On the Jacobi Symbol

The Jacobi Symbol

  is only defined when the bottom is an odd prime. You can extend the definition to allow an odd positive number on the bottom using the Jacobi symbol. Most of the properties of Legendre symbols go through for Jacobi symbols, which makes Jacobi symbols very convenient for computation. We’ll see, however, that there is a price to pay for the greater generality: Euler’s formula no longer works,...

Efficient Algorithms for Computing the Jacobi Symbol

We present two new algorithms for computing the Jacobi Symbol: the right-shift and left-shift k-ary algorithms. For inputs of at most n bits in length, both algorithms take O(n 2 = log n) time and O(n) space. This is asymptotically faster than the traditional algorithm, which is based in Euclid's algorithm for computing greatest common divisors. In practice, we found our new algorithms to be ab...

Kronecker-Jacobi symbol and Quadratic Reciprocity

An O(M(n) logn) Algorithm for the Jacobi Symbol

The best known algorithm to compute the Jacobi symbol of two n-bit integers runs in time O(M(n) log n), using Schönhage’s fast continued fraction algorithm combined with an identity due to Gauss. We give a different O(M(n) logn) algorithm based on the binary recursive gcd algorithm of Stehlé and Zimmermann. Our implementation — which to our knowledge is the first to run in time O(M(n) logn) — i...

On the Optimality of the Binary Algorithm for the Jacobi Symbol

We establish lower bounds on the complexity of computing the following number-theoretic functions and relations from piecewise linear primitives: (i) the Legendre and Jacobi symbols, (ii) pseudoprimality, and (iii) modular exponentiation. As a corollary to the lower bound obtained for (i), an algorithm of Shallit and Sorenson is optimal (up to a multiplicative constant).