Abstract
New Number Field Sieves (NFS) attacks on the discrete logarithm problem have led to increasing the key size of pairing-based cryptography on most popular curves like BN. To ensure 128-bit security level, recent cost estimations between several pairing friendly elliptic curves recommend to switch for BLS24 curves. However, implementing a pairing on a BLS24 curve requires arithmetic operation over \(\mathbb {F}_{p^4}\). In this paper, we transpose previous work on multiplication over extension fields using Newton’s interpolation to construct a new formula for multiplication in \(\mathbb {F}_{p^4}\) and propose efficient hardware implementation of this operation. Our Arithmetic Logic Unit (ALU) is implemented on Kintex-7 Xilinx\(^{\text{\textregistered }}\) FPGA. The efficiency of our design in terms of \(\boldsymbol{time\times area}\) is almost 3 times better than previous specific architectures for multiplication in \(\mathbb {F}_{p^4}\). We also used this new architecture to estimate full pairing implementations on BLS24 and KSS16 for 128 and 192-bit security level. These estimations highlight that the new \(\mathbb {F}_{p^4}\) formula allows a performance increase between 5 and 10%. And especially for KSS16 which appears to provide the best performance at the 128-bit level.
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References
Barbulescu R, Duquesne S (2018) Updating key size estimations for pairings. J Cryptol. https://hal.archives-ouvertes.fr/hal-01534101
Barbulescu R, El Mrabet N, Ghammam L (2019) A taxonomy of pairings, their security, their complexity. IACR Cryptology ePrint Archive 2019:485
Barreto PSLM, Lynn B, Scott M (2002) Constructing elliptic curves with prescribed embedding degrees. In: SCN. Lecture notes in computer science, vol 2576. Springer, pp 257–267
Barreto PSLM, Naehrig M (2005) Pairing-friendly elliptic curves of prime order. In: Selected areas in cryptography. Lecture notes in computer science, vol 3897. Springer, pp 319–331
Ben-Sasson E, Chiesa A, Tromer E, Virza M (2014) Succinct non-interactive zero knowledge for a von neumann architecture. In: Proceedings of the 23rd USENIX conference on security symposium. SEC’14, USENIX Association, USA, pp 781–796
Boneh D, Franklin M (2001) Identity-based encryption from the weil pairing. In: Kilian J (ed) Advances in cryptology – CRYPTO 2001. Springer, Berlin, pp 213–229
Boneh D, Lynn B, Shacham H (2001) Short signatures from the weil pairing. In: Boyd C (ed) Advances in cryptology – ASIACRYPT 2001. Springer, Berlin, pp 514–532
El Mrabet N, Guillermin N, Ionica S (2009) A study of pairing computation for elliptic curves with embedding degree 15. IACR Cryptology ePrint Archive 2009:370
El Mrabet N, Guillevic A, Ionica S (2011) Efficient multiplication in finite field extensions of degree 5. In: Nitaj A, Pointcheval D (eds) Progress in cryptology - AFRICACRYPT 2011. Springer, Berlin, pp 188–205
Freeman D, Scott M, Teske E (2010) A taxonomy of pairing-friendly elliptic curves. J Cryptol 23(2):224–280. https://doi.org/10.1007/s00145-009-9048-z
Guillevic A (2020) A short-list of pairing-friendly curves resistant to special TNFS at the 128-bit security level. In: Public Key Cryptography (2). Lecture Notes in Computer Science, vol 12111. Springer, pp 535–564
Hankerson D, Menezes A, Scott M (2009) Software implementation of pairings. In: Identity-based cryptography, cryptology and information security series, vol 2. IOS Press, Amsterdam, pp 188–206
Huang M, Gaj K, El-Ghazawi T (2011) New hardware architectures for montgomery modular multiplication algorithm. IEEE Trans Comput 60(7):923–936. https://doi.org/10.1109/TC.2010.247
Kachisa EJ, Schaefer EF, Scott M (2008) Constructing brezing-weng pairing-friendly elliptic curves using elements in the cyclotomic field. In: Pairing. Lecture notes in computer science, vol 5209. Springer, pp 126–135
Kim T, Barbulescu R (2016) Extended tower number field sieve: a new complexity for the medium prime case. In: Robshaw M, Katz J (eds) Advances in cryptology - CRYPTO 2016. Springer, Berlin, pp 543–571
Knuth DE (1997) The art of computer programming, vol 1, 3rd edn. Fundamental algorithms. Addison Wesley Longman Publishing Co., Inc., USA
Menezes A (2009) An introduction to pairing-based cryptography. In: Recent trends in cryptography, vol 477, pp 47–65
Pollard JM (1978) Monte Carlo methods for index computation mod \(p\). Math Comput 32:918–924
Vercauteren F (2010) Optimal pairings. IEEE Trans Inf Theory 56(1):455–461. https://doi.org/10.1109/TIT.2009.2034881
Wang AT, Guo BW, Wei CJ (2019) Highly-parallel hardware implementation of optimal ate pairing over barreto-naehrig curves. Integration 64:13–21. https://doi.org/10.1016/j.vlsi.2018.04.013, http://www.sciencedirect.com/science/article/pii/S0167926018300336
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Lavice, A., El Mrabet, N., Berzati, A., Rigaud, JB. (2022). Hardware Implementation of Multiplication over Quartic Extension Fields. In: Giri, D., Raymond Choo, KK., Ponnusamy, S., Meng, W., Akleylek, S., Prasad Maity, S. (eds) Proceedings of the Seventh International Conference on Mathematics and Computing . Advances in Intelligent Systems and Computing, vol 1412. Springer, Singapore. https://doi.org/10.1007/978-981-16-6890-6_43
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DOI: https://doi.org/10.1007/978-981-16-6890-6_43
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