Fast integer multiplication using modular arithmetic

We give an O(N • log N • 2O(log*N)) algorithm for multiplying two N-bit integers that improves the O(N • log N • log log N) algorithm by Schönhage-Strassen. Both these algorithms use modular arithmetic. Recently, Fürer gave an O(N • log N • 2O(log*N)) algorithm which however uses arithmetic over complex numbers as opposed to modular arithmetic. In this paper, we use multivariate polynomial multiplication along with ideas from Fürer's algorithm to achieve this improvement in the modular setting. Our algorithm can also be viewed as a p-adic version of Fürer's algorithm. Thus, we show that the two seemingly different approaches to integer multiplication, modular and complex arithmetic, are similar.

[1]  W. Browder,et al.  Annals of Mathematics , 1889 .

[2]  E. C. Titchmarsh A divisor problem , 1930 .

[3]  E. Wright,et al.  An Introduction to the Theory of Numbers , 1939 .

[4]  E. T. An Introduction to the Theory of Numbers , 1946, Nature.

[5]  Anatolij A. Karatsuba,et al.  Multiplication of Multidigit Numbers on Automata , 1963 .

[6]  Gary L. Miller,et al.  Riemann's Hypothesis and tests for primality , 1975, STOC.

[7]  David Burton Elementary Number Theory , 1976 .

[8]  Matti Jutila On Linnik's constant. , 1977 .

[9]  A. Bearn A new series , 1978 .

[10]  M. Rabin Probabilistic algorithm for testing primality , 1980 .

[11]  Arnold Schönhage,et al.  Asymptotically Fast Algorithms for the Numerical Multiplication and Division of Polynomials with Complex Coeficients , 1982, EUROCAM.

[12]  I. Shafarevich,et al.  Basic notions of algebra , 1989 .

[13]  Cai Tianxin Primes representable by polynomials and the lower bound of the least primes in arithmetic progressions , 1990 .

[14]  Wang Wei On the least prime in an arithmetic progression , 1991 .

[15]  D. R. Heath-Brown Zero-free regions for Dirichlet $L$-functions, and the least prime in an arithmetic progression , 1992 .

[16]  Hongze Li Zero-free regions for Dirichlet L-functions , 1999 .

[17]  David Thomas,et al.  The Art in Computer Programming , 2001 .

[18]  Helmut Hasse,et al.  Number Theory , 2020, An Introduction to Probabilistic Number Theory.

[19]  A. Khrennikov,et al.  P-Adic Numbers and P-Adic Analysis , 2004 .

[20]  Manindra Agrawal,et al.  PRIMES is in P , 2004 .

[21]  Dock Bumpers,et al.  Volume 2 , 2005, Proceedings of the Ninth International Conference on Computer Supported Cooperative Work in Design, 2005..

[22]  Martin Fürer Faster integer multiplication , 2007, STOC '07.

[23]  Triantafyllos Xylouris On Linnik's constant , 2009, 0906.2749.

[24]  D. Passman,et al.  Character Theory of Finite Groups , 2010 .