Abstract
This paper deals with the problem of generating RSA moduli having a predetermined set of bits. It would appear to be of practical interest if one could construct their modulus so that, for example, some of the bits are the ASCII representation of their identification information (i.e., name, address, etc.). This could lead to a savings in both bandwidth for data transmission and storage. A theoretical question which arises in connection with this is to determine the maximum number of bits which can be specified so that the modulus can be determined in polynomial time and, of course, security is maintained.
Article PDF
Similar content being viewed by others
References
G. B. Agnew, R. C. Mullin, and S. A. Vanstone, An implementation of an elliptic curve cryptosystem over F2 155, IEEE Journal on Selected Areas in Communications, Vol. 6, 1993, pp. 3–13.
R. Anderson, A practical RSA trapdoor, Electronics Letters, Vol. 29, No. 11, 1993, p. 995.
P. Beauchemin, G. Brassard, C. Crépeau, C. Goutier, and C. Pomerance, The generation of random numbers that are probably prime, Journal of Cryptology, Vol. 1, No. 2, 1988, pp. 53–64.
D. M. Bressoud, Factorization and Primality Testing, Berlin: Springer-Verlag, 1989.
E. R. Canfield, P. Erdös, and C. Pomerance, On a problem of Oppenheim concerning “Factorisatio Numerorum,” Journal of Number Theory, Vol. 17, No. 1, Aug. 1983, pp. 1–28.
R. D. Carmichael, On composite numbers P which satisfy the Fermat congruence a P−1 ≡ 1 (mod P), American Mathematical Monthly, Vol. 19, 1912, pp. 22–27.
H. Cohen, A Course in Computational Algebraic Number Theory, Berlin: Springer-Verlag, 1993.
W. Diffie and M. E. Hellman, New directions in cryptography, IEEE Transactions on Information Theory, Vol. 22, No. 6, 1976, pp. 644–654.
T. El Gamal, A public key cryptosystem and a signature scheme based on discrete logarithms, IEEE Transactions on Information Theory, Vol. 31, No. 4, 1985, pp. 469–472.
G. Harper, A. J. Menezes, and S. A. Vanstone, Public key cryptosystems with very small key size, Advances in Cryptology—EUROCRYPT '92, Lecture Notes in Computer Science, Vol. 658, Berlin: Springer-Verlag, 1993, pp. 163–173.
B. S. Kaliski Jr., Anderson's RSA trapdoor can be broken (preprint).
D. E. Knuth and L. T. Pardo, Analysis of a simple factorization algorithm, Theoretical Computer Science, Vol. 3, 1976, pp. 321–348.
E. Kranakis, Primality and Cryptography, Stuttgart: Teubner; New York: Wiley, 1986.
Ch. J. de la Vallée Poussin, Démonstration simplifiée du théorème de Dirichlet sur la progression arithmétique, Mémoires Couronnés et autres Mémoires (80 Ed.), Vol. 53, 1895–96, No. 3, p. 59.
A. K. Lenstra, H. W. Lenstra Jr., M. S. Manasse, and J. M. Pollard, The number field sieve, Proceedings of the 22nd ACM Symposium on Theory of Computing, pp. 464–572, 1990.
H. W. Lenstra, Jr., Factoring with elliptic curves, Annals of Mathematics, Vol. 126, 1987, pp. 649–673.
U. M. Maurer, Factoring with an oracle, Advances in Cryptology—EUROCRYPT '92, Lecture Notes in Computer Science, Vol. 658, Berlin: Springer-Verlag, pp. 429–436.
U. M. Maurer, Fast generation of prime numbers and secure public-key cryptographic parameters, Journal of Cryptology (to appear).
A. Menezes and S. Vanstone, Elliptic curve cryptosystems and their implementation, Journal of Cryptology, Vol. 6, No. 4, 1994, pp. 209–224.
G. L. Miller, Riemann's hypothesis and tests for primality, Journal of Computer and System Sciences, Vol. 13, No. 3, Dec. 1976, pp. 300–317.
P. C. van Oorschot, A comparison of practical public key cryptosystems based on integer factorization and discrete logarithms, in Contemporary Cryptology—The Science of Information Integrity, G. J. Simmons, ed., New York: IEEE Press, 1991.
H. C. Pocklington, The determination of the prime or composite nature of large numbers by Fermat's theorem, Proceeding of the Cambridge Philosophical Society, Vol. 18, 1914–1916, pp. 29–30.
J. M. Pollard, Theorems on factorization and primality testing, Proceedings of the Cambridge Philosophical Society, Vol. 76, 1974, pp. 521–528.
C. Pomerance, Analysis and comparison of some integer factoring algorithms, in Computational Methods in Number Theory, H. W. Lenstra, Jr., and R. Tijdeman, eds., Mathematical Centre Tracts, Vol. 154, Amsterdam: Mathematisch Centrum, 1982, pp. 89–139.
M. O. Rabin, Probabilistic algorithms for testing primality, Journal of Number Theory, Vol. 12, 1980, pp. 128–138.
R. L. Rivest and A. Shamir, Efficient factoring based on partial information, Advances in Cryptology—EUROCRYPT '85, Lecture Notes in Computer Science, Vol. 219, Berlin: Springer-Verlag, 1986, pp. 31–34.
R. L. Rivest, A. Shamir, and L. Adleman, A method for obtaining digital signatures and public-key cryptosystems, Communications of the ACM, Vol. 21, No. 2, 1978, pp. 120–126.
Author information
Authors and Affiliations
Additional information
Communicated by Johannes Buchmann
Rights and permissions
About this article
Cite this article
Vanstone, S.A., Zuccherato, R.J. Short RSA keys and their generation. J. Cryptology 8, 101–114 (1995). https://doi.org/10.1007/BF00190758
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00190758