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Reducing the Elliptic Curve Cryptosystem of MeyerMüller to the Cryptosystem of RabinWilliams
, 1996
"... . At Eurocrypt'96, Meyer and Muller presented a new Rabintype cryptosystem based on elliptic curves. In this paper, we will show that this cryptosystem may be reduced to the cryptosystem of WilliamsRabin. 1 Introduction In 1991, Koyama, Maurer, Okamoto and Vanstone [15] pointed out the existence o ..."
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. At Eurocrypt'96, Meyer and Muller presented a new Rabintype cryptosystem based on elliptic curves. In this paper, we will show that this cryptosystem may be reduced to the cryptosystem of WilliamsRabin. 1 Introduction In 1991, Koyama, Maurer, Okamoto and Vanstone [15] pointed out the existence of new oneway trapdoor functions similar to the RSA on elliptic curves over a ring. At Eurocrypt'96, Meyer and Muller presented an other elliptic RSAtype cryptosystem with a public encryption exponent equal to 2. We will show that this cryptosystem may be reduced to the cryptosystem of RabinWilliams [20, 22]. The remainder of the paper is organized as follows. Section 2 describes the cryptosystem of Meyer and Muller. In Section 3, we show how it may be reduced to the cryptosystem of RabinWilliams. Finally, we conclude in Section 4. CG1996/4 c fl1996 by UCL Crypto Group For more informations, see http://www.dice.ucl.ac.be/crypto/techreports.html Presented at the rump session of Eurocr...
Cryptosystem of Chua and Ling
, 1997
"... Introduction: At Eurocrypt '96, Meyer and M uller [1] presented a new Rabintype scheme based on elliptic curves. In [2], this system was reduced to the system of RabinWilliams [3, 4]. Using the same technique, we show that the system of Chua and Ling [5] can also be reduced. ChuaLing's cryptosys ..."
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Introduction: At Eurocrypt '96, Meyer and M uller [1] presented a new Rabintype scheme based on elliptic curves. In [2], this system was reduced to the system of RabinWilliams [3, 4]. Using the same technique, we show that the system of Chua and Ling [5] can also be reduced. ChuaLing's cryptosystem: This cryptosystem is based on a singular cubic curve of the form Cn (b) : y + bx (mod n). To setup the system, each user chooses two large primes p and q both congruent to 11 modulo 12. Then, he publishes the value of n = pq. Suppose Alice wants to send a message m to Bob. Then she chooses # Z/nZ{0,1} and sets P = (m ). Next, she computes c = # mod n and b = (# 1)m mod n, and sends the ciphertext consisting of c, b, xQ = x([2]P), t = y ([2]P) n and u = lsb ( y ([2]P)). To recover the plaintext m, Bob computes the unique yQ satisfying y Q Q + bx Q (mod n) with Jacobi symbol t and lsb u. He sets Q = (xQ , yQ ). Letting Qp = Q mod p and Qq
Note on the Preliminary Version of the MeyerMüller's Cryptosystem
, 1996
"... . After the introduction of the RSA cryptosystem [5], Rabin [4] proposed to use even public exponents. The resulting function was four to one, and was to be proved as intractable as factorization. Shortly after, Williams [6] showed how to transform the Rabin's function to a one to one. The drawback ..."
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. After the introduction of the RSA cryptosystem [5], Rabin [4] proposed to use even public exponents. The resulting function was four to one, and was to be proved as intractable as factorization. Shortly after, Williams [6] showed how to transform the Rabin's function to a one to one. The drawback of his method is the bitlength extension of the message. This shortcoming was later eliminated by Guillou and Quisquater [2, 3]. Very recently, Meyer and Muller [1] proposed an analogous cryptosystem based on elliptic curves over a ring. We shall show that their cryptosystem is equivalent to the Williams' one. 1 MeyerMuller cryptosystem In this section, we shall not review in details the MeyerMuller cryptosystem. For a full description, we refer to the original paper [1]. 1.1 Encryption of a message m Each user chooses n, the product of two large prime numbers p and q such that p; q j 7 (mod 8). Then, the protocol goes as follows. Note on the Preliminary Version of the MeyerMuller's ...
Public Key Cryptosystems using Elliptic Curves
, 1997
"... This report is a survey on public key cryptosystems that use the theory of elliptic curves. A considerable part will be about the theory of elliptic curves. Encryption systems, digital signature schemes and key agreement schemes using elliptic curves will be described. Their workload and bandwidth w ..."
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This report is a survey on public key cryptosystems that use the theory of elliptic curves. A considerable part will be about the theory of elliptic curves. Encryption systems, digital signature schemes and key agreement schemes using elliptic curves will be described. Their workload and bandwidth will be addressed and some attacks will be described. For all systems the security is based either on the elliptic curve discrete logarithm problem or on the difficulty of factorization. The differences between conventional and elliptic curve systems shall be addressed. Systems based on the elliptic curve discrete logarithm problem can be used with shorter keys to provide the same security, compared to similar conventional systems. Elliptic curve systems based on factoring are slightly more resistant as conventional systems against some attacks.
Topics in PublicKey Cryptography II
, 1999
"... 6> Vn(P; Q) from Dickson polynomials Vn(P; Q) = [ n 2 ] X i=0 n n \Gamma i ` n \Gamma i i ' (\GammaQ) i P n\Gamma2i Fact: Vn(V k (P; Q); Q k ) = V nk (P; Q). In particular, if Q = 1, then Vn(V k (P; 1); 1) = V nk (P; 1) = V k (Vn(P; Q); 1). The above fact forms the bas ..."
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6> Vn(P; Q) from Dickson polynomials Vn(P; Q) = [ n 2 ] X i=0 n n \Gamma i ` n \Gamma i i ' (\GammaQ) i P n\Gamma2i Fact: Vn(V k (P; Q); Q k ) = V nk (P; Q). In particular, if Q = 1, then Vn(V k (P; 1); 1) = V nk (P; 1) = V k (Vn(P; Q); 1). The above fact forms the basis for many RSA and ElGamal type cryptosystems based on Lucas sequences. Observe th
A Survey of Elliptic Curve Cryptosystems, Part I: Introductory
, 2003
"... The theory of elliptic curves is a classical topic in many branches of algebra and number theory, but recently it is receiving more attention in cryptography. An elliptic curve is a twodimensional (planar) curve defined by an equation involving a cubic power of coordinate x and a square power of co ..."
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The theory of elliptic curves is a classical topic in many branches of algebra and number theory, but recently it is receiving more attention in cryptography. An elliptic curve is a twodimensional (planar) curve defined by an equation involving a cubic power of coordinate x and a square power of coordinate y. One class of these curves is