Attacking OpenSSL ECDSA with a small amount of side-channel information

Abstract

In this work, we mount a lattice attack on the ECDSA signatures implemented by the latest version of OpenSSL which uses the windowed non-adjacent form method to implement the scalar multiplication. We first develop a new way of extracting information from the side-channel results of the ECDSA signatures. Just given a small fraction of the information about a side-channel result denoted as double-and-add chain, we take advantage of the length of the chain together with positions of two non-zero digits to recover information about the ephemeral key. Combining the information of both the most significant digits and the least significant bits, we are able to gain more information about the ephemeral key. The problem of recovering ECDSA secret key is then translated to the hidden number problem which can be solved by lattice reduction algorithms. Our attack is mounted to the series secp256k1 curve, and the result shows that 85 signatures would be enough to recover the secret key, which is better than the result that previous attack gained only utilizing the information extracted from the least significant bits, using about 200 signatures to recover the secret key.