Fast evaluation of iterated multiplication of very large polynomials: An application to chinese remainder theory

Abstract

We consider the problem of exactly computing the number of integers in a Chinese Remainder Representation (CRR) whose pseudorank does not equal the rank. We call this number the census. The rank is key in developing CRR-intrinsic methods for comparing integers in CRR, a problem known to be notoriously difficult. Pseudorank can be computed in highly restrictive computation models. We have developed and implemented a fast, efficient algorithm for computing the census based on using a variant of the FFT to compute iterated products of polynomials of very large degree, and with arbitrary size integer coefficients. Experimental census results are tabulated. This census information makes possible a new approach to exploring the fine structure of CRR. References J. Hee. Fast convolution using polynomial transforms. http://jenshee.dk/signalprocessing/polytrans.pdf , 2004. W. Hesse. Division is in uniform tc0. In ICALP '01: Proceedings of the 28th International Colloquium on Automata, Languages and Programming , pages 104--114, London, UK, 2001. Springer--Verlag. Also available as http://people.clarkson.edu/ whesse/div.ps . D. Knuth. Seminumerical Algorithms , volume 1 of The Art of Computer Programming , section 4.3.2. Addison--Wesley, third edition, 1997. W. Kuich and A. Salomaa. Semirings,Automata,Languages . Springer-Verlag, 1986. B. Litow and D. Laing. A census algorithm for chinese remainder pseudorank with experimental results. James Cook University, School of IT Tech Report http://www.cs.jcu.edu.au/ftp/pub/techreports/2005-3.pdf , 2005. H. J. Nussbaumer. Fast Fourier Transform and Convolution Algorithms . Springer, 1982. 2nd ed. A. Salomaa and S. Soittola. Automata Theoretic Aspects of Formal Power Series . Springer--Verlag, 1978. M. P. Schutzenberger. On a theorem of {Jungen}. Proc. Am. Math. Soc. , 13:885--890, 1962. R. Tanaka and N. Szabo. Residue Arithmetic and its Application to Computer Technology . McGraw--Hill, 1968. S. P. Tarasov and M. N. Vyalyi. Semidefinite programming and arithmetic circuit evaluation. Technical report, 2005. Available as http://arxiv.org/abs/cs/0512035v1 . G. Davida A. Chiu and B. Litow. Division in logspace-uniform NC$^1$. Theoretical Informatics and Applications , 35:259--275, 2001. doi:10.1051/ita:2001119 I. M. Vinagradov. Elements of Number Theory . Dover, 1954. S. A. Cook and S. O. Aanderaa. On the minimum computation time of functions. Transactions of the American Mathematical Society , 142:291--314, Aug 1969. K. Culik and J. Kari. Handbook of Formal Languages , chapter 10, pages 599--616. Springer, 1997. G. Davida and B. Litow. Fast parallel arithmetic via modular representation. SIAM J. Comp. , 20,4:756--765, 1991. doi:10.1137/0220048