Efficient bit-parallel multipliers over finite fields GF(2 m ) q Chiou-Yng Lee a, * , Pramod Kumar Meher b,1 a Department of Computer Information and Network Engineering, Lunghwa University of Science and Technology, Taoyuan County 333, Taiwan, ROC b Department of Communication Systems, Institute for Infocomm Research, 1 Fusionopolis Way, Singapore 138632, Singapore article info Article history: Received 23 December 2008 Received in revised form 11 August 2009 Accepted 5 January 2010 Available online 11 March 2010 Keywords: Composite field Trinomial Pentanomial Like-polynomial Like-trinomial Permutation polynomial abstract Hardware implementation of multiplication in finite field GF(2 m ) based on sparse polyno- mials is found to be advantageous in terms of space-complexity as well as the time-com- plexity. In this paper, we present a new permutation method to construct the irreducible like-trinomials of the form (x + 1) m +(x + 1) n + 1 for the implementation of efficient bit-par- allel multipliers. For implementing the multiplications based on such polynomials, we have defined a like-polynomial basis (LPB) as an alternative to the original polynomial basis of GF(2 m ). We have shown further that the modular arithmetic for the binary field based on like-trinomials is equivalent to the arithmetic for the field based on trinomials. In order to design multipliers for composite fields, we have found another permutation polynomial to convert irreducible polynomials into like-trinomials of the forms (x 2 + x + 1) m +(x 2 + x + 1) n + 1, (x 2 + x) m +(x 2 + x) n + 1 and (x 4 + x + 1) m +(x 4 + x + 1) n + 1. The proposed bit-parallel multiplier over GF(2 4m ) is found to offer a saving of about 33% multiplications and 42.8% additions over the corresponding existing architectures. Ó 2010 Elsevier Ltd. All rights reserved. 1. Introduction Efficient design and implementation of finite field multipliers have received high attention in recent years because of their applications in elliptic curve cryptography (ECC) and error-control coding. Recently it has been shown that pairings over elliptic curves [1] are functions from two points on an elliptic curve to an element over a finite field. Tate pairing is a popular pairing and Miller algorithm [2,3] is the first efficient algorithm for computing this pairing. The calculations in this algorithm requires finite field multiplication in the extension field GF(p km ) where k is the security multiplier and its value depends on the characteristic p of the underlying finite field. A popular choice for implementing cryptographic schemes is p = 2 as it leads to more efficient implementations of the finite field arithmetic. For evaluating the Tate pairing, the exten- sion field GF(p km ) is required by km > 1000, which makes the computing systems too large and too slow. In finite field GF(2 m ), there are three major basis of representations, such as dual basis (DB), normal basis (NB) and poly- nomial basis (PB). Several PB multipliers have been suggested in the literature, followed by the first parallel PB multiplier by Bartee and Schneider [4]. A systematic method is proposed in Ref. [5] for the modified Mastrovito multiplication in the Galois fields based on general irreducible polynomials. The choice of a primitive polynomial used for constructing the field, how- ever, has a great impact on the time- and space-complexities of the resulting multiplier. For example, Sunar and Koc [6] have presented a Mastrovito multiplication algorithm, where the space-complexity of the multiplier for irreducible trinomials of 0045-7906/$ - see front matter Ó 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.compeleceng.2010.01.001 q Reviews processed and proposed for publication to the Editor-in-Chief by Associate Editor Dr. M. Lukowiak. * Corresponding author. Tel.: +886 2 82093211x7725; fax: +886 2 82094650. E-mail addresses: pp010@mail.lhu.edu.tw, lchiou@ieee.org (C.-Y. Lee), pkmeher@i2r.a-star.edu.sg (P.K. Meher). URL: http://www1.i2r.a-star.edu.sg/~pkmeher/ (P.K. Meher). 1 Tel.: +65 64082201; fax: +65 67761378. Computers and Electrical Engineering 36 (2010) 955–968 Contents lists available at ScienceDirect Computers and Electrical Engineering journal homepage: www.elsevier.com/locate/compeleceng