2.1.5; see Exercise 2.17. But we can do better, and we do not even need to
recognize primes.
Algorithm 2.3.5 (Calculation of Legendre/Jacobi symbol). Given positive
odd integer m, and integer a, this algorithm returns the Jacobi symbol
a
m ,which
for m an odd prime is also the Legendre symbol.
1. [Reduction loops]
a = a mod m;
t =1;
while(a = 0) {
while(a even) {
a = a/2;
if(m mod 8 ∈{3, 5}) t = −t;
}
(a, m) =(m, a); // Swap variables.
if(a ≡ m ≡ 3(mod4)) t = −t;
a = a mod m;
}
2. [Termination]
if(m == 1) return t;
return 0;
It is clear that this algorithm does not take materially longer than using
Algorithm 2.1.2 to find gcd(a, m), and so runs in O ln 2 m bit operations
when |a|