Architectural review of polynomial bases finite field multipliers over GF(2m)

In elliptic curve cryptography (ECC), hardware architectures of finite field (FF) multipliers are frequently proposed for polynomial as well as for normal bases representations over GF(2m). Although the polynomial bases provide efficient FF multiplication as compared to normal bases, the performance of the entire elliptic cryptosystem mainly depends upon its FF multiplier. Consequently, this paper provides a comparative overview of the recent hardware architectures of FF multipliers for polynomial bases over GF(2m). This is achieved by classifying the most recent state-of-the-art research practices into three categories: bit-serial, bit-parallel and digit-serial multipliers. The comparison of multiple techniques in this article enables the designer to select a suitable multiplier according to different application requirements such as high speed/performance, constrained environments and high throughput/area applications.

[1]  Jean-Pierre Deschamps,et al.  Efficient Elliptic Curve Point Multiplication Using Digit-Serial Binary Field Operations , 2013, IEEE Transactions on Industrial Electronics.

[2]  Shuguo Li,et al.  High-Performance Pipelined Architecture of Elliptic Curve Scalar Multiplication Over GF( ${2}^{m}$ ) , 2016, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[3]  Indranil Sengupta,et al.  High-Speed Unified Elliptic Curve Cryptosystem on FPGAs Using Binary Huff Curves , 2012, VDAT.

[4]  Ludovic Apvrille,et al.  Evaluation of ASIPs Design with LISATek , 2008, SAMOS.

[5]  Ludovic Apvrille,et al.  Application Specific Processors for Multimedia Applications , 2008, 2008 11th IEEE International Conference on Computational Science and Engineering.

[6]  Chiou-Yng Lee,et al.  Scalable Gaussian Normal Basis Multipliers over GF(2m) Using Hankel Matrix-Vector Representation , 2012, Journal of Signal Processing Systems.

[7]  N. Koblitz.  Elliptic curve cryptosystems , 1987 .

[8]  Arash Reyhani-Masoleh,et al.  A New Bit-Serial Architecture for Field Multiplication Using Polynomial Bases , 2008, CHES.

[9]  Yu Zhang,et al.  A high performance ECC hardware implementation with instruction-level parallelism over GF(2163) , 2010, Microprocess. Microsystems.

[10]  B. Preneel,et al.  Serial multiplier architectures over GF(2/sup n/) for elliptic curve cryptosystems , 2004, Proceedings of the 12th IEEE Mediterranean Electrotechnical Conference (IEEE Cat. No.04CH37521).

[11]  Muhammad Rashid,et al.  Comparative analysis of flexible cryptographic implementations , 2016, 2016 11th International Symposium on Reconfigurable Communication-centric Systems-on-Chip (ReCoSoC).

[12]  Nele Mentens,et al.  Side-channel evaluation of FPGA implementations of binary Edwards curves , 2010, 2010 17th IEEE International Conference on Electronics, Circuits and Systems.

[13]  Xuecheng Zou,et al.  Design of highly efficient elliptic curve crypto-processor with two multiplications over GF(2163) , 2009 .

[14]  M. Anwar Hasan,et al.  Low complexity word-level sequential normal basis multipliers , 2005, IEEE Transactions on Computers.

[15]  Ingrid Verbauwhede,et al.  On the Implementation of Unified Arithmetic on Binary Huff Curves , 2013, CHES.

[16]  J. Grossschadl,et al.  A low-power bit-serial multiplier for finite fields GF(2/sup m/) , 2001, ISCAS 2001. The 2001 IEEE International Symposium on Circuits and Systems (Cat. No.01CH37196).

[17]  Berk Sunar,et al.  An Efficient Optimal Normal Basis Type II Multiplier , 2001, IEEE Trans. Computers.

[18]  Jongin Lim,et al.  Modified Sequential Normal Basis Multipliers for Type II Optimal Normal Bases , 2005, ICCSA.

[19]  José Luis Imaña.  Low Latency $GF(2^{m})$ Polynomial Basis Multiplier , 2011, IEEE Transactions on Circuits and Systems I: Regular Papers.

[20]  Mehdi Tibouchi,et al.  Huff's Model for Elliptic Curves , 2010, ANTS.

[21]  Victor S. Miller,et al.  Use of Elliptic Curves in Cryptography , 1985, CRYPTO.

[22]  Belgacem Bouallegue,et al.  Area-Time Efficient Hardware Implementation of Elliptic Curve Cryptosystem , 2015, IACR Cryptol. ePrint Arch..

[23]  Mohammed Benaissa,et al.  Throughput/Area-efficient ECC Processor Using Montgomery Point Multiplication on FPGA , 2015, IEEE Transactions on Circuits and Systems II: Express Briefs.

[24]  Chester Rebeiro,et al.  Pushing the Limits of High-Speed GF(2 m ) Elliptic Curve Scalar Multiplication on FPGAs , 2012, CHES.

[25]  Antonio García,et al.  Minimum-clock-cycle Itoh-Tsujii algorithm hardware implementation for cryptography applications over GF(2m) fields , 2012 .

[26]  Vijay K. Bhargava,et al.  Division and bit-serial multiplication over GF(qm) , 1992 .

[27]  Arash Reyhani-Masoleh,et al.  New Bit-Level Serial GF (2^m) Multiplication Using Polynomial Basis , 2015, 2015 IEEE 22nd Symposium on Computer Arithmetic.

[28]  Hui Li,et al.  Load balance in hierarchical routing network , 2009 .

[29]  Gang Zhou,et al.  Complexity Analysis and Efficient Implementations of Bit Parallel Finite Field Multipliers Based on Karatsuba-Ofman Algorithm on FPGAs , 2010, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[30]  Indranil Sengupta,et al.  Performance Modelling and Acceleration of Binary Edwards Curve Processor on FPGAs , 2015 .

[31]  Dieter Gollmann,et al.  Algorithm engineering for public key algorithms , 1989, IEEE J. Sel. Areas Commun..

[32]  Massoud Masoumi,et al.  Efficient Hardware Implementation of an Elliptic Curve Cryptographic Processor Over GF (2 163) , 2012 .

[33]  Massoud Masoumi,et al.  Novel Architecture for Efficient FPGA Implementation of Elliptic Curve Cryptographic Processor Over ${\rm GF}(2^{163})$ , 2013, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[34]  Reza Azarderakhsh,et al.  Efficient FPGA Implementations of Point Multiplication on Binary Edwards and Generalized Hessian Curves Using Gaussian Normal Basis , 2012, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.

[35]  Seok-Bum Ko,et al.  Parallelization of scalable elliptic curve cryptosystem processors in GF(2m) , 2016, Microprocess. Microsystems.

[36]  Muhammad Rashid,et al.  Hardware design and implementation of scalar multiplication in elliptic curve cryptography (ECC) over GF(2163) on FPGA , 2015, 2015 International Conference on Information and Communication Technologies (ICICT).

[37]  Alfred Menezes,et al.  Guide to Elliptic Curve Cryptography , 2004, Springer Professional Computing.

[38]  Kimmo Järvinen,et al.  Optimized FPGA-based elliptic curve cryptography processor for high-speed applications , 2011, Integr..

[39]  Tanja Lange,et al.  Binary Edwards Curves , 2008, CHES.

[40]  Reza Azarderakhsh,et al.  High-Performance Implementation of Point Multiplication on Koblitz Curves , 2013, IEEE Transactions on Circuits and Systems II: Express Briefs.

[41]  Kimmo Järvinen,et al.  On Parallelization of High-Speed Processors for Elliptic Curve Cryptography , 2008, IEEE Trans. Very Large Scale Integr. Syst..

[42]  Mohammed Benaissa,et al.  High-Speed and Low-Latency ECC Processor Implementation Over GF( $2^{m})$ on FPGA , 2017, IEEE Transactions on Very Large Scale Integration (VLSI) Systems.