Elliptic curve cryptosystems are usually implemented over fields of characteristic two or over (large) prime fields. For large prime fields, projective coordinates are more suitable as they reduce the computational workload in a point multiplication. In this case, choosing for parameter a the value −3 further reduces the workload. Over Fp, not all elliptic curves can be rescaled through isomorp...

The isogeny for elliptic curve cryptosystems was initially used for the efficient improvement of order counting methods. Recently, Smart proposed the countermeasure using isogeny for resisting the refined differential power analysis by Goubin (Goubin’s attack). In this paper, we examine the countermeasure using isogeny against zero-value point (ZVP) attack that is generalization of Goubin’s att...

Elliptic curves in Hesse form admit more suitable arithmetic than ones in Weierstrass form. But elliptic curve cryptosystems usually use Weierstrass form. It is known that both those forms are birationally equivalent. Birational equivalence is relatively hard to compute. We prove that elliptic curves in Hesse form and in Weierstrass form are linearly equivalent over initial field or its small e...

The Diffie-Hellman problem as a cryptographic primitive plays an important role in modern cryptology. The Bit Security or Hard-Core Bits of Diffie-Hellman problem in arbitrary finite cyclic group is a long-standing open problem in cryptography. Until now, only few groups have been studied. Hyperelliptic curve cryptography is an alternative to elliptic curve cryptography. Due to the recent crypt...

Elliptic Curve Cryptography (ECC) is a branch of Cryptography that can be used for encrypting data, generating digital signatures or exchanging keying material during the initial phases of a secure communication. Regarding encryption, the best-known scheme based on ECC is the Elliptic Curve Integrated Encryption Scheme (ECIES). A Java implementation of ECIES is presented in this paper, showing ...

We survey the rings of low multiplicative complexity and the redundant representation of finite fields. The construction is originally due to Ito and Tsujii [3]. We give the important results of Silverman’s works in [1], [2]. Moreover, we note that the fields constructed with Silverman’s method are not suitable for elliptic curve cryptography while Silverman suggests those curves can be used in...

Compared to other public key cryptography counterparts like Diffie-Hellman (DH) and Rivest Shamir Adleman (RSA), Elliptic Curve Cryptography (ECC) is known to provide equivalent level of security with lower number of bits used. Reduced bit usage implies less power and logic area are required to implement this cryptographic scheme. This is particularly important in wireless networks, where a hig...

Elliptical curve cryptography (ECC) is a public key encryption technique based on elliptic curve theory that can be used to create faster, smaller, and more efficient cryptographic keys. ECC generates keys through the properties of the elliptic curve equation instead of the traditional method of generation as the product of very large prime numbers. Because ECC helps to establish equivalent sec...

It is generally accepted that data encryption is the key role in current and future technologies. Many Public key cryptography schemes were presented, divided into different classes depending on a specific mathematical problem. Cryptography plays an important task in accomplishing information security. It is used for encrypting or signing data at the source before transmission, and then decrypt...