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Efficient Circuitry for Computing τ adic NonAdjacent Form
 in Proceedings of the 13th IEEE International Conference on Electronics, Circuits and Systems, (ICECS 2006) . IEEE
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Improved errordetection and faulttolerance in ECSM using input randomization,” CACR
 University of Waterloo
, 2006
"... Verification of the correctness of computations is very important to provide resistance against faultbased attacks. In elliptic curve cryptography (ECC), point validation (PV) alone is not sufficient against all fault analysis attacks. In this report errordetecting and faulttolerant schemes for e ..."
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Cited by 5 (3 self)
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Verification of the correctness of computations is very important to provide resistance against faultbased attacks. In elliptic curve cryptography (ECC), point validation (PV) alone is not sufficient against all fault analysis attacks. In this report errordetecting and faulttolerant schemes for elliptic curve scalar multiplication (ECSM) are considered. For the purpose of detecting errors, we present structures based on recomputation and parallel computation along with PV. These structures use encoding techniques that rely on proprieties of elliptic curves and provide a high probability of detection of errors caused by faults occurred naturally or injected by an attacker. Additionally, we show that using parallel computation along with either PV or recomputation, it is possible to have faulttolerant structures for the ECSM. Prototypes of the proposed structures for error detection and fault tolerance have been implemented and experimental results have been presented.
N.: Parallel Itoh—Tsujii multiplicative inversion algorithm for a special class of trinomials. Des
 Codes Cryptography 45(1) 19–37 (2007) Solinas, J.A.: Efficient Arithmetic on Koblitz Curves. Designs, Codes and Cryptography
, 2000
"... In this contribution, we derive a novel parallel formulation of the standard ItohTsujii algorithm for multiplicative inverse computation overGF(2m). The main building blocks used by our algorithm are: field multiplication, field squaring and field square root operators. It achieves its best perfor ..."
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In this contribution, we derive a novel parallel formulation of the standard ItohTsujii algorithm for multiplicative inverse computation overGF(2m). The main building blocks used by our algorithm are: field multiplication, field squaring and field square root operators. It achieves its best performance when using a special class of irreducible trinomials, namely, P (X) = Xm +Xk + 1, withm and k odd numbers and when implemented in hardware platforms. Under these conditions, our experimental results show that our parallel version of the ItohTsujii algorithm yields a speedup of about 30 % when compared with the standard version of it. Implemented in a Virtex 3200E FPGA device, our design is able to compute multiplicative inversion over GF(2193) after 20 clock cycles in about 0.94µS. Key words: Cryptography, Multiplicative Inversion, polynomial basis, FPGA de
ErrorDetecting and FaultTolerant Structures for ECC
"... For constrained devices, elliptic curve cryptography (ECC) is an attractive choice because it achieves the same level of security with a much smaller key size in comparison with other schemes such as those that are based on integer factorization or discrete logarithm. For security reasons, especiall ..."
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For constrained devices, elliptic curve cryptography (ECC) is an attractive choice because it achieves the same level of security with a much smaller key size in comparison with other schemes such as those that are based on integer factorization or discrete logarithm. For security reasons, especially to provide resistance against faultbased attacks, it is very important to verify the correctness of computations in ECC applications. In this report, faulttolerant and errordetecting elliptic curve cryptosystems are considered. Error detection may be a sufficient countermeasure for many security applications. However, faulttolerant characteristic enables a system to perform its normal operation in spite of faults. This will result in more reliable systems where faults may occur due to natural causes. For the purpose of detecting errors due to faults, a number of schemes based on the pointonthecurve checking, time redundancy, and hardware redundancy are presented. A combination of the pointonthecurve checking and time or hardware redundancy can be used for detecting errors with a very high probability during the computation of the elliptic curve scalar multiplication (ECSM). Additionally, we show that using dual modular redundancy (DMR) and the pointonthecurve checking, it is possible to have a faulttolerant structure for the ECSM. If certain conditions are met, this scheme is more efficient than others such as the wellknown triple modular redundancy. 1 1
A Hardware Architecture for Elliptic Curve Cryptography and Lossless Data Compression
"... We present a hardware architecture that combines Elliptic Curve Cryptography (ECC) and lossless data compression in a single chip. Input data is compressed using a dictionarybased lossless data compressor before encryption, then; two elliptic curve cryptographic algorithms can be applied to the com ..."
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We present a hardware architecture that combines Elliptic Curve Cryptography (ECC) and lossless data compression in a single chip. Input data is compressed using a dictionarybased lossless data compressor before encryption, then; two elliptic curve cryptographic algorithms can be applied to the compressed data: ECIES for encryption or ECDSA for digital signature. Applying data compression presents three advantages: first, the improvement in the cryptographic module throughput by reducing the amount of data to be encrypted; second, the higher utilization of the available bandwidth if encrypted data is transmitted across a public network and third, the increment of the difficulty to recover the original information. The architecture was described in VHDL and synthesized for a Xilinx FPGA device. The results achieved show that it is possible to combine these two algorithms in a single chip while gathering the advantages of compression and cryptography. This work is novel in the sense that no such algorithm combination has been reported neither a hardware implementation of elliptic curve cryptographic schemes. 1.
Elliptic Curve Scalar Multiplication using Point Halving on Reconfigurable Hardware Platforms
"... Abstract — In this paper, a FPGA arithmetic logic unit architecture for computing elliptic curve scalar multiplication over the binary extension field GF (2 163) is presented. Proposed architecture implements a parallel version of the mixedcoordinate point addition and point doubling formulae. This ..."
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Abstract — In this paper, a FPGA arithmetic logic unit architecture for computing elliptic curve scalar multiplication over the binary extension field GF (2 163) is presented. Proposed architecture implements a parallel version of the mixedcoordinate point addition and point doubling formulae. This way, our design can perform elliptic curve point addition, point doubling and point halving efficiently in terms of area resources and timing performance. In fact, our experimental results show that our proposed design can perform an elliptic curve scalar multiplication in about 25µS. Index Terms — elliptic curve, scalar multiplication, point halving, reconfigurable hardware. I.
Hardware Acceleration of Elliptic Curve Based Cryptographic Algorithms: Design and Simulation
, 2008
"... Elliptic curve cryptography (ECC) is an alternative to traditional public key cryptographic systems. Even though, RSA (RivestShamirAdleman) was the most prominent cryptographic scheme, it is being replaced by ECC in many systems. This is due to the fact that ECC gives higher security with shorter ..."
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Elliptic curve cryptography (ECC) is an alternative to traditional public key cryptographic systems. Even though, RSA (RivestShamirAdleman) was the most prominent cryptographic scheme, it is being replaced by ECC in many systems. This is due to the fact that ECC gives higher security with shorter bit length than RSA. In Elliptic curve based algorithms elliptic curve point multiplication is the most computationally intensive operation. Therefore implementing point multiplication using hardware makes ECC more attractive for high performance servers and small devices. In this thesis FPGA accelerator for point multiplication over GF(2^163) is proposed. We designed and synthesized the point accelerator using Xilinx XCV2000 FPGA. Binary field arithmetic units from which the point accelerator is built are also designed and synthesized. Experimental results show that a single point multiplication executes in 47µs. This is a 161 fold speed up over software implementation. And it is also better than the fastest hardware accelerator published in the literature.
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©2005 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE. A Hardware Architecture for Elliptic Curve Cryptography and Lossless Data
© 2008 Elsevier Science Reprinted with permission from Elsevier. es
"... ari uire re a on o liza pr nd 1 II FP curves in publickey cryptography in 1985. Since then elliptic curve cryptography has attained considera in the cryptographic research community r the a sed for ickey actoriz yptogr with 1 at ellip system is called point multiplication. Much effort has been all ..."
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ari uire re a on o liza pr nd 1 II FP curves in publickey cryptography in 1985. Since then elliptic curve cryptography has attained considera in the cryptographic research community r the a sed for ickey actoriz yptogr with 1 at ellip system is called point multiplication. Much effort has been allocated in developing methods for its efficient computation because it acts as the bottleneck in elliptic curve cryptosystems. A comprehensive review can be found in [22], for example. Point multiplication is commonly known to be an operation which is hard to parallelize because of data dependencies and much of the critical path is essential. The fastest method for computing point utilized in Koblitz curve point multiplication with the existing methods [14] whereas Montgomery point multiplication can efficiently use up to four multipliers [18,13]. Hence, the more multipliers are available the smaller is the benefit of using Koblitz curves. Koblitz curves are faster than general curves even if parallel field multipliers are available, but the difference becomes smaller which makes Koblitz curves less attractive. The contributions of our paper are twofold: (1) We present a simple and efficient method for speeding up Koblitz curve computations when parallel field multipliers