Abstract
The generation of prime numbers underlies the use of most public-key cryptosystems, essentially as a primitive needed for the creation of RSA key pairs. Surprisingly enough, despite decades of intense mathematical studies on primality testing and an observed progressive intensification of cryptography, prime number generation algorithms remain scarcely investigated and most real-life implementations are of dramatically poor performance.
We show simple techniques that substantially improve all algorithms previously suggested or extend their capabilities. We derive fast implementations on appropriately equipped portable devices like smart-cards embedding a cryptographic coprocessor. This allows onboard generation of RSA keys featuring a very attractive (average) processing time.
Our motivation here is to help transferring this task from terminals where this operation usually took place so far, to portable devices themselves in near future for more confidence, security, and compliance with network-scaled distributed protocols such as electronic cash or mobile commerce.
Keywords
- Public-key cryptography
- RSA
- primality testing
- prime number generation
- embedded software
- efficient implementations
- cryptoprocessors
- smart cards
- PDAs
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References
ANSI X9.31. Public-key cryptography using RSA for the financial services industry. American National Standard for Financial Services, draft (1995)
Atkin, A.O.L., Morain, F.: Elliptic curves and primality proving. Mathematics of Computation 61, 29–68 (1993)
Boneh, D., Franklin, M.: Efficient generation of shared RSA keys. In: Kaliski Jr., B.S. (ed.) CRYPTO 1997. LNCS, vol. 1294, pp. 425–439. Springer, Heidelberg (1997)
Bosma, W., van der Hulst, M.-P.: Faster primality testing. In: Brassard, G. (ed.) CRYPTO 1989. LNCS, vol. 435, pp. 652–656. Springer, Heidelberg (1990)
Brandt, J., Damgård, I.: On generation of probable primes by incremental search. In: Brickell, E.F. (ed.) CRYPTO 1992. LNCS, vol. 740, pp. 358–370. Springer, Heidelberg (1993)
Brandt, J., Damgård, I., Landrock, P.: Speeding up prime number generation. In: Matsumoto, T., Imai, H., Rivest, R.L. (eds.) ASIACRYPT 1991. LNCS, vol. 739, pp. 440–449. Springer, Heidelberg (1993)
Carmichael, R.D.: Introduction to the Theory of Groups of Finite Order. Dover, Mineola (1956)
Couvreur, C., Quisquater, J.-J.: An introduction to fast generation of large prime numbers. Philips Journal of Research 37, 231–264 (1982)
Ding, C., Pei, D., Salomaa, A.: Chinese Remainder Theorem. Word Scientific, Singapore (1996)
Gallagher, P.X.: On the distribution of primes in short intervals. Mathematica 23, 4–9 (1976)
Hardy, G.H., Littlewood, J.E.: Some problems of ‘Partitio Numerorum’ III: On the expression of a number as a sum of primes. Acta Mathematica 44, 1–70 (1922)
Joye, M., Paillier, P.: Fast generation of prime numbers on portable devices: An update. Extended version of this work, Available on: http://eprint.iacr.org
Joye, M., Paillier, P., Vaudenay, S.: Efficient generation of prime numbers. In: Paar, C., Koç, Ç.K. (eds.) CHES 2000. LNCS, vol. 1965, pp. 340–354. Springer, Heidelberg (2000)
Knuth, D.E.: The Art of Computer Programming - Seminumerical Algorithms, 2nd edn., vol. 2. Addison-Wesley, Reading (1981)
Lu, C., Dos Santos, A.L.M.: A note on efficient implementation of prime generation in small portable devices. Computer Networks 49, 476–491 (2005)
Lu, C., Dos Santos, A.L.M., Pimentel, F.R.: Implementation of fast RSA key generation on smart cards. In: 17th ACM Symposium on Applied Computing, pp. 214–221. ACM Press, New York (2002)
Maurer, U.: Fast generation of prime numbers and secure public-key cryptographic parameters. Journal of Cryptology 8, 123–155 (1995)
Menezes, A.J., van Oorschot, P.C., Vanstone, S.A.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1997)
Monier, L.: Evaluation and comparison of two efficient probabilistic primality testing algorithms. Theoretical Computer Science 12, 97–108 (1980)
Pocklington, C.: The determination of the prime or composite nature of large numbers by Fermat’s theorem. In: Proc. of the Cambridge Philosophical Society, vol. 18, pp. 29–30 (1914)
Quisquater, J.-J., Couvreur, C.: Fast decipherment algorithm for RSA public-key cryptosystem. Electronics Letters 18, 905–907 (1982)
Riesel, H.: Prime Numbers and Computer Methods for Factorization, Birkhäuser (1985)
Rivest, R.L., Shamir, A., Adleman, L.M.: A method for obtaining digital signatures and public-key cryptosystems. Communications of the ACM 21, 120–126 (1978)
Silverman, R.D.: Fast generation of random, strong RSA primes. Cryptobytes 3, 9–13 (1997)
Solovay, R., Strassen, V.: A fast Monte-Carlo test for primality. SIAM Journal on Computing 6, 84–85 (1977)
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Joye, M., Paillier, P. (2006). Fast Generation of Prime Numbers on Portable Devices: An Update. In: Goubin, L., Matsui, M. (eds) Cryptographic Hardware and Embedded Systems - CHES 2006. CHES 2006. Lecture Notes in Computer Science, vol 4249. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11894063_13
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DOI: https://doi.org/10.1007/11894063_13
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