Adapting Belief Propagation to Counter Shuffling of NTTs

Abstract

The Number Theoretic Transform (NTT) is a major building block in recently introduced lattice based post-quantum (PQ) cryptography. The NTT was target of a number of recently proposed Belief Propagation (BP)-based Side Channel Attacks (SCAs). Ravi et al. have recently proposed a number of countermeasures mitigating these attacks. In 2021, Hamburg et al. presented a chosen-ciphertext enabled SCA improving noise-resistance, which we use as a starting point to state our findings. We introduce a pre-processing step as well as a new factor node which we call shuffle node. Shuffle nodes allow for a modified version of BP when included into a factor graph. The node iteratively learns the shuffling permutation of fine shuffling within a BP run. We further expand our attacker model and describe several matching algorithms to find inter-layer connections based on shuffled measurements. Our matching algorithm allows for either mixing prior distributions according to a doubly stochastic mix matrix or to extract permutations and perform an exact un-matching of layers. We additionally discuss the usage of sub-graph inference to reduce uncertainty and improve un-shuffling of butterflies. Based on our results, we conclude that the proposed countermeasures of Ravi et al. are powerful and counter Hamburg et al., yet could lead to a false security perception – a powerful adversary could still launch successful attacks. We discuss on the capabilities needed to defeat shuffling in the setting of Hamburg et al. using our expanded attacker model. Our methods are not limited to the presented case but provide a toolkit to analyze and evaluate shuffling countermeasures in BP-based attack scenarios.

Keywords

• Number Theoretic Transform
• Shuffling
• Kyber
• CCA
• Belief Propagation
• SASCA