Abstract
We propose a cryptosystem modulo p k q based on the RSA cryptosystem. We choose an appropriate modulus p k q which resists two of the fastest factoring algorithms, namely the number field sieve and the elliptic curve method. We also apply the fast decryption algorithm modulo p k proposed in [22]. The decryption process of the proposed cryptosystems is faster than the RSA cryptosystem using Chinese remainder theorem, known as the Quisquater-Couvreur method [17]. For example, if we choose the 768-bit modulus p 2 q for 256-bit primes p and q, then the decryption process of the proposed cryptosystem is about 3 times faster than that of RSA cryptosystem using Quisquater-Couvreur method.
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References
L. M. Adleman and K. S. McCurley,“Open problems in number theoretic complexity, II” proceedings of ANTS-I, LNCS 877, (1994), pp.291–322.
G. R. Blakley and I. Borosh, “Rivest-Shamir-Adleman public key cryptosystems do not always conceal messages,” Comput. & Maths. with Appls., 5, (1979), pp.169–178.
D. Coppersmith, M. Franklin, J. Patarin and M. Reiter, “Low-exponent RSA with related messages,” Advances in Cryptology — EUROCRYPT '96, LNCS 1070, (1996), pp.1–9.
D. Coppersmith, “Finding a small root of a univariate modular equation,” Advances in Cryptology — EUROCRYPT '96, LNCS 1070, (1996), pp.155–165.
J. Cowie, B. Dodson, R. Elkenbracht-Huizing, A. K. Lenstra, P. L. Montgomery, J. Zayer; “A world wide number field sieve factoring record: on to 512 bits,” Advances in Cryptology — ASIACRYPT '96, LNCS 1163, (1996), pp.382–394.
J. Håstad, “Solving simultaneous modular equations of low degree,” SIAM Journal of Computing, 17, (1988), pp.336–341.
B. S. Kaliski Jr. and M. Robshaw, “Secure use of RSA,” CRYPTOBYTES, 1 (3), (1995), pp.7–13.
ECMNET Project; http://www.loria.fr/~zimmerma/records/ecmnet.html
D. Hühnlein, M. J. Jacobson, S. Paulus, and T. Takagi, “A cryptosystem based on non-maximal imaginary quadratic orders with fast decryption.” Advances in Cryptology — EUROCRYPT '98, LNCS 1403, (1998), pp.294–307.
H. W. Lenstra, Jr., “Factoring integers with elliptic curves”, Annals of Mathematics, 126, (1987), pp.649–673.
A. K. Lenstra and H. W. Lenstra, Jr. (Eds.), “The development of the number field sieve,” Lecture Notes in Mathematics, 1554, Springer, (1991).
U. M. Maurer; “Fast generation of prime numbers and secure public-key cryptographic parameters,” Journal of Cryptology, Vol.8, (1995), pp.123–155.
A. J. Menezes, P. C. van Oorschot and S. A. Vanstone, “Handbook of applied cryptography,” CRC Press, (1996).
T. Okamoto, “A fast signature scheme based on congruential polynomial operations,” IEEE Transactions on Information Theory, IT-36, (1990), pp.47–53.
T. Okamoto and S. Uchiyama; “A new public-key cryptosystem as secure as factoring,” Advances in Cryptology — EUROCRYPT '98, LNCS 1403, (1998), pp.308–318.
R. Peralta and E. Okamoto, “Faster factoring of integers of a special form,” IEICE Trans. Fundamentals, Vol.E79-A, No.4, (1996), pp.489–493.
J.-J. Quisquater and C. Couvreur, “Fast decipherment algorithm for RSA public-key cryptosystem,” Electronic Letters, 18, (1982), pp.905–907.
M. O. Rabin, “Digitalized signatures and public-key functions as intractable as factorization”, Technical Report No.212, MIT, Laboratory of Computer Science, Cambridge (1979), pp.1–16.
R. Rivest, A. Shamir and L. M. Adleman, “A method for obtaining digital signatures and public-key cryptosystems,” Communications of the ACM, 21(2), (1978), pp.120–126.
R. Rivest and R. D. Silverman, “Are 'strong’ primes needed for RSA,” The 1997 RSA Laboratories Seminar Series, Seminars Proceedings, (1997).
A. Shamir; “RSA for paranoids,” CryptoBytes, 1, Autumn, (1995), pp. 1–4.
T. Takagi, “Fast RSA-type cryptosystem using n-adic expansion,” Advances in Cryptology — CRYPTO '97, LNCS 1294, (1997), pp.372–384.
E. R. Verheul and H. C. A. van Tilborg, “Cryptanalysis of ‘less short’ RSA secret exponents,” Applicable Algebra in Engineering, Communication and Computing, 8, (1997), pp.425–435.
M. J. Wiener, “Cryptanalysis of short RSA secret exponents,” IEEE Transactions on Information Theory, IT-36, (1990), pp.553–558.
H. C. Williams and B. Schmid, “Some remarks concerning the M.I.T. public-key cryptosystem,” BIT 19, (1979), pp.525–538.
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Takagi, T. (1998). Fast RSA-type cryptosystem modulo p k q . In: Krawczyk, H. (eds) Advances in Cryptology — CRYPTO '98. CRYPTO 1998. Lecture Notes in Computer Science, vol 1462. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0055738
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DOI: https://doi.org/10.1007/BFb0055738
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