This Article 
   
 Share 
   
 Bibliographic References 
   
 Add to: 
 
Digg
Furl
Spurl
Blink
Simpy
Google
Del.icio.us
Y!MyWeb
 
 Search 
   
Concurrent Error Detection and Correction in Gaussian Normal Basis Multiplier over GF(2^m)
June 2009 (vol. 58 no. 6)
pp. 851-857
ASCII Text
x 
 
Che Wun Chiou, Chin-Cheng Chang, Chiou-Yng Lee, Ting-Wei Hou, Jim-Min Lin, "Concurrent Error Detection and Correction in Gaussian Normal Basis Multiplier over GF(2^m)," IEEE Transactions on Computers, vol. 58, no. 6, pp. 851-857, June, 2009.
 
 
BibTex
x
 
@article{ 10.1109/TC.2008.226,
author = {Che Wun Chiou and Chin-Cheng Chang and Chiou-Yng Lee and Ting-Wei Hou and Jim-Min Lin},
title = {Concurrent Error Detection and Correction in Gaussian Normal Basis Multiplier over GF(2^m)},
journal ={IEEE Transactions on Computers},
volume = {58},
number = {6},
issn = {0018-9340},
year = {2009},
pages = {851-857},
doi = {http://doi.ieeecomputersociety.org/10.1109/TC.2008.226},
publisher = {IEEE Computer Society},
address = {Los Alamitos, CA, USA},
}
 
 
RefWorks Procite/RefMan/Endnote
x 
 
TY - JOUR
JO - IEEE Transactions on Computers
TI - Concurrent Error Detection and Correction in Gaussian Normal Basis Multiplier over GF(2^m)
IS - 6
SN - 0018-9340
SP851
EP857
EPD - 851-857
A1 - Che Wun Chiou,
A1 - Chin-Cheng Chang,
A1 - Chiou-Yng Lee,
A1 - Ting-Wei Hou,
A1 - Jim-Min Lin,
PY - 2009
KW - Finite field multiplication
KW - Gaussian normal basis
KW - elliptic curve cryptosystem
KW - fault-based cryptanalysis
KW - concurrent error detection
KW - concurrent error correction.
VL - 58
JA - IEEE Transactions on Computers
ER -
 
Che Wun Chiou, Ching Yun University, Chung-Li
Chin-Cheng Chang, Feng Chia University, Taichung City
Chiou-Yng Lee, LungHwa University, Taoyuan County
Ting-Wei Hou, National Cheng Kung University, Tainan City
Jim-Min Lin, Feng Chia University, Taichung City
Fault-based cryptanalysis has been developed to effectively break both private-key and public-key cryptosystems, making robust finite field multiplication a very important research topic in recent years. However, no robust normal basis multiplier has been proposed in the literature. Therefore, this investigation presents a semisystolic Gaussian normal basis multiplier. Based on the proposed Gaussian normal basis multiplier, both concurrent error detection and correction capabilities can be easily achieved using time redundancy technology with no hardware modification.

[1] F.J. MacWilliams and N.J.A. Sloane, The Theory of Error-Correcting Codes. North Holland, 1977.
[2] R. Lidl and H. Niederreiter, Introduction to Finite Fields and Their Applications. Cambridge Univ. Press, 1994.
[3] R.E. Blahut, Fast Algorithms for Digital Signal Processing. Addison-Wesley, 1985.
[4] T.C. Bartee and D.J. Schneider, “Computation with Finite Fields,” Information and Computing, vol. 6, pp.79-98, Mar. 1963.
[5] E.D. Mastrovito, “VLSI Architectures for Multiplication over Finite Field ${\rm GF}(2^{\rm m})$ ,” Proc. Sixth Int'l Conf. Applied Algebra, Algebraic Algorithms, and Error-Correcting Codes. (AAECC-6), T. Mora, ed., pp.297-309, July 1988.
[6] Ç.K. Koç and B. Sunar, “Low-Complexity Bit-Parallel Canonical and Normal Basis Multipliers for a Class of Finite Fields,” IEEE Trans. Computers, vol. 47, no. 3, pp.353-356, Mar. 1998.
[7] T. Itoh and S. Tsujii, “Structure of Parallel Multipliers for a Class of Fields ${\rm GF}(2^{\rm m})$ ,” Information and Computation, vol. 83, pp.21-40, 1989.
[8] C.Y. Lee, E.H. Lu, and J.Y. Lee, “Bit-Parallel Systolic Multipliers for ${\rm GF}(2^{\rm m})$ Fields Defined by All-One and Equally-Spaced Polynomials,” IEEE Trans. Computers, vol. 50, no. 5, pp.385-393, May 2001.
[9] 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.856-861, July 1996.
[10] H. Wu, “Bit-Parallel Finite Field Multiplier and Squarer Using Polynomial Basis,” IEEE Trans. Computers, vol. 51, no. 7, pp.750-758, July 2002.
[11] H. Fan and M.A. Hasan, “A New Approach to Subquadratic Space Complexity Parallel Multipliers for Extended Binary Fields,” IEEE Trans. Computers, vol. 56, no. 2, pp.224-233, Feb. 2007.
[12] H. Wu, M.A. Hasan, and I.F. Blake, “New Low-Complexity Bit-Parallel Finite Field Multipliers Using Weakly Dual Bases,” IEEE Trans. Computers, vol. 47, no. 11, pp.1223-1234, Nov. 1998.
[13] S.T.J. Fenn, M. Benaissa, and D. Taylor, “${\rm GF}(2^{\rm m})$ Multiplication and Division over the Dual Basis,” IEEE Trans. Computers, vol. 45, no. 3, pp.319-327, Mar. 1996.
[14] M. Wang and I.F. Blake, “Bit Serial Multiplication in Finite Fields,” SIAM J. Discrete Math., vol. 3, no. 1, pp.140-148, Feb. 1990.
[15] E.R. Berlekamp, “Bit-Serial Reed-Solomon Encoder,” IEEE Trans. Information Theory, vol. 28, no. 6, pp.869-874, Nov. 1982.
[16] C.Y. Lee and C.W. Chiou, “Efficient Design of Low-Complexity Bit-Parallel Systolic Hankel Multipliers to Implement Multiplication in Normal and Dual Bases of ${\rm GF}(2^{\rm m})$ ,” IEICE Trans. Fundamentals of Electronics, Comm. and Computer Science, vol. E88-A, no. 11, pp.3169-3179, Nov. 2005.
[17] J.L. Massey and J.K. Omura, Computational Method and Apparatus for Finite Field Arithmetic, US patent 4,587,627, May 1986.
[18] C.C. Wang, T.K. Truong, H.M. Shao, L.J. Deutsch, J.K. Omura, and I.S. Reed, “VLSI Architectures for Computing Multiplications and Inverses in ${\rm GF}(2^{\rm m})$ ,” IEEE Trans. Computers, vol. 34, no. 8, pp.709-717, Aug. 1985.
[19] A. Reyhani-Masoleh, “Efficient Algorithms and Architectures for Field Multiplication Using Gaussian Normal Bases,” IEEE Trans. Computers, vol. 55, no. 1, pp.34-47,Jan. 2006.
[20] C.W. Chiou and C.Y. Lee, “Multiplexer-Based Double-Exponentiation for Normal Basis of GF ($2^{\rm m}$ ),” Computers and Security, vol. 24, no. 1, pp.83-86, 2005.
[21] G.B. Agnew, R.C. Mullin, I.M. Onyszchuk, and S.A. Vanstone, “An Implementation for a Fast Public-Key Cryptosystem,” J. Cryptology, vol. 3, pp.63-79, 1991.
[22] M.A. Hasan, M.Z. Wang, and V.K. Bhargava, “A Modified Massey-Omura Parallel Multiplier for a Class of Finite Fields,” IEEE Trans. Computers, vol. 42, no. 10, pp.1278-1280, Oct. 1993.
[23] S. Kwon, “A Low Complexity and a Low Latency Bit Parallel Systolic Multiplier over ${\rm GF}(2^{\rm m})$ Using an Optimal Normal Basis of Type II,” Proc. 16th IEEE Symp. Computer Arithmetic, pp.196-202, June 2003.
[24] H. Fan and M.A. Hasan, “Subquadratic Computational Complexity Schemes for Extended Binary Field Multiplication Using Optimal Normal Bases,” IEEE Trans. Computers, vol. 56, no. 10, pp.1435-1437, Oct. 2007.
[25] D.W. Ash, I.F. Blake, and S.A. Vanstone, “Low Complexity Normal Bases,” Discrete Applied Math., vol. 25, pp.191-210, 1989.
[26] ANSI X.962, Public Key Cryptography for the Financial Services Industry: The Elliptic Curve Digital Signature Algorithm (ECDSA), Am. Nat'l Standards Inst., 1999.
[27] FIPS 186-2, Digital Signature Standard (DSS), Federal Information Processing Standards Publication 186-2, Nat'l Inst. of Standards and Tech nology, 2000.
[28] IEEE Standard 1363-2000, IEEE Standard Specifications for Public-Key Cryptography, Jan. 2000.
[29] D. Boneh, R. DeMillo, and R. Lipton, “On the Importance of Checking Cryptographic Protocols for Faults,” Proc. Ann. Int'l Conf. Eurocrypt, pp.37-51, 1997.
[30] E. Biham and A. Shamir, “Differential Fault Analysis of Secret Key Cryptosystems,” Proc. Int'l Conf. Cryptology, pp.513-525, 1997.
[31] J. Kelsey, B. Schneier, D. Wagner, and C. Hall, “Side-Channel Cryptanalysis of Product Ciphers,” Proc. European Symp. Research in Computer Security (ESORICS), pp.97-110, Sept. 1998.
[32] R.J. Anderson and M. Kuhn, “Low Cost Attack on Tamper Resistant Devices,” Proc. Fifth Int'l Workshop Security Protocols, 1997.
[33] I. Biehl, B. Meyer, and V. Müller, “Differential Fault Attacks on Elliptic Curve Cryptosystems,” Proc. Int'l Conf. Cryptology 2000, pp.131-146, 2000.
[34] M. Ciet and M. Joye, “Elliptic Curve Cryptosystems in the Presence of Permanent and Transient faults,” Cryptology ePrint Archive, 2003/028, http://eprint.iacr.org/2003028.pdf, 2003.
[35] J. Blömer, M. Otto, and J.-P. Seifert, “Sign Change Fault Attacks on Elliptic Curve Cryptosystems,” Proc. Int'l Workshop Fault Diagnosis and Tolerance in Cryptography (FDTC '06), pp.36-52, 2006.
[36] R. Karri, G. Kuznetsov, and M. Goessel, “Parity-Based Concurrent Error Detection of Substitution-Permutation Network Block Ciphers,” Proc. Int'l Workshop Cryptographic Hardware and Embedded Systems (CHES '03), pp.113-124, 2003.
[37] G. Bertoni, L. Breveglieri, I. Koren, P. Maistri, and V. Piuri, “Error Analysis and Detection Procedures for a Hardware Implementation of the Advanced Encryption Standard,” IEEE Trans. Computers, vol. 52, no. 4, pp.492-505, Apr. 2003.
[38] M. Joye, A.K. Lenstra, and J.-J. Quisquater, “Chinese Remaindering Based Cryptosystems in the Presence of Faults,” J. Cryptology, vol. 12, pp.241-245, 1999.
[39] D. Boneh, R.A. DeMillo, and R.J. Lipton, “On the Importance of Eliminating Errors in Cryptographic Computations,” J. Cryptology, vol. 14, pp.101-119, 2001.
[40] S. Fenn, M. Gossel, M. Benaissa, and D. Taylor, “On-Line Error Detection for Bit-Serial Multipliers in ${\rm GF}(2^{\rm m})$ ,” J. Electronic Testing: Theory and Applications, vol. 13, pp.29-40, 1998.
[41] A. Reyhani-Masoleh and M.A. Hasan, “Error Detection in Polynomial Basis Multipliers over Binary Extension Fields,” Proc. Int'l Workshop Cryptographic Hardware and Embedded Systems (CHES '02), pp.515-528, 2003.
[42] A. Reyhani-Masoleh and M.A. Hasan, “Fault Detection Architectures for Field Multiplication Using Polynomial Bases,” IEEE Trans. Computers, vol. 55, no. 9, pp.1089-1103, Sept. 2006.
[43] C.-Y. Lee, C.W. Chiou, and J.-L. Lin, “Concurrent Error Detection in a Bit-Parallel Systolic Multiplier for Dual Basis of ${\rm GF}(2^{\rm m})$ ,” J. Electronic Testing: Theory and Applications, vol. 21, no. 5, pp.539-549, 2005.
[44] C.W. Chiou, “Concurrent Error Detection in Array Multipliers for ${\rm GF}(2^{\rm m})$ Fields,” IEE Electronics Letters, vol. 38, no. 14, pp.688-689, July 2002.
[45] C.W. Chiou, C.Y. Lee, and J.M. Lin, “Concurrent Error Detection in a Polynomial Basis Multiplier over ${\rm GF}(2^{\rm m})$ ,” J. Electronic Testing: Theory and Applications, vol. 22, no. 2, pp.143-150, Apr. 2006.
[46] C.W. Chiou, C.Y. Lee, A.W. Deng, and J.M. Lin, “Concurrent Error Detection in Montgomery Multiplication over ${\rm GF}(2^{\rm m})$ ,” IEICE Trans. Fundamentals of Electronics, Comm., and Computer Science, vol. E89-A, no. 2, pp.566-574, Feb. 2006.
[47] J.H. Patel and L.Y. Fung, “Concurrent Error Detection in ALU's by Recomputing with Shifted Operands,” IEEE Trans. Computers, vol. 31, no. 7, pp.589-595, July 1982.
[48] J.H. Patel and L.Y. Fung, “Concurrent Error Detection in Multiply and Divide Arrays,” IEEE Trans. Computers, vol. 32, no. 4, pp.417-422, Apr. 1983.
[49] A.J. Menezes, Applications of Finite Fields. Kluwer Academic Publications, 1993.
[50] I.F. Blake, R.M. Roth, and G. Seroussi, “Efficient Arithmetic in ${\rm GF}(2^{\rm m})$ through Palindromic Representation,” Technical Report HPL-98-134, http://www.hpl.hp.com/techreports/98HPL-98-134.html , 1998.
[51] H.Y. Kim, J.Y. Park, J.H. Cheon, J.H. Park, J.H. Kim, and S.G. Hahn, “Fast Elliptic Curve Point Counting Using Gaussian Normal Basis,” Proc. Ann. Int'l Conf. EUROCRYPT 2002, pp.14-28, 2002.
[52] N. Weste and K. Eshraghian, Principles of CMOS VLSI Design: A System Perspective. Addison-Wesley, 1985.
[53] M74HC86, Quad Exclusive OR Gate, STMicroelectronics, http://www.st.com/stonline/books/pdf/docs 2006.pdf, 2001.
[54] M74HC08, Quad 2-Input AND Gate, STMicroelectronics, http://www.st.com/stonline/books/pdf/docs 1885.pdf, 2001.
[55] M74HC279, Quad ${\rm\bar S}-{\rm\bar R}$ Latch, STMicroelectronics, http://www.st.com/stonline/books/pdf/docs 1937.pdf, 2001.
[56] M74HC32: Quad 2-Input OR Gate, STMicroelectronics, http://www.st.com/stonline/books/pdf/docs 1944.pdf, 2001.

Index Terms:
Finite field multiplication, Gaussian normal basis, elliptic curve cryptosystem, fault-based cryptanalysis, concurrent error detection, concurrent error correction.
Citation:
Che Wun Chiou, Chin-Cheng Chang, Chiou-Yng Lee, Ting-Wei Hou, Jim-Min Lin, "Concurrent Error Detection and Correction in Gaussian Normal Basis Multiplier over GF(2^m)," IEEE Transactions on Computers, vol. 58, no. 6, pp. 851-857, June 2009, doi:10.1109/TC.2008.226
Usage of this product signifies your acceptance of the Terms of Use.