Abstract
RSA is a popular public key algorithm. Its private key operation is modular exponentiation with a composite 2k-bit modulus that is the product of two k-bit primes. Computing 2k-bit modular exponentiation can be sped up four fold with the Chinese Remainder Theorem (CRT), requiring two k-bit modular exponentiations (plus recombination). Multi-prime RSA is the generalization to the case where the modulus is a product of r ≥ 3 primes of (roughly) equal bit-length, 2k/r. Here, CRT trades 2k-bit modular exponentiation with r modular exponentiations, with 2k/r-bit moduli (plus recombination). This paper discusses multi-prime RSA with key lengths (=2k) of 2048/3072/4096 bits, and r = 3 or r = 4 primes. With these parameters, the security of multi-prime RSA is comparable to that of classical RSA. We show how to optimize multi-prime RSA on modern processors, by parallelizing r modular exponentiations and leveraging “vector” instructions, achieving performance gains of up to 5.07x.
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References
Jonsson, J., Kaliski, B.: Public-Key Cryptography Standards (PKCS) #1: RSA Cryptography Specifications Version 2.1. In RFC 3447, Network Working Group, IETF (2003). https://www.ietf.org/rfc/rfc3447.txt
Dierks, T., Rescorla, E.: The Transport Layer Security (TLS) Protocol Version 1.2. In RFC5246, Network Working Group, IETF (2008). https://www.ietf.org/rfc/rfc5246.txt
Barker, E., Roginsky, A.: Transitions: Recommendation for Transitioning the Use of Cryptographic Algorithms and Key Lengths. In NIST Special Publication 800-131A, p. 5 (2011). http://csrc.nist.gov/publications/nistpubs/800-131A/sp800-131A.pdf
NSA: Cryptography Today (Accessed September 2015). https://www.nsa.gov/ia/programs/suiteb_cryptography/index.shtml
Lenstra, A.K., Lenstra Jr., H.W. (eds.): The Development of the Number Field Sieve, vol. 1554. Lecture Notes in Mathematics. Springer, Berlin (1993)
Lenstra, A.K.: Unbelievable security matching AES security using public key systems. In: Advances in Cryptology, ASIACRYPT 2001, pp. 67–86. Springer, Heidelberg (2001)
Lesntra Jr., H.W.: Factoring integers with elliptic curves. The Annals of Mathematics 126(3), 649–673 (1987)
Thorsten, K., et al.: Factorization of a 768-Bit RSA modulus. In: Proceedings of the 30th Annual Cryptology Conference on Advances in Cryptology, CRYPTO 2010, Santa Barbara, CA, USA, 15-19 August 2010, pp. 333–350 (2010)
Langley, A.G.: Multi-prime RSA trade offs. In ImperialViolet (blog) (2011). https://www.imperialviolet.org/2011/04/09/multiprime.html
Buxton, M.: Haswell New Instruction Descriptions Now Available! Intel Corporation (2011). http://software.intel.com/en-us/blogs/2011/06/13/haswell-new-instruction-descriptions-now-available/
Gueron, S., Krasnov, V.: Software implementation of modular exponentiation, using advanced vector instructions architectures. In: Proceedings of the 4th International Conference on Arithmetic of Finite Fields, WAIFI 2012, pp. 119–135 (2012)
OpenSSL: The Open Source toolkit for SSL/TLS. http://www.openssl.org/
Gueron, S., Krasnov, V.: OpenSSL multi-prime patch. https://github.com/vkrasnov/multiprime
Gueron, S., Krasnov, V.: [PATCH] Efficient, and side channel analysis resistant 1024-bit modular exponentiation, for optimizing RSA2048 on AVX2 capable x86_64 platforms. OpenSSL patch, posted July 2012. https://rt.openssl.org/Ticket/Display.html?id=2850
Intel Corportaion: Intel® Architecture Instruction Set Extensions Programming Reference. Intel, August 2015. https://software.intel.com/sites/default/files/managed/07/b7/319433-023.pdf
Gueron, S., Krasnov, V.: [PATCH] Efficient 1024-bit and 2048-bit modular exponentiation for AVX512 capable x86_64. OpenSSL patch, posted January 2014. https://rt.openssl.org/Ticket/Display.html?id=3240
Gueron, S., Krasnov, V.: [PATCH] Fast modular exponentiation with the new VPMADD52 instructions. OpenSSL patch, posted November 2014. https://rt.openssl.org/Ticket/Display.html?id=3590
Gueron, S., Drucker, N.: [PATCH] Fast 1536-bit modular exponentiation with the new VPMADD52 instructions. OpenSSL patch, posted September 2015. https://rt.openssl.org/Ticket/Display.html?id=4032
Acknowledgements
This research was supported by the PQCRYPTO project, which was partially funded by the European Commission Horizon 2020 research Programme, grant #645622.
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Gueron, S., Krasnov, V. (2016). Speed Records for Multi-prime RSA Using AVX2 Architectures. In: Latifi, S. (eds) Information Technology: New Generations. Advances in Intelligent Systems and Computing, vol 448. Springer, Cham. https://doi.org/10.1007/978-3-319-32467-8_22
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DOI: https://doi.org/10.1007/978-3-319-32467-8_22
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