Abstract
The security of the RSA system with the prime pairs of some special form is investigated. A new special-purpose algorithm for factoring RSA numbers is proposed. The basic idea of the method is to factor RSA numbers by factoring a well-chosen quadratic polynomial with integral coefficients. When viewed as a general-purpose algorithm, the new algorithm has a high computational complexity. It is shown that the RSA numbern=pq can be easily factored ifp andq have the special form ofp=as+b, q=cs+d, wherea, b, c, d are relatively small numbers. Such prime pairs (p, q) are the weak keys of RSA, so when we generate RSA modulus, we should avoid using such prime pairs (p, q).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Rivest R, Shamir A, Adleman L. A Method for Obtaining Digital Signatures and Public-Key Cryptosystems[J].Communications of the ACM, 1978,21:120–126.
Okamoto T, Uchiyama S. A New Public Key Cryptosystem as Secure as Factoring [C] //Proceedings of EUROCRYPT 1998. Berlin: Springer, 1998:308–318.
Pollard J M. Theorems on Factorization and Primality Tes ting [J]//Proceedings of the Cambridge Philosophical Society. 1974,76:521–528.
Williams H C, Ap+1 Method of Factoring[J].Mathematics of Computation, 1982,39:225–234.
Boneh D, Durfee G, Howgrave-Graham N. FactoringN=p′q for Larger [C]//Proceedings of CRYPTO 1999. Berlin: Springer, 1999: 326–337.
Lenstra H W. Factoring Integers with Elliptic Curves [J].The Annal of Mathematics, 1987,126:649–673.
Lenstra A K, Lenstra H W, Manasse M S,et al. The Number Field Sieve [C]//Proceedings of the 22nd ACM Symposium on Theory of Computing. Baltimore, Maryland, May, 1990:564–572.
Pomerance C. The Quadratic Sieve Factoring Algorithm [C]//Proceedings of EUROCRYPT 1984. Berlin: Springer, 1984:169–182.
Williams H C, Schmid B. Some Remarks Concerning the MIT Public Key Cryptosystem[J].BIT, 1979,19:525–538.
Gordon J. Strong RSA Keys[J].Electronics Letters, 1984,20:514–516.
Author information
Authors and Affiliations
Corresponding author
Additional information
Foundation item: Supported by the National Natural Science Foundation of China (60473029)
Biography: WANG Baocang (1979), male, Ph.D. candidate research direction: cryptology and information security.
Rights and permissions
About this article
Cite this article
Baocang, W., Shuanggen, L. & Yupu, H. New weak keys in RSA. Wuhan Univ. J. Nat. Sci. 11, 1529–1532 (2006). https://doi.org/10.1007/BF02831813
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02831813