Abstract
Following the previous work by Bajard-Didier-Kornerup, McLaughlin, Mihailescu and Bajard-Imbert-Jullien, we present an algorithm for modular polynomial multiplication that implements the Montgomery algorithm in a residue basis; here, as in Bajard et al.’s work, the moduli are trinomials over \({\mathbb{F}}_2\). Previous work used a second residue basis to perform the final division. In this paper, we show how to keep the same residue basis, inspired by l’Hospital rule. Additionally, applying a divide-and-conquer approach to the Chinese remaindering, we obtain improved estimates on the number of additions for some useful degree ranges.
Chapter PDF
Similar content being viewed by others
References
Montgomery, P.L.: Modular multiplication without trial division. Mathematics of Computation 44, 519–521 (1985)
Koç, C.K., Acar, T.: Montgomery multiplication in GF(2k). Designs, Codes and Cryptography 14, 57–69 (1998)
von zur Gathen, J., Gerhard, J.: Modern computer algebra. Cambridge University Press, Cambridge (1999)
Schönhage, A.: Schnelle Multiplikation von Polynomen über Körpern der Charakteristik 2. Acta Informatica 7, 395–398 (1977)
Cantor, D.G.: On arithmetical algorithms over finite fields. J. Combin. Theory Ser. A 50, 285–300 (1989)
Bajard, J.C., Didier, L.S., Kornerup, P.: An RNS Montgomery modular multiplication algorithm. IEEE Transactions on Computers 47, 766–776 (1998)
McLaughlin Jr., P.: New frameworks for Montgomery’s modular multiplication method. Mathematics of Computation 73, 899–906 (2004)
Bajard, J.C., Imbert, L., Jullien, G.A.: Parallel Montgomery multiplication in GF(2k) using trinomial residue arithmetic. In: 17th IEEE Symposium on Computer Arithmetic, pp. 164–171. IEEE, Los Alamitos (2005)
Mihailescu, P.: Fast convolutions meet Montgomery. Mathematics of Computation 77, 1199–1221 (2008)
Sunar, B.: A generalized method for constructing subquadratic complexity GF(2k) multipliers. IEEE Transactions on Computers 53, 1097–1105 (2004)
Fan, H., Hasan, M.: A new approach to subquadratic space complexity parallel multipliers for extended binary fields. IEEE Transactions on Computers 56, 224–233 (2007)
Giorgi, P., Nègre, C., Plantard, T.: Subquadratic binary field multiplier in double polynomial system. In: SECRYPT 2007 (2007)
Wu, H.: Low complexity bit-parallel finite field arithmetic using polynomial basis. In: Koç, Ç.K., Paar, C. (eds.) CHES 1999. LNCS, vol. 1717, pp. 280–291. Springer, Heidelberg (1999)
Ernst, M., Jung, M., Madlener, F., Huss, S., Blümel, R.: A reconfigurable system on chip implementation for elliptic curve cryptography over GF(2n). In: Kaliski Jr., B.S., Koç, Ç.K., Paar, C. (eds.) CHES 2002. LNCS, vol. 2523, pp. 381–399. Springer, Heidelberg (2003)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Schost, É., Hariri, A. (2009). Subquadratic Polynomial Multiplication over GF(2m) Using Trinomial Bases and Chinese Remaindering. In: Avanzi, R.M., Keliher, L., Sica, F. (eds) Selected Areas in Cryptography. SAC 2008. Lecture Notes in Computer Science, vol 5381. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-04159-4_23
Download citation
DOI: https://doi.org/10.1007/978-3-642-04159-4_23
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-04158-7
Online ISBN: 978-3-642-04159-4
eBook Packages: Computer ScienceComputer Science (R0)