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Guide to Elliptic Curve Cryptography
, 2004
"... Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves ..."
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Cited by 382 (17 self)
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Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves also figured prominently in the recent proof of Fermat's Last Theorem by Andrew Wiles. Originally pursued for purely aesthetic reasons, elliptic curves have recently been utilized in devising algorithms for factoring integers, primality proving, and in publickey cryptography. In this article, we aim to give the reader an introduction to elliptic curve cryptosystems, and to demonstrate why these systems provide relatively small block sizes, highspeed software and hardware implementations, and offer the highest strengthperkeybit of any known publickey scheme.
Selecting Cryptographic Key Sizes
 TO APPEAR IN THE JOURNAL OF CRYPTOLOGY, SPRINGERVERLAG
, 2001
"... In this article we offer guidelines for the determination of key sizes for symmetric cryptosystems, RSA, and discrete logarithm based cryptosystems both over finite fields and over groups of elliptic curves over prime fields. Our recommendations are based on a set of explicitly formulated parameter ..."
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Cited by 257 (6 self)
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In this article we offer guidelines for the determination of key sizes for symmetric cryptosystems, RSA, and discrete logarithm based cryptosystems both over finite fields and over groups of elliptic curves over prime fields. Our recommendations are based on a set of explicitly formulated parameter settings, combined with existing data points about the cryptosystems.
Lower Bounds for Discrete Logarithms and Related Problems
, 1997
"... . This paper considers the computational complexity of the discrete logarithm and related problems in the context of "generic algorithms"that is, algorithms which do not exploit any special properties of the encodings of group elements, other than the property that each group element is encoded a ..."
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Cited by 223 (11 self)
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. This paper considers the computational complexity of the discrete logarithm and related problems in the context of "generic algorithms"that is, algorithms which do not exploit any special properties of the encodings of group elements, other than the property that each group element is encoded as a unique binary string. Lower bounds on the complexity of these problems are proved that match the known upper bounds: any generic algorithm must perform\Omega (p 1=2 ) group operations, where p is the largest prime dividing the order of the group. Also, a new method for correcting a faulty DiffieHellman oracle is presented. 1 Introduction The discrete logarithm problem plays an important role in cryptography. The problem is this: given a generator g of a cyclic group G, and an element g x in G, determine x. A related problem is the DiffieHellman problem: given g x and g y , determine g xy . In this paper, we study the computational power of "generic algorithms" that is, ...
The Decision DiffieHellman Problem
, 1998
"... The Decision DiffieHellman assumption (ddh) is a gold mine. It enables one to construct efficient cryptographic systems with strong security properties. In this paper we survey the recent applications of DDH as well as known results regarding its security. We describe some open problems in this are ..."
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Cited by 197 (6 self)
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The Decision DiffieHellman assumption (ddh) is a gold mine. It enables one to construct efficient cryptographic systems with strong security properties. In this paper we survey the recent applications of DDH as well as known results regarding its security. We describe some open problems in this area. 1 Introduction An important goal of cryptography is to pin down the exact complexity assumptions used by cryptographic protocols. Consider the DiffieHellman key exchange protocol [12]: Alice and Bob fix a finite cyclic group G and a generator g. They respectively pick random a; b 2 [1; jGj] and exchange g a ; g b . The secret key is g ab . To totally break the protocol a passive eavesdropper, Eve, must compute the DiffieHellman function defined as: dh g (g a ; g b ) = g ab . We say that the group G satisfies the Computational DiffieHellman assumption (cdh) if no efficient algorithm can compute the function dh g (x; y) in G. Precise definitions are given in the next sectio...
The gapproblems: a new class of problems for the security of cryptographic schemes
 Proceedings of PKC 2001, volume 1992 of LNCS
, 1992
"... Abstract. This paper introduces a novel class of computational problems, the gap problems, which can be considered as a dual to the class of the decision problems. We show the relationship among inverting problems, decision problems and gap problems. These problems find a nice and rich practical ins ..."
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Cited by 123 (11 self)
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Abstract. This paper introduces a novel class of computational problems, the gap problems, which can be considered as a dual to the class of the decision problems. We show the relationship among inverting problems, decision problems and gap problems. These problems find a nice and rich practical instantiation with the DiffieHellman problems. Then, we see how the gap problems find natural applications in cryptography, namely for proving the security of very efficient schemes, but also for solving a more than 10year old open security problem: the Chaum’s undeniable signature.
A Generalized Birthday Problem
 In CRYPTO
, 2002
"... We study a kdimensional generalization of the birthday problem: given k lists of nbit values, nd some way to choose one element from each list so that the resulting k values xor to zero. For k = 2, this is just the extremely wellknown birthday problem, which has a squareroot time algorithm ..."
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Cited by 93 (0 self)
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We study a kdimensional generalization of the birthday problem: given k lists of nbit values, nd some way to choose one element from each list so that the resulting k values xor to zero. For k = 2, this is just the extremely wellknown birthday problem, which has a squareroot time algorithm with many applications in cryptography.
Predicate Encryption Supporting Disjunctions, Polynomial Equations, and Inner Products
"... Abstract. Predicate encryption is a new paradigm generalizing, among other things, identitybased encryption. In a predicate encryption scheme, secret keys correspond to predicates and ciphertexts are associated with attributes; the secret key SKf corresponding to a predicate f can be used to decryp ..."
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Cited by 78 (15 self)
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Abstract. Predicate encryption is a new paradigm generalizing, among other things, identitybased encryption. In a predicate encryption scheme, secret keys correspond to predicates and ciphertexts are associated with attributes; the secret key SKf corresponding to a predicate f can be used to decrypt a ciphertext associated with attribute I if and only if f(I) = 1. Constructions of such schemes are currently known for relatively few classes of predicates. We construct such a scheme for predicates corresponding to the evaluation of inner products over ZN (for some large integer N). This, in turn, enables constructions in which predicates correspond to the evaluation of disjunctions, polynomials, CNF/DNF formulae, or threshold predicates (among others). Besides serving as a significant step forward in the theory of predicate encryption, our results lead to a number of applications that are interesting in their own right. 1
A General Framework for Subexponential Discrete Logarithm Algorithms in Groups of Unknown Order
, 2000
"... We develop a generic framework for the computation of logarithms in nite class groups. The model allows to formulate a probabilistic algorithm based on collecting relations in an abstract way independently of the specific type of group to which it is applied, and to prove a subexponential running ti ..."
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Cited by 56 (9 self)
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We develop a generic framework for the computation of logarithms in nite class groups. The model allows to formulate a probabilistic algorithm based on collecting relations in an abstract way independently of the specific type of group to which it is applied, and to prove a subexponential running time if a certain smoothness assumption is verified. The algorithm proceeds in two steps: First, it determines the abstract group structure as a product of cyclic groups; second, it computes an explicit isomorphism, which can be used to extract discrete logarithms.
How to win the clonewars: efficient periodic ntimes anonymous authentication
 In ACM Conference on Computer and Communications Security
, 2006
"... We create a credential system that lets a user anonymously authenticate at most n times in a single time period. A user withdraws a dispenser of n etokens. She shows an etoken to a verifier to authenticate herself; each etoken can be used only once, however, the dispenser automatically refreshes ..."
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Cited by 55 (11 self)
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We create a credential system that lets a user anonymously authenticate at most n times in a single time period. A user withdraws a dispenser of n etokens. She shows an etoken to a verifier to authenticate herself; each etoken can be used only once, however, the dispenser automatically refreshes every time period. The only prior solution to this problem, due to Damg˚ard et al. [30], uses protocols that are a factor of k slower for the user and verifier, where k is the security parameter. Damg˚ard et al. also only support one authentication per time period, while we support n. Because our construction is based on ecash, we can use existing techniques to identify a cheating user, trace all of her etokens, and revoke her dispensers. We also offer a new anonymity service: glitch protection for basically honest users who (occasionally) reuse etokens. The verifier can always recognize a reused etoken; however, we preserve the anonymity of users who do not reuse etokens too often. 1
Security of Signed ElGamal Encryption
 In Asiacrypt ’2000, LNCS 1976
, 2000
"... . Assuming a cryptographically strong cyclic group G of prime order q and a random hash function H, we show that ElGamal encryption with an added Schnorr signature is secure against the adaptive chosen ciphertext attack, in which an attacker can freely use a decryption oracle except for the target c ..."
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Cited by 41 (3 self)
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. Assuming a cryptographically strong cyclic group G of prime order q and a random hash function H, we show that ElGamal encryption with an added Schnorr signature is secure against the adaptive chosen ciphertext attack, in which an attacker can freely use a decryption oracle except for the target ciphertext. We also prove security against the novel onemoredecyption attack. Our security proofs are in a new model, corresponding to a combination of two previously introduced models, the Random Oracle model and the Generic model. The security extends to the distributed threshold version of the scheme. Moreover, we propose a very practical scheme for private information retrieval that is based on blind decryption of ElGamal ciphertexts. 1 Introduction and Summary We analyse a very practical public key cryptosystem in terms of its security against the strong adaptive chosen ciphertext attack (CCA) of [RS92], in which an attacker can access a decryption oracle on arbitrary ciphertexts (ex...