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Special Purpose Parallel Computing
 Lectures on Parallel Computation
, 1993
"... A vast amount of work has been done in recent years on the design, analysis, implementation and verification of special purpose parallel computing systems. This paper presents a survey of various aspects of this work. A long, but by no means complete, bibliography is given. 1. Introduction Turing ..."
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A vast amount of work has been done in recent years on the design, analysis, implementation and verification of special purpose parallel computing systems. This paper presents a survey of various aspects of this work. A long, but by no means complete, bibliography is given. 1. Introduction Turing [365] demonstrated that, in principle, a single general purpose sequential machine could be designed which would be capable of efficiently performing any computation which could be performed by a special purpose sequential machine. The importance of this universality result for subsequent practical developments in computing cannot be overstated. It showed that, for a given computational problem, the additional efficiency advantages which could be gained by designing a special purpose sequential machine for that problem would not be great. Around 1944, von Neumann produced a proposal [66, 389] for a general purpose storedprogram sequential computer which captured the fundamental principles of...
Open Problems in Number Theoretic Complexity, II
"... this paper contains a list of 36 open problems in numbertheoretic complexity. We expect that none of these problems are easy; we are sure that many of them are hard. This list of problems reflects our own interests and should not be viewed as definitive. As the field changes and becomes deeper, new ..."
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this paper contains a list of 36 open problems in numbertheoretic complexity. We expect that none of these problems are easy; we are sure that many of them are hard. This list of problems reflects our own interests and should not be viewed as definitive. As the field changes and becomes deeper, new problems will emerge and old problems will lose favor. Ideally there will be other `open problems' papers in future ANTS proceedings to help guide the field. It is likely that some of the problems presented here will remain open for the forseeable future. However, it is possible in some cases to make progress by solving subproblems, or by establishing reductions between problems, or by settling problems under the assumption of one or more well known hypotheses (e.g. the various extended Riemann hypotheses, NP 6= P; NP 6= coNP). For the sake of clarity we have often chosen to state a specific version of a problem rather than a general one. For example, questions about the integers modulo a prime often have natural generalizations to arbitrary finite fields, to arbitrary cyclic groups, or to problems with a composite modulus. Questions about the integers often have natural generalizations to the ring of integers in an algebraic number field, and questions about elliptic curves often generalize to arbitrary curves or abelian varieties. The problems presented here arose from many different places and times. To those whose research has generated these problems or has contributed to our present understanding of them but to whom inadequate acknowledgement is given here, we apologize. Our list of open problems is derived from an earlier `open problems' paper we wrote in 1986 [AM86]. When we wrote the first version of this paper, we feared that the problems presented were so difficult...
Efficient modular division implementation: Ecc over gf(p) affine coordinates application
 Programmable Logic and Application (FPL 2004), volume 3203 of Lecture Notes in Computer Science
, 2004
"... Abstract. Elliptic Curve Public Key Cryptosystems (ECPKC) are becoming increasingly popular for use in mobile appliances where bandwidth and chip area are strongly constrained. For the same level of security, ECPKC use much smaller key length than the commonly used RSA. The underlying operation of a ..."
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Abstract. Elliptic Curve Public Key Cryptosystems (ECPKC) are becoming increasingly popular for use in mobile appliances where bandwidth and chip area are strongly constrained. For the same level of security, ECPKC use much smaller key length than the commonly used RSA. The underlying operation of affine coordinates elliptic curve point multiplication requires modular multiplication, division/inversion and addition/substraction. To avoid the critical division/inversion operation, other coordinate systems may be chosen, but this implies more operations and a strong increase in memory requirements. So, in area and memory constrained devices, affine coordinates should be preferred, especially over GF(p). This paper presents a powerful reconfigurable hardware implementation of the Takagi modular divider algorithm. Resulting 256bit circuits achieved a ratio throughput/area improved by at least 900 % of the only known design in Xilinx VirtexE technology. Comparison with typical modular multiplication performance is carried out to suggest the use of affine coordinates also for speed reason. 1
9 1990 SpringerVerlag New York Inc. An Improved Parallel Algorithm for Integer GCD
"... Abstract. We present a simple parallel algorithm for computing the greatest common divisor (gcd) of two nbit integers in the Common version of the CRCW model of computation. The runtime of the algorithm in terms of bit operations is O(n/log n), using n ~+ ~ processors, where e is any positive cons ..."
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Abstract. We present a simple parallel algorithm for computing the greatest common divisor (gcd) of two nbit integers in the Common version of the CRCW model of computation. The runtime of the algorithm in terms of bit operations is O(n/log n), using n ~+ ~ processors, where e is any positive constant. This improves on the algorithm of Kannan, Miller, and Rudolph, the only sublinear algorithm known previously, both in run time and in number of processors; they require O(n log log n/log n), n 2 log 2 n, respectively, in the same CRCW model. We give an alternative implementation of our algorithm in the CREW model. Its runtime is O(n log log n/log n), using n ~+ ~ processors. Both implementations can be modified to yield the extended gcd, within the same complexity bounds. Key Words. Greatest common divisor, Parallel algorithms.