Research on Elliptic Curve Crypto System with Bitcoin Curves – SECP256k1, NIST256p, NIST521p and LLL

Abstract

Very recent attacks like ladder leak demonstrated feasibility to recover private key 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. In 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 ECDSA signature to sign a message using 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 way in which the intruder can discover the private key if the signer leaks any one of the nonce value. 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 analysis is performed by considering all the three curves for implementation of Elliptic Curve Digital Signature Algorithm (ECDSA).The comparative analysis for each of the selected curves in terms of computational time is done with leak of nonce and with 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 to zero depicts the strength of algorithm. The average computational costs to break ECDSA with curves SECP256K1 and NIST256p using LLL are 2.9 and 3.4 respectively.