Subscribe

Issue No.04 - April (2008 vol.57)

pp: 472-480

DOI Bookmark: http://doi.ieeecomputersociety.org/10.1109/TC.2007.70822

ABSTRACT

In this contribution we introduce a low-complexity bit-parallel algorithm for computing square roots over binary extension fields. Our proposed method can be applied for any type of irreducible polynomials. We derive explicit formulae for the space and time complexities associated to the square root operator when working with binary extension fields generated using irreducible trinomials. We show that for those finite fields, it is possible to compute the square root of an arbitrary field element with equal or better hardware efficiency than the one associated to the field squaring operation. Furthermore, a practical application of the square root operator in the domain of field exponentiation computation is presented. It is shown that by using as building blocks squarers, multipliers and square root blocks, a parallel version of the classical square-and-multiply exponentiation algorithm can be obtained. A hardware implementation of that parallel version may provide a speedup of up to 50% percent when compared with the traditional version.

INDEX TERMS

Computations in finite fields, Computer arithmetic, Algorithms

CITATION

Francisco Rodr?guez Henr?quez, Guillermo Morales-Luna, Julio López, "Low-Complexity Bit-Parallel Square Root Computation over GF(2^{m}) for All Trinomials",

*IEEE Transactions on Computers*, vol.57, no. 4, pp. 472-480, April 2008, doi:10.1109/TC.2007.70822REFERENCES

- [1] “IEEE P1363: Standard Specifications for Public Key Cryptography”
IEEE Standards documents, Draft Version D18. IEEE, http://grouper.ieee.org/groups1363/, Nov. 2004.- [2] J. Daemen and V. Rijmen, “The Design of Rijndael,”
AES: The Advance Encryption Standard. Springer-Verlag, 2002.- [3] D. Hankerson, A. Menezes, and S. Vanstone,
Guide to Elliptic Cryptography. Springer-Verlag, 2004.- [4] R. Schroeppel, C. Beaver, R. Gonzales, R. Miller, and T. Draelos, “A Low-Power Design for an Elliptic Curve Digital Signature Chip,”
Proc. Fourth Int'l Workshop Cryptographic Hardware and Embedded Systems, B. Kaliski, Ç. Koç, and C. Paar, eds., pp. 366-380, Aug. 2002.- [5] G. Orlando and C. Paar, “An Efficient Squaring Architecture for ${\rm GF}(2^{m})$ and Its Applications in Cryptographic Systems,”
IEE Electronic Letters, vol. 36, no. 13, pp. 1116-1117, June 2000.- [6] D.E. Knuth,
The Art of Computer Programming, third ed. Addison-Wesley, 1997.- [7] R. Avanzi, “Another Look at Square Roots and Traces (and Quadratic Equations) in Fields of Even Characteristic,” Cryptology ePrint Archive Report 2007/103, http:/eprint.iacr.org/, 2007.
- [8] M. Scott, “Optimal Irreducible Polynomials for ${\rm GF}(2^{m})$ Arithmetic,” Cryptology ePrint Archive Report 2007/192, http:/eprint.iacr.org/, 2007.
- [9] D. Hankerson and F. Rodríguez-Henríquez, “Parallel Formulation of Scalar Multiplication on Koblitz Curves,” Report CACR 2007-17, Center for Applied Cryptographic Research, http:/www.cacr.math.uwaterloo.ca/, 2007.
- [10] F. Rodríguez-Henríquez, G. Morales-Luna, N. Saqib, and N. Cruz-Cortés, “Parallel Itoh-Tsujii Multiplicative Inversion Algorithm for a Special Class of Trinomials,” Cryptology ePrint Archive Report 2006/035, http:/eprint.iacr.org/, 2006.
- [12] R. Dahab, D. Hankerson, F. Hu, M. Long, J. Lopez, and A. Menezes, “Software Multiplication Using Normal Bases,” Technical Report CACR 2004-12, Dept. of Combinatorics and Optimization, Univ. of Waterloo, p. 21, 2004.
- [13] F. Rodríguez-Henríquez, G. Morales-Luna, and J. López-Hernández, “Low-Complexity Bit-Parallel Square Root Computation over ${\rm GF}(2^{m})$ for All Trinomials,” Cryptology ePrint Archive Report 2006/133, http://eprint.iacr.org/, 2006.
- [14] A. Menezes, P. van Oorschot, and S. Vanstone,
Handbook of Applied Cryptography. CRC Press, Oct. 1996.- [15] H. Wu, “Low Complexity Bit-Parallel Finite Field Arithmetic Using Polynomial Basis,”
Proc. First Int'l Workshop Cryptographic Hardware and Embedded Systems, Ç. Koç and C. Paar, eds., pp. 280-291, Aug. 1999.- [16] H. Wu, “On Complexity of Squaring Using Polynomial Basis in ${\rm GF}(2^{m})$ ,”
Proc. Seventh Ann. Int'l Workshop Selected Areas in Cryptography, pp. 118-129, Sept. 2000.- [17]
Recommended Elliptic Curves for Federal Government Use, special publication, Nat'l Inst. Standards and Tech nology, http://csrc.nist.gov/csrcfedstandards.html , July 1999.- [18] R. Schroeppel, “Elliptic Curve Point Ambiguity Resolution Apparatus and Method,” Int'l Application Number PCT/US00/31014, 9 Nov. 2000.
- [19] E.W. Knudsen, “Elliptic Scalar Multiplication Using Point Halving,”
Advances in Cryptology—Proc. Fifth Int'l Conf. Theory and Applications of Cryptology and Information Security, pp. 135-149, 1999. |