Citations: Elliptic curve cryptosystems using curves of smooth order over the ring Zn

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S. Vanstone and R. Zuccherato, Elliptic Curve Cryptosystem Using Curves of Smooth Order Over the Ring Zn , IEEE Trans. Inf. Theory, Vol. 43, No 4, July 1997.

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CryptoComputing with rationals - Fouque, Stern, Wackers (2002) (Correct)

....in section 4. Finally, we describe practical parameters and our spreadsheet application. 2 Preliminary tools 2. 1 The Paillier cryptosystem Various cryptosystems based on randomized encryption schemes E(M) which encrypt a message M by raising a basis g to the power M have been proposed so far [13, 4, 7, 23, 15 17]. Their security is based on the intractability of computing discrete logarithm in basis g without a secret element, the secret key. Given this secret as a trapdoor, the computation becomes easy. We call those cryptosystems trapdoor discrete logarithm schemes. As an important consequence of this ....

S. Vanstone and R. Zuccherato. Elliptic Curve Cryptosystem Using Curves of Smooth Order Over the Ring Zn . IEEE Transaction on Information Theory, IT-43, 1997. 11

On the Difficulty of Breaking the Diffie-Hellman Protocol - Maurer, Wolf (1995) (Correct)

....of order p 1 are explicitly constructable. We will show later that the subgroup of order p 1 of F p 2 works for all p. The following statements about the orders of the curves of the form (3) in the case they are not p 1 can be proved with algebraic number theoretic methods (see [15] [37]) If p j 1 (mod 4) p has a unique representation in the ring Z[i] of Gaussian integers: p = a bi) a Gamma bi) a 2 b 2 ; j 1 (mod 2 2i) The curves y 2 = x 3 Gamma Dx have the orders p 1 Sigma 2a; p 1 Sigma 2b; 4) and the four orders occur equally often. Let now ....

....how groups can be built such that the side string S is known to the group designer or provably exists. We show how to construct large prime numbers p such that an appropriate auxiliary group H p over F p is either constructable or at least provably exists. We generalize a method, presented in [37], of constructing a large prime p such that either a quarter of the curves y 2 = x 3 Gamma Dx or every sixth curve of the form y 2 = x 3 D have smooth order. First, we construct primes p = a 2 (k Sigma 1) 2 (for a fixed k with l digits) such that a 2 k 2 , which is then one ....

S.A. Vanstone and R.J. Zuccherato, Elliptic curve cryptosystems using curves of smooth order over the ring Zn , Preliminary version, 1994.

Diffie-Hellman Oracles - Maurer, Wolf (1996) (15 citations) (Correct)

....more involved. Such a group G can be obtained by choosing a large smooth number m and using the method of Lay and Zimmer [10] for constructing a prime p together with an elliptic curve of order m. We now consider efficient constructions for the first case. We generalize a method, presented in [20] by Vanstone and Zuccherato, for constructing a large prime p such that either a quarter of the curves y 2 = x 3 Gamma Dx or every sixth curve of the form y 2 = x 3 D have smooth order. We show how to construct primes p = a 2 (k Sigma 1) 2 (for a fixed k with l digits) such that ....

S.A. Vanstone and R.J. Zuccherato, Elliptic curve cryptosystems using curves of smooth order over the ring Zn , Preliminary version, 1994.

Sharing Decryption in the Context of Voting or Lotteries - Fouque, Poupard, Stern (2000) (26 citations) (Correct)

....b = H = h s = g as . Consequently G and H have the same discrete logarithm in the respective basis g and h. 3. 3 The Paillier cryptosystem Various cryptosystems based on randomized encryption schemes E(M) which encrypt a message M by raising a basis g to the power M have been proposed so far [11, 1, 4, 22, 12 14]. Their security is based on the intractability of computing discrete logarithm in the basis g without a secret data, the secret key, and easy using this trapdoor. We call those cryptosystems trapdoor discrete logarithm schemes. As an important consequence of this encryption technique, those ....

....the most interesting trapdoor discrete logarithm cryptosystem, according to its efficiency and to its large bandwidth. In order to study the security of our proposal, we have defined semantic security for threshold cryptosystems. The distribution of other trapdoor discrete logarithm cryptosystems [11, 1, 4, 22, 12, 13] still remains an open problem. ....

S. Vanstone and R. Zuccherato. Elliptic Curve Cryptosystem Using Curves of Smooth Order Over the Ring Zn . IEEE Transaction on Information Theory, IT43, 1997.

On the Complexity of Breaking the Diffie-Hellman Protocol - Maurer, Wolf (1996) (1 citation) (Correct)

....more involved. Such a group G can be obtained by choosing a large smooth number m and using the method of Lay and Zimmer [19] for constructing a prime p together with an elliptic curve of order m. We now consider efficient constructions for the first case. We generalize a method, presented in [37] by Vanstone and Zuccherato, for constructing a large prime p such that either a quarter of the curves y 2 = x 3 Gamma Dx or every sixth curve of the form y 2 = x 3 D have smooth order. First, we construct primes p = a 2 (k Sigma 1) 2 (for a fixed k with l digits) such that a 2 ....

S.A. Vanstone and R.J. Zuccherato, Elliptic curve cryptosystems using curves of smooth order over the ring Zn , Preliminary version, 1994.

On a cryptosystem of Vanstone and Zuccherato - James Mckee And (1998) (5 citations) Self-citation (Vanstone Zuccherato) (Correct)

....On a cryptosystem of Vanstone and Zuccherato James McKee and Richard Pinch James McKee is at Pembroke College, Oxford. Email address: mckee maths.ox.ac.uk Richard Pinch is at Queens College, Cambridge. Email address: rgep cam.ac. uk February 14, 1998 DRAFT 2 Abstract In [1], Vanstone and Zuccherato proposed a public key elliptic curve cryptosystem in which the public key consists of an integer N and an elliptic curve E defined over the ring Z=NZ. Here N is a product of two secret primes p and q, each of special form, and the order of E modulo N is smooth. We present ....

....each of which factors the modulus N and hence breaks the cryptosystem. The first attack exploits the special form of p and q; the second exploits the smoothness of the elliptic curve; and the third attack breaks a proposed application of the system to user authentication. For parameters as in [1], the modulus can be factored within a fraction of a second. Keywords Cryptography, public key, authentication, discrete logarithm, elliptic curves, factoring. I. The proposed cryptosystem In a recent cryptosystem proposed by Vanstone and Zuccherato [1] part of the public key is an integer N ....

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Scott A. Vanstone and Robert J. Zuccherato, "Elliptic curve cryptosystems using curves of smooth order over the ring Zn ," IEEE Trans. Inform. Theory, vol. 43, no. 4, pp. 1231--1237, 1997.

Public-Key Cryptosystems Based on Composite - Degree Residuosity Classes (Correct)