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A Digital Signature Scheme Based on Pell Equation

Aditya Mani Mishra1
SRF, Department of Mathematics,Motilal Nehru NIT, Allahabad, U.P., India
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Elliptic curve digital signature algorithm (ECDSA) is well established digital signature scheme based on the discrete logarithms problems. On the other hand, many cryptosystem has been designed using the Pell Equation. We apply the idea of ECDSA on the solution space of Pell equation to design a digital signature scheme. In this paper we compare the security of our signatures scheme to DSA and ECDSA. Our scheme is as secure as conventional DSA. We show that the signatures scheme based on Pell equation is more efficient than its analogue to elliptic curve i.e. ECDSA.


DSA, ECDSA, Pell Equation,Discrete Logarithm Problem.


In 1985, ElGamal [6] proposed a public key cryptosystem and a digital signature scheme based on the difficulty of solving the Discrete Logarithm Problem in the multiplicative group of an appropriate finite field. In 1991, the National Institute of Standards and Technology of USA proposed the Digital Signature Algorithm (DSA) which is an efficient variant of the ElGamal digital signature scheme [14, 17]. The security of DSA is based on discrete logarithms problem. The Elliptic Curve Digital Signature Algorithm (ECDSA) is being proposed as an ANSI standard. Unlike the normal discrete logarithm problem (DLP) and the integer factorization problem (IFP), the elliptic curve discrete logarithm problem (ECDLP) has no sub exponential-time algorithm. For this reason, the strength-per-key-bit is substantially greater in an algorithm that uses elliptic curves. An original ECDSA was proposed in 1992 by Vanstone [18], and its three variants were given in [21, 4]. These signature schemes are basically the analogues of the corresponding ElGamal digital signature schemes [16]. Many variants of ECDSA have been given in [20]. The Pell Equation has existence since a very ancient time in number theory [3]. It has a number of applications in mathematics [7, 19]. In Algebra, it can be used to find regulator of group [2, 11]. Pell Equation has infinite solution and its solution space forms a cyclic group under appropriate group operation. On the basis of this group, many variants of cryptosystem have been designed [19, 5, 15, 8]. In [5] author proposed fast RSA type scheme based on Pell equation in which encryption speed is 1.5 times faster than then standard RSA and the decryption in 2 times faster. In [15] author defined a new operation on the solution space of Pell Equation and proposed three RSA type cryptosystems. In [8] author proposed a PKC whose security is based on DLP. To the best of our knowledge no signature scheme is found in the literature over the Pell's equation. So, our motivation to this article is to construct a digital signature scheme over Pell's equation.


In this section we brief the Pell's equation and Elliptic Curve.
A. Pell Equation: Suppose ������+ is a square free integer. The Diophantine equation ��2 − �� ��2 = 1 ... (2.1.1) is called Pell Equation. The equation has infinite solutions in the real field. The solution of this equation can be found by continued fraction method [19]. By Lagrange theorem [19], once the fundamental solution of Pell Equation is known, one can find its all solutions. Hence, the solution space of (2.1.1) forms a cyclic group, in fact an infinite cyclic group. Now consider equation (2.1.1) under modulo system. Let p be a positive prime number. Consider


The DSA was proposed in August 1991 by the U.S. National Institute of Standards and Technology (NIST) and became a U.S. Federal Information Processing Standard (FIPS 186) in 1993. It was the first digital signature scheme accepted as legally binding by a government [17]. The protocol of DSA is as below.


E. Security
If we compare the addition operation defined in Elliptic curve and Pell's equation, we see that, the addition operation defined in Elliptic curve is more complicated than the operation defined in Pell's equation. One addition in Elliptic curve requires one inversion and two multiplication where as in Pell's equation requires five multiplication. Since one inversion is computationally equivalent to six multiplication. Due to this point of view we can see that proposed scheme over Pell Equation is more efficient then ECDSA.


In this article we proposed a digital signatures scheme based on Pell's equation. The proposed scheme is as secure as standard DSA. We have shown that the proposed scheme is two times faster than standard DSA if 2 log p bits are signed at a time and also the proposed scheme is more efficient than the standard ECDSA.


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