Advertisement

Polynomial multiplication over binary finite fields: new upper bounds

  • Alessandro De PiccoliEmail author
  • Andrea ViscontiEmail author
  • Ottavio Giulio RizzoEmail author
Regular Paper
  • 4 Downloads

Abstract

When implementing a cryptographic algorithm, efficient operations have high relevance both in hardware and in software. Since a number of operations can be performed via polynomial multiplication, the arithmetic of polynomials over finite fields plays a key role in real-life implementations—e.g., accelerating cryptographic and cryptanalytic software (pre- and post-quantum) (Chou in Accelerating pre-and post-quantum cryptography. Ph.D. thesis, Technische Universiteit Eindhoven, 2016). One of the most interesting papers that addressed the problem has been published in 2009. In Bernstein (in: Halevi (ed) Advances in Cryptology—CRYPTO 2009: 29th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 16–20, 2009. Proceedings, pp 317–336. Springer Berlin Heidelberg, Berlin, Heidelberg, 2009), Bernstein suggests to split polynomials into parts and presents a new recursive multiplication technique which is faster than those commonly used. In order to further reduce the number of bit operations (Bernstein in High-speed cryptography in characteristic 2: minimum number of bit operations for multiplication, 2009. http://binary.cr.yp.to/m.html) required to multiply n-bit polynomials, researchers adopt different approaches. In CMT: Circuit minimization work. http://www.cs.yale.edu/homes/peralta/CircuitStuff/CMT.html a greedy heuristic has been applied to linear straight-line sequences listed in Bernstein (High-speed cryptography in characteristic 2: minimum number of bit operations for multiplication, 2009. http://binary.cr.yp.to/m.html). In 2013, D’angella et al. (Applied computing conference, 2013. ACC’13. WEAS. pp. 31–37. WEAS, 2013) skip some redundant operations of the multiplication algorithms described in Bernstein (in: Halevi (ed) Advances in Cryptology—CRYPTO 2009: 29th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 16–20, 2009. Proceedings, pp 317–336. Springer Berlin Heidelberg, Berlin, Heidelberg, 2009). In 2015, Cenk et al. (J Cryptogr Eng 5(4):289–303, 2015) suggest new multiplication algorithms. In this paper, (a) we present a “k-1”-level recursion algorithm that can be used to reduce the effective number of bit operations required to multiply n-bit polynomials, and (b) we use algebraic extensions of \(\mathbb {F}_2\) combined with Lagrange interpolation to improve the asymptotic complexity.

Keywords

Polynomial multiplication Karatsuba Two-level seven-way recursion algorithm Binary fields Fast software implementations 

Notes

References

  1. 1.
    Abdulrahman, E.A.H., Reyhani-Masoleh, A.: High-speed hybrid-double multiplication architectures using new serial-out bit-level mastrovito multipliers. IEEE Trans. Comput. 65(6), 1734–1747 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agnew, G.B., Beth, T., Mullin, R.C., Vanstone, S.A.: Arithmetic operations in \(GF(2^m)\). J. Cryptol. 6(1), 3–13 (1993)CrossRefzbMATHGoogle Scholar
  3. 3.
    Berlekamp, E.R.: Algebraic Coding Theory, vol. 111. McGraw-Hill, New York (1968)zbMATHGoogle Scholar
  4. 4.
    Bernstein, D.J.: Curve25519: new diffie-hellman speed records. In: Yung, M., Dodis, Y., Kiayias, A., Malkin, T. (eds.) Public Key Cryptography—PKC 2006: 9th International Conference on Theory and Practice in Public-Key Cryptography, New York, NY, USA, April 24–26, 2006. Proceedings, pp. 207–228. Springer Berlin Heidelberg, Berlin, Heidelberg (2006)Google Scholar
  5. 5.
    Bernstein, D.J.: Batch binary edwards. In: Halevi, S. (ed.) Advances in Cryptology—CRYPTO 2009: 29th Annual International Cryptology Conference, Santa Barbara, CA, USA, August 16–20, 2009. Proceedings, pp. 317–336. Springer Berlin Heidelberg, Berlin, Heidelberg (2009)Google Scholar
  6. 6.
    Bernstein, D.J.: High-speed cryptography in characteristic 2: minimum number of bit operations for multiplication (2009). http://binary.cr.yp.to/m.html
  7. 7.
    Bernstein, D.J., Duif, N., Lange, T., Schwabe, P., Yang, B.Y.: High-speed high-security signatures. J. Cryptogr. Eng. 2(2), 77–89 (2012)CrossRefzbMATHGoogle Scholar
  8. 8.
    Blahut, R.E.: Theory and Practice of Error Control Codes, vol. 126. Addison-Wesley, Reading (1983)zbMATHGoogle Scholar
  9. 9.
    Blahut, R.E.: Fast Algorithms for Digital Signal Processing. Addison-Wesley Longman Publishing Co., Inc., Reading (1985)zbMATHGoogle Scholar
  10. 10.
    Blake, I., Seroussi, G., Smart, N.: Elliptic Curves in Cryptography, vol. 265. Cambridge University Press, Cambridge (1999)CrossRefzbMATHGoogle Scholar
  11. 11.
    Boyar, J., Peralta, R.: A new combinational logic minimization technique with applications to cryptology. In: Festa, P. (ed.) Experimental Algorithms: 9th International Symposium, SEA 2010, Ischia Island, Naples, Italy, May 20–22, 2010. Proceedings, pp. 178–189. Springer Berlin Heidelberg, Berlin, Heidelberg (2010)Google Scholar
  12. 12.
    Cenk, M., Hasan, M.A.: Some new results on binary polynomial multiplication. J. Cryptogr. Eng. 5(4), 289–303 (2015)CrossRefGoogle Scholar
  13. 13.
    Cenk, M., Negre, C., Hasan, M.A.: Improved three-way split formulas for binary polynomial multiplication. In: Selected Areas in cryptography, pp. 384–398. Springer (2011)Google Scholar
  14. 14.
    Cenk, M., Negre, C., Hasan, M.A.: Improved three-way split formulas for binary polynomial and toeplitz matrix vector products. IEEE Trans. Comput. 62(7), 1345–1361 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Chakraborty, D., Mancillas-López, C., Rodriguez-Henriquez, F., Sarkar, P.: Efficient hardware implementations of brw polynomials and tweakable enciphering schemes. IEEE Trans. Comput. 62(2), 279–294 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Chang, N.S., Kim, C.H., Park, Y.H., Lim, J.: A non-redundant and efficient architecture for Karatsuba-Ofman algorithm. In: Information Security, 8th International Conference, ISC 2005, Singapore, pp. 288–299, Springer (2005)Google Scholar
  17. 17.
    Chou, T.: Accelerating pre-and post-quantum cryptography. Ph.D. thesis, Technische Universiteit Eindhoven (2016)Google Scholar
  18. 18.
  19. 19.
    Cook, S.A.: On the minimum computation time of functions. Ph.D. thesis, Harvard University (1966)Google Scholar
  20. 20.
    D’angella, D., Schiavo, C.V., Visconti, A.: Tight upper bounds for polynomial multiplication. In: Applied Computing Conference, 2013. ACC’13. WEAS. pp. 31–37. WEAS (2013)Google Scholar
  21. 21.
    Fan, H., Sun, J., Gu, M., Lam, K.Y.: Overlap-free Karatsuba-Ofman polynomial multiplication algorithms. IET Inf. Secur. 4(1), 8–14 (2010)CrossRefGoogle Scholar
  22. 22.
    Find, M.G., Peralta, R.: Better circuits for binary polynomial multiplication. IEEE Trans. Comput. 68(4), 624–630 (2019)CrossRefGoogle Scholar
  23. 23.
    von zur Gathen, J., Shokrollahi, J.: Fast arithmetic for polynomials over \(F_2\) in hardware. In: Information Theory Workshop, 2006. ITW’06 Punta del Este. IEEE. pp. 107–111. IEEE (2006)Google Scholar
  24. 24.
    Homma, N., Saito, K., Aoki, T.: Toward formal design of practical cryptographic hardware based on galois field arithmetic. IEEE Trans. Comput. 63(10), 2604–2613 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Imana, J.L.: Fast bit-parallel binary multipliers based on type-i pentanomials. IEEE Trans. Comput. PP(99), 1–1 (2017)Google Scholar
  26. 26.
    Karatsuba, A., Ofman, Y.: Multiplication of multidigit numbers on automata. Soviet Phys. Doklady 7, 595–596 (1963)Google Scholar
  27. 27.
    Li, Y., Ma, X., Zhang, Y., Qi, C.: Mastrovito form of non-recursive Karatsuba multiplier for all trinomials. IEEE Trans. Comput. 66(9), 1573–1584 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    McClellen, J.H., Rader, C.M.: Number Theory in Digital Signal Processing. Prentice Hall Professional Technical Reference, Englewood Cliffs (1979)Google Scholar
  29. 29.
    McEliece, R.J.: Finite Fields for Computer Scientists and Engineers, vol. 23. Kluwer Academic Publishers Boston, Boston (1987)zbMATHGoogle Scholar
  30. 30.
    Menezes, A.J., Van Oorschot, P.C., Vanstone, S.A.: Handbook of Applied Cryptography. CRC Press, Boca Raton (1997)zbMATHGoogle Scholar
  31. 31.
    Orellana, R.: Course notes in discrete mathematics in computer science. https://math.dartmouth.edu/archive/m19w03/public_html/book.html
  32. 32.
    Paar, C.: Optimized arithmetic for Reed-Solomon encoders. In: Proceedings of IEEE International Symposium on Information Theory, p. 250 (1997)Google Scholar
  33. 33.
    Peter, S., Langendorfer, P.: An efficient polynomial multiplier in \(GF(2^m)\) and its application to ECC designs. In: Design, Automation & Test in Europe Conference & Exhibition, 2007. DATE’07. pp. 1–6. IEEE (2007)Google Scholar
  34. 34.
    Rodrıguez-Henrıquez, F., Koç, Ç.: On fully parallel Karatsuba multipliers for \(GF(2^m)\). In: International Conference on Computer Science and Technology (CST 2003), Cancun, Mexico, pp. 405–410 (2003)Google Scholar
  35. 35.
    Rotman, J.J.: An Introduction to the Theory of Groups, vol. 148. Springer, New York (2012)Google Scholar
  36. 36.
    Schönhage, D.D.A., Strassen, V.: Schnelle Multiplikation grosser Zahlen. Computing 7(3–4), 281–292 (1971)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Toom, A.L.: The complexity of a scheme of functional elements realizing the multiplication of integers. Soviet Math. Doklady 3, 714–716 (1963)zbMATHGoogle Scholar
  38. 38.
    Visconti, A., Schiavo, C.V., Peralta, R.: Improved upper bounds for the expected circuit complexity of dense systems of linear equations over GF(2). Inf. Process. Lett. 137, 1–5 (2018)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Computer Science “Giovanni Degli Antoni”Università degli Studi di MilanoMilanItaly
  2. 2.Department of Mathematics “Federigo Enriques”Università degli Studi di MilanoMilanItaly

Personalised recommendations