Security of public-key cryptosystems based on Chebyshev polynomials

Abstract

Chebyshev polynomials have been recently proposed for designing public-key systems. Indeed, they enjoy some nice chaotic properties, which seem to be suitable for use in Cryptography. Moreover, they satisfy a semi-group property, which makes possible implementing a trapdoor mechanism. In this paper we study a public key cryptosystem based on such polynomials, which provides both encryption and digital signature. The cryptosystem works on real numbers and is quite efficient. Unfortunately, from our analysis it comes up that it is not secure. We describe an attack which permits to recover the corresponding plaintext from a given ciphertext. The same attack can be applied to produce forgeries if the cryptosystem is used for signing messages. Then, we point out that also other primitives, a Diffie-Hellman like key agreement scheme and an authentication scheme, designed along the same lines of the cryptosystem, are not secure due to the aforementioned attack. We close the paper by discussing the issues and the possibilities of constructing public key cryptosystems on real numbers.

Security Attacks and Enhancements to Chaotic Map-Based RFID Authentication Protocols

“…Chebyshev polynomials are employed in various cryptographic schemes including key agreement protocols [26,34], password-based authentication schemes [16,20], public key encryption schemes [37,8], and RFID authentication protocols [11,6,3]. The implementation of Chebyshev polynomials uses small resources so that smart cards and RFID tags can utilize them.…”

Efficiency Analysis for Public Key Systems Based on Fractal Functions

“…Public-key systems are considered to be slower than private (symmetric) key systems, so that they are used to encrypt small data items and preferable to be used as a key exchange in symmetric systems to protect the real data (Xiang et al, 2005). All the three encryption algorithms RSA, Rabin and ElGamal are based on their mechanism on the following system:…”

Blind Signature Scheme Based on Chebyshev Polynomials

“…In this section we briefly describe Chebyshev polynomials, since they represent the cornerstone on which the public key cryptosystem, described in [11,15], key agreement protocols, described in [23] and the authentication scheme, described in [14], are built.…”

“…But it was soon found the private key could be quickly recovered from the public key, using trigonometric function substitution [14,16]. In other words, the oneway condition is not satisfied in such cryptosystem.…”