The best known algorithm to compute the Jacobi symbol of two n-bit integers
runs in time O(M(n) log n), using Sch\"onhage's fast continued fraction
algorithm combined with an identity due to Gauss. We give a different O(M(n)
log n) algorithm based on the binary recursive gcd algorithm of Stehl\'e and
Zimmermann. Our implementation - which to our knowledge is the first to run in
time O(M(n) log n) - is faster than GMP's quadratic implementation for inputs
larger than about 10000 decimal digits.Comment: Submitted to ANTS IX (Nancy, July 2010
