Efficient Hardware Implementations for Tripling Oriented Elliptic Curve Crypto-System


(*) Corresponding author


Authors' affiliations


DOI's assignment:
the author of the article can submit here a request for assignment of a DOI number to this resource!
Cost of the service: euros 10,00 (for a DOI)

Abstract


This paper proposes several hardware designs and implementations for Tripling Oriented elliptic curve crypto-system (ECC) over GF (p). Projective coordinates are used to apply ECC's computations to avoid the time-consuming inversion operation. In order to improve the performance even further, parallel hardware designs to implement ECC operations in parallel are proposed. This plays a crucial role in speeding up ECC operations, compared to the known serial design. This research also presents several ECC designs by varying parallelization level for elliptic curve's computations. The purpose of this process is to provide efficient design solutions that fit different security applications according to requirements and available resources for certain application. Moreover, the NAF algorithm is used to perform scalar multiplication operation benefiting from the ability of NAF algorithm to reduce the average number of point addition operations. The proposed designs are implemented using VHDL, validated with ModelSim, and then simulated using Xilinx tool with target FPGA. The 4-PM design using homogenous coordinates achieves the best performance level for Tripling Oriented curve. Such design is significant for security applications that need high-speed ECC. Jacobean coordinates, on the other hand, shows the best performance results when applied with less parallelization levels as well as the serial design
Copyright © 2014 Praise Worthy Prize - All rights reserved.

Keywords


Elliptic Curves Cryptosystem; Time-Consumption Resources; FPGA Hardware Implementation Projective Coordinates; Security Applications

Full Text:

PDF


References


Wade Trappe, Lawrence. c, Introduction to Cryptography with Coding Theory. Washington, Pearson Prentice Hall, 2002.

N. Koblitz, “Elliptic curve cryptosystem”, Mathematics of Computation, Vol. 48, pp. 203-209, 1987.

V. Miller, “Uses of elliptic curves in cryptography”, Lecture Notes in Computer Science, Vol. 218, pp. 417-426, 1986.

Blake, Seroussi, and Smart. Elliptic Curves in Cryptography. Cambridge University Press: New York, 1999

Chung, Sim, and Lee, “Fast Implementation of Elliptic Curve Defined over GF(pm) on CalmRISC with MAC2424 Coprocessor,” Workshop on Cryptographic Hardware and Embedded Systems, CHES 2000, Massachusetts, August 2000.

A. Menezes, E. Teske, A. Weng, “Weak fields for ECC,” in the Cryptographer’s Track at RSA Conference (CT-RSA), LNCS 2964, 2004, pp. 366–386

Tanja Lange, “A note on L´opez-Dahab coordinates”, Faculty of Mathematics, Technical University of Denmark, 2006.

David, Nigel, and Jacques, “Projective coordinates Leak”, Applied research and security center, France.

GuerricMeurice de Dormale, Jean-Jacques Quisquater. High-speed hardware implementations of Elliptic Curve Cryptography: A survey. Journal of Systems Architecture 53 (2007) 72-84, by Elsevier.

A. Satoh, K. Takano, “A scalable dual-field elliptic curve cryptographic processor,” IEEE Transactions Computers 52 (4) (2003) 449–460.

G. Orlando, and C. Paar, "A scalable GF (p) elliptic curve processor architecture for programmable hardware,” Cryptographic Hardware and Embedded Systems - CHES 2001, Paris, France, May 14-15, 2001

Adnan Gutub and Mohammad K. Ibrahim., “High Radix Parallel Architecture For GF(p) Elliptic Curve Processor”, IEEE Conference on Acoustics, Speech, and Signal Processing, ICASSP 2003, Pages: 625- 628, Hong Kong, April 6-10.

Adnan Gutub. Efficient Utilization of Scalable Multipliers in Parallel to Compute GF (p) Elliptic Curve Cryptographic Operations. Kuwait Journal of Science & Engineering (KJSE) 2007, 34(2): 165-182.

L. Tawalbeh and Q. Abu Al-Haija. Speeding up Elliptic Curve Cryptography Computations by Adopting Edwards Curves over GF (P). International Journal of Security (IJS) 2009, CSC Journals, Malaysia, Vol.3, Issue.4, IJS-19.

Q. Abu Al-Haija and Mohammad Al-Khatib. Parallel Hardware Algorithms & Designs for Elliptic Curves Cryptography to Improve Point Operations Computations Using New Projective Coordinates. Journal of Information Assurance and Security (JIAS) 2010, By Dynamic Publishers Inc., USA, Vol.4, No.1, Paper 6: (588-594).

Adnan Abdul-Aziz Gutub and Mohammad K. Ibrahim, "High performance elliptic curve GF (2k) crypto-processor architecture for multimedia", IEEE International Conference on Multimedia & Expo, ICME 2003, pages 81- 84, Baltimore, Maryland, USA, July 6-9, 2003.

Adnan Gutub, “High Speed Low Power GF (2k) Elliptic Curve Cryptography Processor Architecture”, IEEE 10th Annual Technical Exchange Meeting, KFUPM, Dhahran, Saudi Arabia, March 23-24, 2003.

L. Tawalbeh and Q. Abu Al-Haija. Enhanced FPGA Implementations for Doubling Oriented and Jacobi-Quartics Elliptic Curves Cryptography. Journal of Information Assurance and Security (JIAS), Volume 6, pp. 167-175

Khatib, M., Al-Haija, Q.A., Jaafar, A., Hardware architectures & designs for projective Elliptic curves point addition operation using variable levels of parallelism, (2011) International Review on Computers and Software (IRECOS), 6 (2), pp. 227-236.

Mohammad Al-Khatib, Q. Abu Al-Haija, and Ramlan Mahmud. Performance Evaluation of Projective Binary Edwards Elliptic Curve Computations with Parallel Architectures. Journal of Information Assurance and Security (JIAS) 2011, By Dynamic Publishers Inc., USA, Vol.6, No.1, Paper1: (001-009).

Mohammad Al-khatib, AzmiJaafar, and Q. Sbu Al-Haija. Choices on Designing GF (p) Elliptic Curve Coprocessor Benefiting from Mapping Homogeneous Curves in Parallel Multiplications. International Journal on computer science and engineering 2011, Vol.3, No.2, Paper 2: (467-480).

Mohammad Alkhatib, Azmi B. Jaafar, ZuriatiZukarnain, and Mohammad Rushdan. On the Design of Projective Binary Edwards Elliptic Curves over GF (p) Benefiting from Mapping Elliptic Curves Computations to Variable Degree of Parallel Design. International Journal on computer science and engineering 2011, Vol.3, No.4, Paper 44: (1697-1712).

Alkhatib, M., Jaafar, A., Zukarnain, Z., Rushdan, M., Trade-off between area and speed for projective edwards elliptic curves crypto-system over GF (p) using parallel hardware designs and architectures, (2011) International Review on Computers and Software (IRECOS), 6 (4), pp. 615-625.

Al-Khatib, M., Jaafar, A., Zukarnain, Z., Rushdan, M., Hardware designs and architectures for projective Montgomery ECC over GF (p) Benefiting from mapping elliptic curve computations to different degrees of parallelism, (2011) International Review on Computers and Software (IRECOS), 6 (6), pp. 1059-1070.


Refbacks

  • There are currently no refbacks.



Please send any question about this web site to info@praiseworthyprize.com
Copyright © 2005-2022 Praise Worthy Prize