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Low Redundancy in Dictionaries with O(1) Worst Case Lookup Time
 IN PROC. 26TH INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES AND PROGRAMMING (ICALP
, 1998
"... A static dictionary is a data structure for storing subsets of a finite universe U , so that membership queries can be answered efficiently. We study this problem in a unit cost RAM model with word size ze jU j), and show that for nelement subsets, constant worst case query time can be obtain ..."
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Cited by 18 (0 self)
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A static dictionary is a data structure for storing subsets of a finite universe U , so that membership queries can be answered efficiently. We study this problem in a unit cost RAM model with word size ze jU j), and show that for nelement subsets, constant worst case query time can be obtained using B +O(log log jU j) + o(n) bits of storage, where B = dlog jU j e is the minimum number of bits needed to represent all such subsets. The solution for dense subsets uses B + O( jU j log log jU j log jU j ) bits of storage, and supports constant time rank queries. In a dynamic setting, allowing insertions and deletions, our techniques give an O(B) bit space usage.
Constructing Elliptic Curve Cryptosystems in Characteristic 2
, 1998
"... Since the group of an elliptic curve defined over a finite field F_q... The purpose of this paper is to describe how one can search for suitable elliptic curves with random coefficients using Schoof's algorithm. We treat the important special case of characteristic 2, where one has certain simplific ..."
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Cited by 17 (1 self)
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Since the group of an elliptic curve defined over a finite field F_q... The purpose of this paper is to describe how one can search for suitable elliptic curves with random coefficients using Schoof's algorithm. We treat the important special case of characteristic 2, where one has certain simplifications in some of the algorithms.
Lecture Notes on Cryptography
, 2001
"... This is a set of lecture notes on cryptography compiled for 6.87s, a one week long course on cryptography taught at MIT by Shafi Goldwasser and Mihir Bellare in the summers of 1996–2001. The notes were formed by merging notes written for Shafi Goldwasser’s Cryptography and Cryptanalysis course at MI ..."
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This is a set of lecture notes on cryptography compiled for 6.87s, a one week long course on cryptography taught at MIT by Shafi Goldwasser and Mihir Bellare in the summers of 1996–2001. The notes were formed by merging notes written for Shafi Goldwasser’s Cryptography and Cryptanalysis course at MIT with notes written for Mihir Bellare’s Cryptography and network security course at UCSD. In addition, Rosario Gennaro (as Teaching Assistant for the course in 1996) contributed Section 9.6, Section 11.4, Section 11.5, and Appendix D to the notes, and also compiled, from various sources, some of the problems in Appendix E. Cryptography is of course a vast subject. The thread followed by these notes is to develop and explain the notion of provable security and its usage for the design of secure protocols. Much of the material in Chapters 2, 3 and 7 is a result of scribe notes, originally taken by MIT graduate students who attended Professor Goldwasser’s Cryptography and Cryptanalysis course over the years, and later edited by Frank D’Ippolito who was a teaching assistant for the course in 1991. Frank also contributed much of the advanced number theoretic material in the Appendix. Some of the material in Chapter 3 is from the chapter on Cryptography, by R. Rivest, in the Handbook of Theoretical Computer Science. Chapters 4, 5, 6, 8 and 10, and Sections 9.5 and 7.4.6, were written by Professor Bellare for his Cryptography and network security course at UCSD.
A Lower Bound for Primality
, 1999
"... Recent work by Bernasconi, Damm and Shparlinski proved lower bounds on the circuit complexity of the squarefree numbers, and raised as an open question if similar (or stronger) lower bounds could be proved for the set of prime numbers. In this short note, we answer this question affirmatively, by s ..."
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Cited by 11 (5 self)
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Recent work by Bernasconi, Damm and Shparlinski proved lower bounds on the circuit complexity of the squarefree numbers, and raised as an open question if similar (or stronger) lower bounds could be proved for the set of prime numbers. In this short note, we answer this question affirmatively, by showing that the set of prime numbers (represented in the usual binary notation) is not contained in AC 0 [p] for any prime p. Similar lower bounds are presented for the set of squarefree numbers, and for the problem of computing the greatest common divisor of two numbers. 1 Introduction What is the computational complexity of the set of prime numbers? There is a large body of work presenting important upper bounds on the complexity of the set of primes (including [AH87, APR83, Mil76, R80, SS77]), but  Supported in part by NSF grant CCR9734918. y Supported in part by NSF grant CCR9700239. z Supported in part by ARC grant A69700294. as was pointed out recently in [BDS98a, BDS9...
A Short History of Computational Complexity
 IEEE CONFERENCE ON COMPUTATIONAL COMPLEXITY
, 2002
"... this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quit ..."
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this article mention all of the amazing research in computational complexity theory. We survey various areas in complexity choosing papers more for their historical value than necessarily the importance of the results. We hope that this gives an insight into the richness and depth of this still quite young eld
Implementation Of The AtkinGoldwasserKilian Primality Testing Algorithm
 Rapport de Recherche 911, INRIA, Octobre
, 1988
"... . We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual impl ..."
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Cited by 9 (7 self)
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. We describe a primality testing algorithm, due essentially to Atkin, that uses elliptic curves over finite fields and the theory of complex multiplication. In particular, we explain how the use of class fields and genus fields can speed up certain phases of the algorithm. We sketch the actual implementation of this test and its use on testing large primes, the records being two numbers of more than 550 decimal digits. Finally, we give a precise answer to the question of the reliability of our computations, providing a certificate of primality for a prime number. IMPLEMENTATION DU TEST DE PRIMALITE D' ATKIN, GOLDWASSER, ET KILIAN R'esum'e. Nous d'ecrivons un algorithme de primalit'e, principalement du `a Atkin, qui utilise les propri'et'es des courbes elliptiques sur les corps finis et la th'eorie de la multiplication complexe. En particulier, nous expliquons comment l'utilisation du corps de classe et du corps de genre permet d'acc'el'erer les calculs. Nous esquissons l'impl'ementati...
An Approximation Algorithm for the Number of Zeros of Arbitrary Polynomials over GF[q]
 PROC. FOCS
, 1991
"... We design the first polynomial time (for an arbitrary and fixed field GF [q]) (ffl; ffi )approximation algorithm for the number of zeros of arbitrary polynomial f(x 1 ; : : : ; x n ) over GF [q]. It gives the first efficient method for estimating the number of zeros and nonzeros of multivariate po ..."
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Cited by 7 (4 self)
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We design the first polynomial time (for an arbitrary and fixed field GF [q]) (ffl; ffi )approximation algorithm for the number of zeros of arbitrary polynomial f(x 1 ; : : : ; x n ) over GF [q]. It gives the first efficient method for estimating the number of zeros and nonzeros of multivariate polynomials over small finite fields other than GF [2] (like GF [3]), the case important for various circuit approximation techniques. The algorithm is based on the estimation of the number of zeros of an arbitrary polynomial f(x 1 ; : : : ; x n ) over GF [q] in the function on the number m of its terms. The bounding ratio number is proved to be m (q\Gamma1) log q which is the main technical contribution of this paper and could be of independent algebraic interest.
R SN 1tt (NP) distinguishes robust manyone and Turing completeness
 In Proceedings of the 3rd Italian Conference on Algorithms and Complexity
, 1998
"... Do complexity classes have manyone complete sets if and only if they have Turingcomplete sets? We prove that there is a relativized world in which a relatively natural complexity class—namely a downward closure of NP, RSN 1tt (NP)—has Turingcomplete sets but has no manyone complete sets. In fact ..."
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Cited by 7 (4 self)
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Do complexity classes have manyone complete sets if and only if they have Turingcomplete sets? We prove that there is a relativized world in which a relatively natural complexity class—namely a downward closure of NP, RSN 1tt (NP)—has Turingcomplete sets but has no manyone complete sets. In fact, we show that in the same relativized world this class has 2truthtable complete sets but lacks 1truthtable complete sets. As part of the groundwork for our result, we prove that RSN 1tt(NP) has many equivalent forms having to do with ordered and parallel access to NP and NP ∩ coNP. 1
Elliptic Curves, Primality Proving And Some Titanic Primes
, 1989
"... We describe how to generate large primes using the primality proving algorithm of Atkin. Figure 1: The Titanic . 1. Introduction. During the last ten years, primality testing evolved at great speed. Motivated by the RSA cryptosystem [3], the first deterministic primality proving algorithm was de ..."
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Cited by 5 (3 self)
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We describe how to generate large primes using the primality proving algorithm of Atkin. Figure 1: The Titanic . 1. Introduction. During the last ten years, primality testing evolved at great speed. Motivated by the RSA cryptosystem [3], the first deterministic primality proving algorithm was designed by Adleman, Pomerance and Rumely [2] and made practical by Cohen, H. W. Lenstra and A. K. Lenstra (see [9, 10] and more recently [5]). It was then proved that the time needed to test an arbitrary integer N for primality is O((log N) c log log log N ) for some positive constant c ? 0. When implemented on a huge computer, the algorithm was able to test 200 digit numbers in about 10 minutes of CPU time. A few years ago, Goldwasser and Kilian [11], used the theory of elliptic curves over finite fields to give the first primality proving algorithm whose running time is polynomial in log N (assuming a plausible conjecture in number theory). Atkin [4] used the theory of complex multiplicat...