Modular arithmetic and finite field theory: A tutorial

The paradigm of algorithm analysis has achieved major pre-eminence in the field of symbolic and algebraic manipulation in the last few years. A major factor in its success has been the use of modular arithmetic. Application of this technique has proved effective in reducing computing times for algorithms covering a wide variety of symbolic mathematical problems. This paper is intended to review the basic theory underlying modular arithmetic. In addition, attention will be paid to certain practical problems which arise in the construction of a modular arithmetic system. A second area of importance in symbol manipulation is the theory of finite fields. A recent algorithm for polynomial factorization over a finite field has led to faster algorithms for factorization over the field of rationals. Moreover, the work in modular arithmetic often consists of manipulating elements in a finite field. Hence, this paper will outline some of the major theorems for finite fields, hoping to provide a basis from which an easier grasp of these new algorithms can be made.

[1]  E. Berlekamp.  Factoring polynomials over large finite fields* , 1970, SYMSAC '71.

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

[3]  Donald E. Knuth,et al.  The Art of Computer Programming, Volume I: Fundamental Algorithms, 2nd Edition , 1997 .

[4]  G. E. Collins.  Computing Multiplicative Inverses in GF(p) , 1969 .

[5]  L. E. Heindel,et al.  The SAC-1 Modular Arithmetic System , 1972 .

[6]  A. B. BASSET,et al.  Modern Algebra , 1905, Nature.

[7]  W. S. Brown.  On Euclid's algorithm and the computation of polynomial greatest common divisors , 1971, SYMSAC '71.

[8]  Jo Ann Howell,et al.  An algorithm for solving linear algebraic equations using residue arithmetic I , 1969 .

[9]  L. E. Dickson.  Introduction to the theory of numbers , 1933 .

[10]  Jo Ann Howell,et al.  An algorithm for solving linear algebraic equations using residue arithmetic II , 1969 .

[11]  George E. Collins,et al.  The Calculation of Multivariate Polynomial Resultants , 1971, JACM.

[12]  Richard I. Tanaka,et al.  Residue arithmetic and its applications to computer technology , 1967 .

[13]  Jo Ann Howell,et al.  Solving linear equations using residue arithmetic — Algorithm II , 1970 .

[14]  Aviezri S. Fraenkel.  The Use of Index Calculus and Mersenne Primes for the Design of a High-Speed Digital Multiplier , 1961, JACM.

[15]  G. E. Collins.  Computing time analyses for some arithmetic and algebraic algorithms , 1969 .

[16]  H. A. Feldman.  Some symbolic computations in finite fields , 1969 .

[17]  Harvey L. Garner,et al.  RESIDUE NUMBER SYSTEM ENHANCEMENTS FOR PROGRAMMABLE PROCESSORS , 2008 .

[18]  Elwyn R. Berlekamp,et al.  Algebraic coding theory , 1984, McGraw-Hill series in systems science.

[19]  I. Borosh,et al.  Exact solutions of linear equations with rational coefficients by congruence techniques , 1966 .

[20]  Donald Ervin Knuth,et al.  The Art of Computer Programming , 1968 .

[21]  E. Horowitz.  Algorithms for partial fraction decomposition and rational function integration , 1971, SYMSAC '71.