A Tale of Snakes and Horses: Amplifying Correlation Power Analysis on Quadratic Maps

Abstract

We study the success probabilities of two variants of Correlation Power Analysis (CPA) to retrieve multiple secret bits. The target is a permutation-based symmetric cryptographic construction with a quadratic map as an S-box. More precisely, we focus on the nonlinear mapping X used in the Xoodoo and Keccak-p permutations, which is affine equivalent to the nonlinear mapping of Ascon. We thus consider three-bit and five-bit S-boxes. Our leakage model is the difference in power consumption of register cells before and after one round. It reflects that, >in hardware, the aforesaid cryptographic algorithms are usually implemented by deploying a round-based architecture. The power consumption difference depends on whether the targeted bits in the register flip. In particular, we describe two attacks based on the CPA methodology. First, we start with a standard CPA approach, for which, to the best of our knowledge, we are the first to point out the differences between attacking a three-bit and a five-bit S-box. For CPA, the highest correlation coefficient is the most likely secret hypothesis. We improve on this with our novel combined Correlation Power Analysis (combined CPA), or Snake attack, which uses quadratic map cryptanalysis (e.g., of the function X) to achieve a better attack in terms of the number of traces required and computational complexity. For the Snake attack, sums of absolute or squared values of correlation coefficients are used to determine the most likely guess. As a result, we effectively show that our proposed Snake attack can recover the secret in n2² (or n2²+n) intermediate results, compared to 2²ⁿ for the CPA, where n is the length of the targeted S-Box. We collected power measurements from a hardware setup to demonstrate practical attack success probabilities according to the rank of the correct secret hypothesis both for Xoodoo and Keccak-p. In addition, we explain our success probabilities thanks to the Henery model developed for horse races. In short, after performing 16,896 attacks, the Snake attack or combined CPA on Xoodoo consistently recovers the correct secret bits with each attack using 43,860 traces on average and with only 12 correlations, compared to 61,380 traces for the standard CPA attack with 64 correlations. The Snake attack requires about one-fifth as many correlation values as the standard CPA. For Keccak-p, the difference is more drastic: the Snake attack recovers the secret bits invariably after 771,600 traces with just 20 correlations. In contrast, the standard CPA attack operates on 1,024 correlations (about fifty times more than the Snake attack) and requires 1,223,400 traces.