This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Look-Up Table-Based Large Finite Field Multiplication in Memory Constrained Cryptosystems
July 2000 (vol. 49 no. 7)
pp. 749-758

Abstract—Many cryptographic systems use multiplication in the finite field GF($2^n$) for their underlying computations. In the recent past, a number of look-up table-based algorithms have been proposed for the software implementation of GF($2^n$) multiplication. Look-up table-based algorithms can provide speed advantages, but they either require a large memory space or do not fully utilize the resources of the processor on which the software is executed. In this work, an algorithm for GF($2^n$) multiplication is proposed which can alleviate this problem. In each iteration of the proposed algorithm, a group of bits of one of the input operands are examined and two look-up tables are accessed. The group size determines the table sizes, but does not affect the utilization of the processor resources. It can be used for both software and hardware realizations and is particularly suitable for implementations in memory constrained environment, such as smart cards and embedded cryptosystems.

[1] M.A. Hasan, “Look-Up Table Based Large Finite Field Multiplication in Memory Constrained Cryptosystems,” Proc. Seventh IMA Conf. Cryptography and Coding, pp. 213-221, 1999.
[2] G.B. Agnew, R.C. Mullin, and S.A. Vanstone, An Implementation of Elliptic Curve Cryptosystems over$F_{2^{155}}$ IEEE J. Selected Areas in Comm., vol. 11, no. 5, pp. 804-813, June 1993.
[3] G. Harper, A. Menezes, and S. Vanstone, “Public-Key Cryptosystems with Very Small Key Lengths,” Proc. Advances in Cryptology—EUROCRYPT '92, pp. 163-173, 1992.
[4] E. De Win, A. Bosselaers, S. Vanderberghe, P. De Gersem, and J. Vandewalle, “A Fast Software Implementation for Arithmetic Operations in$\big. {\rm GF(2^n)}\bigr.$,” Advances in Cryptology, Proc. Asiacrypt '96, K. Kim and T. Matsumoto, eds., pp. 65-76, 1996.
[5] J. Guajardo and C. Paar, “Efficient Algorithms for Elliptic Curve Cryptosystems,” Advances in Cryptology—CRYPTO 97, B.S. Kaliski, ed., pp. 342-356, 1997.
[6] Ç.K. Koç and T. Acar, “Montgomery Multplication in$\big. GF(2^k)\bigr.$,” Design, Codes, and Cryptography, vol. 14, no. 1, pp. 57-69, 1998.
[7] L. Song and K.K. Parhi, “Low Energy Digit-Serial/Parallel Finite Field Multipliers,” J. VLSI Signal Processing, vol. 19, pp. 149-166, June 1998.
[8] E.D. Mastrovito, “VLSI Architectures for Computations in Galois Fields,” PhD thesis, Dept. of Electrical Eng., Linköping Univ., Linköping, Sweden, 1991.
[9] B. Sunar and Ç.K. Koç, Mastrovito Multiplier for All Trinomials IEEE Trans. Computers, vol. 48, no. 5, pp. 522-527, May 1999.
[10] T. Itoh and S. Tsujii, “Structure of Parallel Multipliers for a Class of Finite Fields$GF(2^m)$,” Information and Computation, vol. 83, pp. 21-40, 1989.
[11] M.A. Hasan, M. Wang, and V.K. Bhargava, Modular Construction of Low Complexity Parallel Multipliers for a Class of Finite Fields$GF(2^m)$ IEEE Trans. Computers, vol. 41, no. 8, pp. 962-971, Aug. 1992.
[12] Certicom Research, “GEC1: Recommended Elliptic Curve Domain Parameters,” Standards for Efficient Cryptography Group, http:/www.secg.org, 1999.
[13] C. Paar, “A New Architecture for a Parallel Finite Field Multiplier with Low Complexity Based on Composite Fields,” IEEE Trans. Computers, vol. 45, no. 7, pp. 846-861, July 1996.

Index Terms:
Computer arithmetic, Galois (or finite) field multiplication, cryptographic systems, polynomial basis and look-up tables.
Citation:
M. Anwarul Hasan, "Look-Up Table-Based Large Finite Field Multiplication in Memory Constrained Cryptosystems," IEEE Transactions on Computers, vol. 49, no. 7, pp. 749-758, July 2000, doi:10.1109/12.863045
Usage of this product signifies your acceptance of the Terms of Use.