Solving HNP with One Bit Leakage: An Asymmetric Lattice Sieving Algorithm

The Hidden Number Problem (HNP) was introduced by Boneh and Venkastesan to analyze the bit-security of the Diffie–Hellman key exchange scheme. It is often used to mount a side-channel attack on (EC)DSA. The hardness of HNP is mainly determined by the number of nonce leakage bits and the size of the modulus. With the development of lattice reduction algorithms and lattice sieving, the range of practically vulnerable parameters are extended further. However, 1-bit leakage is still believed to be challenging for lattice attacks. In this paper, we proposed an asymmetric lattice sieving algorithm that can solve HNP with 1-bit leakage. The algorithm is composed of a BKZ pre-processing and a sieving step. The novel part of our lattice sieving algorithm is that the lattice used in these two steps have different dimensions. In particular, in the BKZ step we use more samples to derive a better lattice basis, while we just use truncated lattice basis for the lattice sieving step. To verify our algorithm, we use it to solve HNP with 1-bit leakage and 116-bit modulus.