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Integer Factorization Based on Elliptic Curve Method: Towards Better Exploitation of Reconfigurable Hardware
"... Currently, the best known algorithm for factorizing modulus of the RSA public key cryptosystem is the Number Field Sieve. One of its important phases usually combines a sieving technique and a method for checking smoothness of midsize numbers. For this factorization, the Elliptic Curve Method (ECM) ..."
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Currently, the best known algorithm for factorizing modulus of the RSA public key cryptosystem is the Number Field Sieve. One of its important phases usually combines a sieving technique and a method for checking smoothness of midsize numbers. For this factorization, the Elliptic Curve Method (ECM) is an attractive solution. As ECM is highly regular and many parallel computations are required, hardwarebased platforms were shown to be more costeffective than software solutions. The few papers dealing with implementation of ECM on FPGA are all based on bitserial architectures. They use only generalpurpose logic and lowcost FPGAs which appear as the best performance/cost solution. This work explores another approach, based on the exploitation of embedded multipliers available in modern FPGAs and the use of highperformances FPGAs. The proposed architecture – based on a fully parallel and pipelined modular multiplier circuit – exhibits a 15fold improvement over throughput/hardware cost ratio of previously published results.
Efficient implementation of anonymous credentials on java card smart cards
 In 1st IEEE International Workshop on Information Forensics and Security (WIFS 2009
, 2009
"... The Direct Anonymous Attestation scheme allows to map procedures with an imperative requirement for anonymity, such as voting, to the electronic world while offering provable security. However, the scheme is complex and requires demanding computations to be performed on a tamperproof device. Such ..."
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The Direct Anonymous Attestation scheme allows to map procedures with an imperative requirement for anonymity, such as voting, to the electronic world while offering provable security. However, the scheme is complex and requires demanding computations to be performed on a tamperproof device. Such devices, e.g. secure smart cards, are typically resource constrained. We present the first implementation of the (simplified) Direct Anonymous Attestation protocols suitable for contemporary Java Card smart cards. We point out performance bottlenecks and provide efficient solutions which allow our implementation to terminate within acceptable time. Index Terms — SECPRIV, SYSSOFT, SECINTE 1.
Faster interleaved modular multiplication based on Barrett and Montgomery reduction methods
 1715–1721, 2010, [Online] Available: http://dx.doi.org/10.1109/TC.2010.93
"... IEEE Abstract—This paper proposes two improved interleaved modular multiplication algorithms based on Barrett and Montgomery modular reduction. The algorithms are simple and especially suitable for hardware implementations. Four large sets of moduli for which the proposed methods apply are given and ..."
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IEEE Abstract—This paper proposes two improved interleaved modular multiplication algorithms based on Barrett and Montgomery modular reduction. The algorithms are simple and especially suitable for hardware implementations. Four large sets of moduli for which the proposed methods apply are given and analyzed from a security point of view. By considering stateoftheart attacks on publickey cryptosystems, we show that the proposed sets are safe to use, in practice, for both elliptic curve cryptography and RSA cryptosystems. We propose a hardware architecture for the modular multiplier that is based on our methods. The results show that concerning the speed, our proposed architecture outperforms the modular multiplier based on standard modular multiplication by more than 50 percent. Additionally, our design consumes less area compared to the standard solutions. Index Terms—Modular multiplication, Barrett reduction, Montgomery reduction, publickey cryptography.
Modular inverse algorithms without multiplications for cryptographic applications
 EURASIP Journal on Embedded System
, 2006
"... Hardware and algorithmic optimization techniques are presented to the leftshift, rightshift, and the traditional Euclideanmodular inverse algorithms. Theoretical arguments and extensive simulations determined the resulting expected running time. On many computational platforms these turn out to be ..."
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Hardware and algorithmic optimization techniques are presented to the leftshift, rightshift, and the traditional Euclideanmodular inverse algorithms. Theoretical arguments and extensive simulations determined the resulting expected running time. On many computational platforms these turn out to be the fastest known algorithms for moderate operand lengths. They are based on variants of Euclideantype extended GCD algorithms. On the considered computational platforms for operand lengths used in cryptography, the fastest presented modular inverse algorithms need about twice the time of modular multiplications, or even less. Consequently, in elliptic curve cryptography delaying modular divisions is slower (affine coordinates are the best) and the RSA and ElGamal cryptosystems can be accelerated. Copyright © 2006 Laszlo Hars. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. 1.
Accelerated Modular Multiplication Algorithm of Large Word Length Numbers with a Fixed Module
"... Abstract. A new algorithm is proposed for the software implementation of modular multiplication, which uses precomputations with a constant module. The developed modular multiplication algorithm provides high performance in comparison with the already known algorithms, and is oriented at the variab ..."
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Abstract. A new algorithm is proposed for the software implementation of modular multiplication, which uses precomputations with a constant module. The developed modular multiplication algorithm provides high performance in comparison with the already known algorithms, and is oriented at the variable value of the module, especially with the software implementation on micro controllers and smart cards with a small number of bits.
Accessible Secure Information Society Applications via the Use of Optimised Cryptographic Calculations
"... Abstract. Information Society aims to promote innovation in the context of governmental and enterprise information systems and participation of the majority of the general population. An important prerequisite for the penetration and widespread use of Information Society technologies is to enhance t ..."
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Abstract. Information Society aims to promote innovation in the context of governmental and enterprise information systems and participation of the majority of the general population. An important prerequisite for the penetration and widespread use of Information Society technologies is to enhance the perception of security that these technologies offer without making them inaccessible to users with limited computational resources. In this paper a new algorithm for the software implementation of modular multiplication is proposed, which uses precomputations with a constant modulus to reduce the computational load imposed upon the processor. The developed modular multiplication algorithm provides faster execution on low complexity hardware in comparison with the existing algorithms and is oriented towards the variable value of the modulus, especially with the software implementation on micro controllers and smart cards whose architectures include a small number of bits. The use of the new algorithm in Information Society applications that demand security is investigated. Such applications include eGovernment, eBanking, eCommerce etc. The algorithm is shown to be adequate both for the applications for which it was originally intended, as well as for applications that are much more demanding in the level of security they require, such as Military data processing and communication systems.
Elliptic Curve Factorization Method: Towards Better Exploitation of Reconfigurable Hardware
"... Currently, the best known algorithm for factorizing modulus of the RSA public key cryptosystem is the Number Field Sieve. One of its important phases usually combines a sieving technique and a method for checking smoothness of midsize numbers. For this factorization, the Elliptic Curve Method (ECM) ..."
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Currently, the best known algorithm for factorizing modulus of the RSA public key cryptosystem is the Number Field Sieve. One of its important phases usually combines a sieving technique and a method for checking smoothness of midsize numbers. For this factorization, the Elliptic Curve Method (ECM) is an attractive solution. As ECM is highly regular and many parallel computations are required, hardwarebased platforms were shown to be more costeffective than software solutions. The few papers dealing with implementation of ECM on FPGA are all based on bitserial architectures. They use only generalpurpose logic and lowcost FPGAs which appear as the best performance/cost solution. This work explores another approach, based on the exploitation of embedded multipliers available in modern FPGAs and the use of highperformances FPGAs. The proposed architecture – based on a fully parallel and pipelined modular multiplier circuit – exhibits a 15fold improvement over throughput/hardware cost ratio of previously published results.
Notations
, 2003
"... Abstract Truncated Multiplication computes a truncated product, a contiguous subsequence of the digits of the product of 2 long integers. We review a few truncated multiplication algorithms and adapt them to integers. They are a constant times faster than ndigit full multiplications of time complex ..."
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Abstract Truncated Multiplication computes a truncated product, a contiguous subsequence of the digits of the product of 2 long integers. We review a few truncated multiplication algorithms and adapt them to integers. They are a constant times faster than ndigit full multiplications of time complexity O(n α), with 1< α ≤ 2, important in cryptography. For example, the least significant half products with Karatsuba multiplication need 80 % of the full multiplication time. The faster the multiplication, the less relative time saving we can achieve. Several improved long integer arithmetic algorithms are presented, including integer reciprocals and divisions, 2ndigit modular multiplication on HW for ndigit half products, Barrett and Montgomery multiplications. They get further accelerated with application of fast truncated multiplication, like Montgomery multiplication performed in 2.6 Karatsuba multiplications time.
Applications of Fast Truncated Multiplication in Cryptography Laszlo Hars
"... Abstract. Truncated Multiplications compute Truncated Products, contiguous subsequences of the digits of the products of integers. They are based on the ndigit full multiplication algorithms of time complexity O(n α), with 1< α ≤ 2, but a constant times faster. Applying these fast truncated multi ..."
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Abstract. Truncated Multiplications compute Truncated Products, contiguous subsequences of the digits of the products of integers. They are based on the ndigit full multiplication algorithms of time complexity O(n α), with 1< α ≤ 2, but a constant times faster. Applying these fast truncated multiplications several improved cryptographic long integer arithmetic algorithms are presented, including integer reciprocals, divisions, Barrett and Montgomery multiplications, 2ndigit modular multiplication on HW for ndigit half products. E.g., Montgomery multiplication is performed in 2.6 Karatsuba multiplications time.
Fast Truncated Multiplication for Cryptographic Applications Laszlo Hars
"... Abstract. The Truncated Multiplication computes a truncated product, a contiguous subsequence of the digits of the product of 2 integers. A few truncated polynomial multiplication algorithms are presented and adapted to integers. They are based on the most often used ndigit full multiplication algo ..."
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Abstract. The Truncated Multiplication computes a truncated product, a contiguous subsequence of the digits of the product of 2 integers. A few truncated polynomial multiplication algorithms are presented and adapted to integers. They are based on the most often used ndigit full multiplication algorithms of time complexity O(n α), with 1< α ≤ 2, but a constant times faster. For example, the least significant half products with Karatsuba multiplication need only 80 % of the full multiplication time. The faster the multiplication, the less relative time saving can be achieved.