Formalizing a mathematical theory is a necessary first step to proving the correctness of programs that refer to that theory in their specification. This paper demonstrates how the mathematical theory of elliptic curves and their application to cryptography can be formalized in higher order logic. This formal development is mechanized using the HOL4 theorem prover, resulting in a collection of ...

The strength of public key cryptography utilizing Elliptic Curves relies on the difficulty of computing discrete logarithms in a finite field. Diffie-Hellman key exchange algorithm also relies on the same fact. There are two flavors of this algorithm, one using Elliptic Curves and another without using Elliptic Curves. Both flavors of the algorithm rely on the difficulty of computing discrete l...

Public-key cryptography is based on the intractability of certain mathematical problems. Early public-key systems, such as the RSA algorithm, are secure assuming that it is difficult to factor a large integer composed of two or more large prime factors. For elliptic-curve-based protocols, it is assumed that finding the discrete logarithm of a random elliptic curve element with respect to a publ...

Elliptic curve cryptography has raised attention as it allows for having shorter keys and ciphertexts. For example, to obtain similar security levels with 2048 bit RSA key, it is necessary to use only 256 bit keys using over elliptic curve cryptography. Additionally, developments in the index calculus method for solving a discrete logarithm problem increases the sizes of the keys to keep the se...

The threshold cryptography provides a new approach to building intrusion tolerance applications. In this paper, a threshold decryption scheme based elliptic curve cryptography is presented. A zero-knowledge test approach based on elliptic curve cryptography is designed. The application of these techniques in Web security is studied. Performance analysis shows that our scheme is characterized by...

Elliptic Curve Cryptography (ECC) is a relatively recent branch of cryptography which is based on the arithmetic on elliptic curves and security of the hardness of the Elliptic Curve Discrete Logarithm Problem (ECDLP). Elliptic curve cryptographic schemes are public-key mechanisms that provide encryption, digital signature and key exchange capabilities. Elliptic curve algorithms are also applie...

We present a pseudo random bit generator whose security is based on the intractability of the discrete logarithm problem in the group E(Fp) of rational points on an elliptic curve over a finite prime field Fp. The bit generator is implemented within the framework of the Java Cryptography Architecture (JCA). It uses an elliptic curve E chosen such that both E(Fp) and its twist E (Fp) are of prim...

Cryptography is the study of techniques for ensuring the secrecy and authentication of the information. Public –key encryption schemes are secure only if the authenticity of the public-key is assured. Elliptic curve arithmetic can be used to develop a variety of elliptic curve cryptographic (ECC) schemes including key exchange, encryption and digital signature. The principal attraction of ellip...

This paper provides an overview of elliptic curves and their use in cryptography. The focus of the paper is on the performance advantages obtained in the wireless environments by using elliptic curve cryptography instead of traditional cryptosystems such as RSA. Specific applications to secure messaging and identity-based encryption are also discussed. keywords: elliptic curves, wireless, Digit...