A Modification of ECDSA to Avoid the Rho Method Attack

Abstract

Elliptic Curve Digital Signature Algorithm (ECDSA) is a digital signature algorithm that utilizes an elliptic curve. ECDSA consists of three steps, which are key generation, signature generation, and verification algorithm. ECDSA is used on Bitcoin transactions to generate the user's public key, private key, and signature, and also to verify a Bitcoin user's signature. There are some researches on ECDSA weak randomness which can be exploited by attackers to reveal the user's private key, and causes thefts of the user's money. ECDSA weak randomness is generating a random number that is not cryptographically secure. Some modifications of ECDSA to overcome this problem have been done, such as generating the digital signature by using two private keys. Although those modified algorithms overcome ECDSA weak randomness exploitations, it is not resistant to the Rho method attack which can solve elliptic curve discrete logarithm problem (ECDLP). In case ECDLP can be solved, the user's private key can be revealed. Therefore, in this paper, we modify an ECDSA algorithm that overcomes the exploitation of ECDSA weak randomness and is also resistant to the Rho method attack by using three private keys.