Implementation of Elliptic Curve Cryptosystem with Bitcoin Curves on SECP256k1, NIST256p, NIST521p, and LLL

Abstract

Very recent attacks like ladder leaks demonstrated the feasibility of recovering private keys with side-channel attacks using just one bit of secret nonce. ECDSA nonce bias can be exploited in many ways. Some attacks on ECDSA involve complicated Fourier analysis and lattice mathematics. This paper will enable cryptographers to identify efficient ways in which ECDSA can be cracked on curves NIST256p, SECP256k1, NIST521p, and weak nonce, kind of attacks that can crack ECDSA and how to protect yourself. Initially, we begin with an ECDSA signature to sign a message using the private key and validate the generated signature using the shared public key. Then we use a nonce or a random value to randomize the generated signature. Every time we sign, a new verifiable random nonce value is created, and a way in which the intruder can discover the private key if the signer leaks any one of the nonce values. Then we use Lenstra–Lenstra–Lovasz (LLL) method as a black box, we will try to attack signatures generated from bad nonce or bad random number generator (RAG) on NIST256p, SECP256k1 curves. The combination of nonce generation, post-message signing, and validation in ECDSA helps achieve Uniqueness, Authentication, Integrity, and Non-Repudiation. The analysis is performed by considering all three curves for the implementation of the Elliptic Curve Digital Signature Algorithm (ECDSA). The comparative analysis for each of the selected curves in terms of computational time is done with the leak of nonce and with the Lenstra–Lenstra–Lovasz method to crack ECDSA. The average computational costs to break ECDSA with curves NIST256p, NIST521p, and SECP256k1 are 0.016, 0.34,0.46 respectively which is almost zero depicting the strength of the algorithm. The average computational costs to break ECDSA with curves SECP256K1 and NIST256p using LLL are 2.9 and 3.4 respectively.