Iterative Modular Division over GF(2 m ): Novel Algorithm and Implementations on FPGA

Dormale, Guerric Meurice de; Quisquater, Jean-Jacques

doi:10.1007/11802839_45

Cited by 7 publications

(2 citation statements)

References 20 publications

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“…The annoying large integer comparison is then avoided thanks to the counter δ idea of Brent and Kung. More details can be found in [13]. The chosen serial modular division algorithm is presented in Algorithm 2, where ⊕ stands for the bitwise xor operation.…”

Section: Dividermentioning

confidence: 99%

Low-Cost Elliptic Curve Digital Signature Coprocessor for Smart Cards

Dormale

Ambroise

Bol

et al. 2006

IEEE 17th International Conference on Application-Specific Systems, Architectures and Processors (ASAP'06)

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This paper proposes different low-cost coprocessors forpublic key authentication on 8-bit smart cards. Elliptic curve cryptography is used for its efficiency per bit of key and the Elliptic Curve Digital Signature Algorithm is chosen. For this functionality, an area constrained coprocessor is probably the best approach to perform the most computer-intensive operations at an acceptable speed considering the limited memory and power of the selected platform. For that purpose, the scalar point multiplication in GF(2 m ) in both affine and projective coordinates was implemented in order to compare their performances with the same level of optimization and the same technology. A hardware/software co-design strategy was also used to avoid the need of a dedicated register file.The performances of the arithmetic units of both coprocessors were adjusted in order to achieve the same throughput. They were then implemented on a 0.13 μm CMOS process for the extension degrees 163, 193 and 233. Resource usage and a 10% improvement of silicon area show that the affine version can be an attractive solution.

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Section: Dividermentioning

confidence: 99%

Low-Cost Elliptic Curve Digital Signature Coprocessor for Smart Cards

Dormale

Ambroise

Bol

et al. 2006

IEEE 17th International Conference on Application-Specific Systems, Architectures and Processors (ASAP'06)

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“…2m − 1 cycles are required with an area of 2m slices [25]. A fully unrolled implementation would need the quite huge amount of 4m 2 − m 2 /2 slices.…”

Section: Extended Euclidean Divisionmentioning

confidence: 99%

Collision Search for Elliptic Curve Discrete Logarithm over GF(2 m ) with FPGA

Dormale¹,

Bulens²,

Quisquater³

Cryptographic Hardware and Embedded Systems - CHES 2007

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Abstract. In this last decade, Elliptic Curve Cryptography (ECC) has gained increasing acceptance in the industry and the academic community and has been the subject of several standards. This interest is mainly due to the high level of security with relatively small keys provided by ECC. Indeed, no sub-exponential algorithms are known to solve the underlying hard problem: the Elliptic Curve Discrete Logarithm.The aim of this work is to explore the possibilities of dedicated hardware implementing the best known algorithm for generic curves: the parallelized Pollard's ρ method. This problem has specific constraints and requires therefore new architectures. Four different strategies were investigated with different FPGA families in order to provide the best areatime product, according to the capabilities of the chosen platforms. The approach yielding the best throughput over hardware cost ratio is then fully described and was implemented in order to estimate the cost of an attack. Such results should help to improve the accuracy of the security level offered by a given key size, especially for the shorter parameters proposed for resource constrained devices.

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High-speed hardware implementations of Elliptic Curve Cryptography: A survey

Dormale

Quisquater

2007

Journal of Systems Architecture

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Iterative Modular Division over GF(2 m ): Novel Algorithm and Implementations on FPGA

Cited by 7 publications

References 20 publications

Low-Cost Elliptic Curve Digital Signature Coprocessor for Smart Cards

Low-Cost Elliptic Curve Digital Signature Coprocessor for Smart Cards

Collision Search for Elliptic Curve Discrete Logarithm over GF(2 m ) with FPGA

High-speed hardware implementations of Elliptic Curve Cryptography: A survey

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