The aim is to check if it is possible to break the ECDSA cryptosystem under the following criteria.
- Suppose that each ECDSA signature is generated by using the GLV method for point multiplication (the random nonce $k$ used in each signature is decomposed into $k_1$and $k_2$).
- For every ECDSA signature, I can obtain few bits (least significant bits, most significant bits, ...) of the respective $k_1$ and $k_2$.
Is it possible to carry out a lattice attack to gain the value of the underlying secret key ($\alpha$)? If yes, then can you explain the attack in detail?
PS: I have a beginner level understanding of ECDSA and lattices.