Efficient computation of the Jacobi symbol

Abstract

The family of left-to-right GCD algorithms reduces input numbers by repeatedly subtracting the smaller number, or multiple of the smaller number, from the larger number. This paper describes how to extend any such algorithm to compute the Jacobi symbol, using a single table lookup per reduction. For both quadratic time GCD algorithms (Euclid, Lehmer) and subquadratic algorithms (Knuth, Sch\"onhage, M\"oller), the additional cost is linear, roughly one table lookup per quotient in the quotient sequence. This method was used for the 2010 rewrite of the Jacobi symbol computation in GMP.