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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 369 (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.
Speeding Up The Computations On An Elliptic Curve Using AdditionSubtraction Chains
 Theoretical Informatics and Applications
, 1990
"... We show how to compute x k using multiplications and divisions. We use this method in the context of elliptic curves for which a law exists with the property that division has the same cost as multiplication. Our best algorithm is 11.11% faster than the ordinary binary algorithm and speeds up acco ..."
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Cited by 100 (4 self)
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We show how to compute x k using multiplications and divisions. We use this method in the context of elliptic curves for which a law exists with the property that division has the same cost as multiplication. Our best algorithm is 11.11% faster than the ordinary binary algorithm and speeds up accordingly the factorization and primality testing algorithms using elliptic curves. 1. Introduction. Recent algorithms used in primality testing and integer factorization make use of elliptic curves defined over finite fields or Artinian rings (cf. Section 2). One can define over these sets an abelian law. As a consequence, one can transpose over the corresponding groups all the classical algorithms that were designed over Z/NZ. In particular, one has the analogue of the p \Gamma 1 factorization algorithm of Pollard [29, 5, 20, 22], the Fermatlike primality testing algorithms [1, 14, 21, 26] and the public key cryptosystems based on RSA [30, 17, 19]. The basic operation performed on an elli...
Simultaneous hardcore bits and cryptography against memory attacks
 In TCC
, 2009
"... Abstract. This paper considers two questions in cryptography. Cryptography Secure Against Memory Attacks. A particularly devastating sidechannel attack against cryptosystems, termed the “memory attack”, was proposed recently. In this attack, a significant fraction of the bits of a secret key of a c ..."
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Cited by 74 (8 self)
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Abstract. This paper considers two questions in cryptography. Cryptography Secure Against Memory Attacks. A particularly devastating sidechannel attack against cryptosystems, termed the “memory attack”, was proposed recently. In this attack, a significant fraction of the bits of a secret key of a cryptographic algorithm can be measured by an adversary if the secret key is ever stored in a part of memory which can be accessed even after power has been turned off for a short amount of time. Such an attack has been shown to completely compromise the security of various cryptosystems in use, including the RSA cryptosystem and AES. We show that the publickey encryption scheme of Regev (STOC 2005), and the identitybased encryption scheme of Gentry, Peikert and Vaikuntanathan (STOC 2008) are remarkably robust against memory attacks where the adversary can measure a large fraction of the bits of the secretkey, or more generally, can compute an arbitrary function of the secretkey of bounded output length. This is done without increasing the size of the secretkey, and without introducing any
New PublicKey Schemes Based on Elliptic Curves over the Ring Z_n
, 1991
"... Three new trapdoor oneway functions are proposed that are based on elliptic curves over the ring Z_n. The first class of functions is a naive construction, which can be used only in a digital signature scheme, and not in a publickey cryptosystem. The second, preferred class of function, does not s ..."
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Cited by 46 (0 self)
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Three new trapdoor oneway functions are proposed that are based on elliptic curves over the ring Z_n. The first class of functions is a naive construction, which can be used only in a digital signature scheme, and not in a publickey cryptosystem. The second, preferred class of function, does not suffer from this problem and can be used for the same applications as the RSA trapdoor oneway function, including zeroknowledge identification protocols. The third class of functions has similar properties to the Rabin trapdoor oneway functions. Although the security of these proposed schemes is based on the difficulty of factoring n, like the RSA and Rabin schemes, these schemes seem to be more secure than those schemes from the viewpoint of attacks without factoring such as low multiplier attacks.
An efficient discrete log pseudo random generator
 Proc. of Crypto '98
, 1998
"... Abstract. The exponentiation function in a finite field of order p (a prime number) is believed to be a oneway function. It is well known that O(log log p) bits are simultaneously hard for this function. We consider a special case of this problem, the discrete logarithm with short exponents, which ..."
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Cited by 20 (1 self)
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Abstract. The exponentiation function in a finite field of order p (a prime number) is believed to be a oneway function. It is well known that O(log log p) bits are simultaneously hard for this function. We consider a special case of this problem, the discrete logarithm with short exponents, which is also believed to be hard to compute. Under this intractibility assumption we show that discrete exponentiation modulo a prime p can hide n−ω(log n) bits(n=⌈log p ⌉ and p =2q+1, where q is also a prime). We prove simultaneous security by showing that any information about the n − ω(log n) bits can be used to discover the discrete log of g s mod p where s has ω(log n) bits. For all practical purposes, the size of s can be a constant c bits. This leads to a very efficient pseudorandom number generator which produces n − c bits per iteration. For example, when n = 1024 bits and c = 128 bits our pseudorandom number generator produces a little less than 900 bits per exponentiation. 1
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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Cited by 17 (0 self)
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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.
Generic Groups, Collision Resistance, and ECDSA
 Designs, Codes and Cryptography
, 2002
"... Proved here is the sufficiency of certain conditions to ensure the Elliptic Curve Digital Signature Algorithm (ECDSA) existentially unforgeable by adaptive chosenmessage attacks. The sufficient conditions include (i) a uniformity property and collisionresistance for the underlying hash function, ( ..."
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Cited by 13 (1 self)
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Proved here is the sufficiency of certain conditions to ensure the Elliptic Curve Digital Signature Algorithm (ECDSA) existentially unforgeable by adaptive chosenmessage attacks. The sufficient conditions include (i) a uniformity property and collisionresistance for the underlying hash function, (ii) pseudorandomness in the private key space for the ephemeral private key generator, (iii) generic treatment of the underlying group, and (iv) a further condition on how the ephemeral public keys are mapped into the private key space. For completeness, a brief survey of necessary security conditions is also given. Some of the necessary conditions are weaker than the corresponding sufficient conditions used in the security proofs here, but others are identical.
Elliptic Curve Pseudorandom Sequence Generators
, 1998
"... In this paper, we introduce a new approach to the generation of binary sequences by applying trace functions to elliptic curves over GF(2 m ). We call these sequences elliptic curve pseudorandom sequences (ECsequence). We determine their periods, distribution of zeros and ones, and linear span ..."
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Cited by 12 (1 self)
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In this paper, we introduce a new approach to the generation of binary sequences by applying trace functions to elliptic curves over GF(2 m ). We call these sequences elliptic curve pseudorandom sequences (ECsequence). We determine their periods, distribution of zeros and ones, and linear spans for a class of ECsequences generated from supersingular curves. We exhibit a class of ECsequences which has half period as a lower bound for their linear spans. ECsequences can be constructed algebraically and can be generated efficiently in software or hardware by the same methods that are used for implementation of elliptic curve publickey cryptosystems.
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...
A PublicKey Encryption Scheme with PseudoRandom Ciphertexts
 In ESORICS ’04, LNCS 3193
, 2004
"... Abstract. This work presents a practical publickey encryption scheme that offers security under adaptive chosenciphertext attack (CCA) and has pseudorandom ciphertexts, i.e. ciphertexts indistinguishable from random bit strings. Ciphertext pseudorandomness has applications in steganography. The ..."
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Cited by 9 (0 self)
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Abstract. This work presents a practical publickey encryption scheme that offers security under adaptive chosenciphertext attack (CCA) and has pseudorandom ciphertexts, i.e. ciphertexts indistinguishable from random bit strings. Ciphertext pseudorandomness has applications in steganography. The new scheme features short ciphertexts due to the use of elliptic curve cryptography, with ciphertext pseudorandomness achieved through a new key encapsulation mechanism (KEM) based on elliptic curve DiffieHellman with a pair of elliptic curves where each curve is a twist of the other. The publickey encryption scheme resembles the hybrid DHIES construction; besides by using the new KEM, it differs from DHIES in that it uses an authenticatethenencrypt (AtE) rather than encryptthenauthenticate (EtA) approach for symmetric cryptography. 1