Abstract
In the work at hand we regard the public-key cryptosystems RSA, Dickson, LUC and Williams. The Dickson and LUC systems are, for parameter \(P=a=1\), identical, except for the fact that the LUC system reduces the degrees of the decryption functions by employing ciphertext-dependent decryption parameters. We show that also for the Dickson system with parameter \(a=-1\) the degrees of the decryption functions can be reduced. Furthermore, we emphasize on the implementability of the systems and apply for Dickson and LUC a seemingly rather unknown algorithm proposed by Montgomery to evaluate recurrences of the form \(X_{m+n}=f(X_m,X_n,X_{m-n})\). It turns out that this algorithm reduces the computational efforts of Dickson and LUC compared to commonly applied binary algorithms by about \(10\,\%\). For the Williams system we propose an algorithm which reduces its computational effort to almost one half compared to other proposed algorithms. Finally, we evaluate the computational efforts of the cryptosystems and show that the improvements proposed in this paper reduce the performance gaps between RSA and Dickson, LUC and Williams considerably.
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Notes
It is also possible to evaluate \(g_t(x,a)\) or \(V_t(x,a)\) for arbitrary parameters \(a\). In the following, however, we discuss the case where \(a=+1\).
We note that also Montgomery mentioned this possibility briefly in his unpublished manuscript [16], however, not in relation with the Williams system.
References
Aly, H., Müller, W.: Public-key cryptosystems based on Dickson polynomials. In: Proceedings of Pragocrypt, pp. 493–504 (1996)
Bellare, M., Rogaway, P.: Optimal asymmetric encryption. In: Advances in Cryptology—Eurocrypt ’94, Lecture Notes in Computer Science, vol. 950, pp. 92–111. Springer (1994)
Bernstein, D.: Pippenger’s Exponentiation Algorithm. http://cr.yp.to/papers/pippenger.pdf (2002). Accessed 16 Sept 2012
Bleichenbacher, D., Joye, M., Quisquater, J.J.: A new and optimal chosen-message attack on RSA-type cryptosystems. In: Information and Communications Security, Lecture Notes in Computer Science, vol. 1334, pp. 302–313. Springer (1997)
Boneh, D., Venkatesan, R.: Breaking RSA may not be equivalent to factoring. In: Advances in Cryptology, Eucrocrypt 98, Lecture Notes in Computer Science, vol. 1403, pp. 59–71. Springer (1998)
Intel: Intel(R) 64 and IA-32 architectures optimization reference manual. Technical report, Intel Corporation (2011)
Joye, M., Lenstra, A., Quisquater, J.J.: Protocol failures for RSA-like functions using lucas sequences and elliptic curves. In: Security Protocols, Lecture Notes in Computer Science, vol. 1189, pp. 93–100. Springer (1997)
Katzenbeisser, S.: Recent Advances in RSA Cryptography. Springer, Berlin (2001)
Knuth, D.: The Art of Computer Programming: Seminumerical Algorithms. Addison-Wesely, Reading (1998)
Koblitz, N.: A Course in Number Theory and Cryptography. Springer, Berlin (2006)
Koc, C., Acar, T.: Analyzing and comparing Montgomery multiplication algorithms. IEEE Micro. 16, 26–33 (1996)
Lehmer, D.: Euclid’s algorithm for large numbers. Am. Math. Mon. 45(4), 7 (1938)
Lidl, R., Mullen, G., Turnwald, G.: Dickson Polynomials. Pitman Monographs and Surveys in Pure and Applied Mathematics, vol. 65. Longman Scientific & Technical, London (1993)
Lidl, R., Müller, W., Oswald, A.: Some remarks on strong Fibonacci pseudoprimes. Appl. Algebra Eng. Commun. Comput. 1, 59–65 (1990)
Menezes, A., van Oorshot, P., Vanstone, S.: Handbook of Applied Cryptography. CRC Press, Boca Raton (2001)
Montgomery, P.: Evaluating Recurrences of the Form \(X_{m+n}=f(X_m, X_n, X_{m-n})\) via Lucas Chains. http://research.microsoft.com/en-us/um/people/petmon/Lucas.pdf (1983). Unpublished, Accessed 16 Sept 2012
Montgomery, P.: Modular multiplication without trial division. Math. Comput. 44, 519–521 (1985)
Müller, S.: On the security of a Williams based public key encryption scheme. In: Public Key Cryptography, Lecture Notes in Computer Science, vol. 1992, pp. 1–18. Springer (2001)
Müller, W., Nöbauer, W.: Some remarks on public-key cryptosystems. Stud. Sci. Math. Hung. 16, 71–76 (1981)
Postl, H.: Fast evaluation of Dickson polynomials. In: Contributions to General Algebra 6, pp. 223–225. Verlag Hölder-Pichler-Tempsky, Vienna (1988)
Quisquater, J.J., Couvreur, C.: Fast decipherment algorithm for RSA public-key cryptosystem. Electron. Lett. 18(21), 905–907 (1982)
Rivest, R., Shamir, A., Adleman, L.: A method for obtaining digital signatures and public-key cryptosystems. Commun. ACM 21, 120–126 (1978)
Salomaa, A.: Public-Key Cryptography, 2nd edn. Springer, Berlin (1996)
Shallit, J., Sorenson, J.: A binary algorithm for the Jacobi symbol. ACM SIGSAM Bull. 27, 4–11 (1993)
Smith., P., Lennon, M.: LUC: A new public key system. In: Proceedings of IFIP International Symposium on Computer Security, pp. 97–111 (1993)
The GNU Multiple Precision Arithmetic Library. http://gmplib.org/. Accessed 16 Sept 2012
Turnwald, G.: On Shur’s conjecture. J. Aust. Math. Soc. 58, 312–357 (1995)
Welschenbach, M.: Kryptographie in C und C++. Springer, Berlin (2001)
Williams, H.: Some public key crypto-functions as intractable as factorization. In: Proceedings of Crypto, pp. 66–70 (1984)
Williams, H.: Édouard Lucas and Primality Testing. Wiley, London (1998)
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The author would like to thank W.B. Müller for many helpful suggestions.
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Brandner, G. RSA, Dickson, LUC and Williams: a study on four polynomial-type public-key cryptosystems. AAECC 24, 17–36 (2013). https://doi.org/10.1007/s00200-012-0181-9
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DOI: https://doi.org/10.1007/s00200-012-0181-9