An O(M(n) log n) algorithm for the Jacobi symbol

Richard Brent 1 Paul Zimmermann 2
2 CARAMEL - Cryptology, Arithmetic: Hardware and Software
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : 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 $O(M(n)\log n)$ 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 $O(M(n)\log n)$ --- is faster than GMP's quadratic implementation for inputs larger than about $10000$ decimal digits.
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https://hal.inria.fr/inria-00447968
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Submitted on : Monday, January 18, 2010 - 8:51:26 AM
Last modification on : Tuesday, December 18, 2018 - 4:18:25 PM

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Richard Brent, Paul Zimmermann. An O(M(n) log n) algorithm for the Jacobi symbol. 9th Algorithmic Number Theory Symposium - ANTS IX, Jul 2010, Nancy, France. pp.83-95, ⟨10.1007/978-3-642-14518-6_10⟩. ⟨inria-00447968⟩

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