Factorization Using the Quadratic Sieve Algorithm
Since the cryptosecurity of the RSA two key cryptoalgorithm is no greater than the difficulty of factoring the modulus (product of two secret primes), a code that implements the Quadratic Sieve factorization algorithm on the CRAY I computer has been developed at the Sandia National Laboratories to determine as sharply as possible the current state-of-the-art in factoring. Because all viable attacks on RSA thus far proposed are equivalent to factorization of the modulus, sharper bounds on the computational difficulty of factoring permit improved estimates for the size of RSA parameters needed for given levels of cryptosecurity.
Analysis of the Quadratic Sieve indicates that it may be faster than any previously published general purpose algorithm for factoring large integers. The high speed of the CRAY I coupled with the capability of the CRAY to pipeline certain vectorized operations make this algorithm (and code) the front runner in current factoring techniques.
KeywordsGaussian Elimination Sandia National Laboratory Prime Base Quadratic Residue Single Precision
Unable to display preview. Download preview PDF.
- 1.[BLSTW]J. Brillhart, D. H. Lehmer, J. L. Selfridge, B. Tuckerman, S. S. Wagstaff, Jr., Factorizations of b n ±1 up to High Powers, American Math. Soc., 1983.Google Scholar
- 2.[B]D. M. Burton, Elementary Number Theory, Allyn and Bacon, Inc., Boston, Mass., 1976.Google Scholar
- 3.[K]D. E. Knuth, The Art of Computer Programming, Vol. 2, Semi-Numerical Algorithms, 2nd Edition, AddisonWesley, Reading, Mass., 1981.Google Scholar
- 4.[LV]W. J. LeVeque, Studies in Number Theory, MAA Studies in Mathematics, Vol. 6, 1969.Google Scholar
- 5.[MB]M. A. Morrison, J. Brillhart, “A method of factoring and the factorization of F7,” Math. Comp. 29, 1975, 183–205.Google Scholar
- 6.[PW]D. Parkinson, M. C. Wunderlich, “A memory efficient algorithm for Gaussian elimination over GF(2) on parallel computers,” in preparation.Google Scholar
- 7.[P1]C. Pomerance, “Analysis and comparison of some integer factoring algorithms,” Number Theory and Computers (H. W. Lenstra, Jr., and R. Tijdeman, eds.) Math. Centrum Tracts, Number 154, Part I.Google Scholar
- 8.[P2]C. Pomerance, Personal communication, May 26, 1983.Google Scholar