The integer sub-decomposition method to improve the elliptic elgamal digital signature algorithm

Abstract

The elliptic ElGamal digital signature algorithm (EEDSA) has been created based on the ElGamal public key cryptosystem (EPKC) and the algorithm of the digital signature (DSA), defined on the elliptic curves, which was accepted in several (ANSI, IEEE, NIST and ISO) standards. The processes to generate keys, compute a signature and verify this signature in EEDSA require computing the elliptic curve scalar multiplications kP. This work proposes improving the EEDSA through computing an elliptic curve scalar multiplication by employing the sub-decomposition of integer which is known by ISD method instead of using doubling and addition points on E over a prime field Fp. The proposed method, namely the EEDSA-ISD algorithm, is benefited from the fast computations in the ISD method, which is depended on the sub-decomposition of the scalars in scalar multiplications. The EEDSA-ISD method also depends on speeding the computations of the efficiently computable endomorphisms of elliptic curve E over finite fields in ISD method. On the other hand, the security level of the improved ECDSA-ISD algorithm is determined based on the hardness to solve the elliptic curve discrete logarithm problem (ECDLP) from its sub-decomposition. So, for these reasons, the improved EEDSA-ISD algorithm is considered as more fast and secure to resist the ECDLP attacks. Therefore, it is more efficient in compared to the original EEDSA.