International Association
for Cryptologic Research

Determining bounds on the differential probability of differential trails and the squared correlation contribution of linear trails forms an important part of the security evaluation of a permutation. For Xoodoo, such bounds were proven using the trail core tree search technique, with a dedicated tool (XooTools) that scans the space of all r-round trails with weight below a given threshold Tr. The search space grows exponentially with the value of Tr and XooTools appeared to have reached its limit, requiring huge amounts of CPU time to push the bounds a little further. The bottleneck was the phase called trail extension where short trails are extended to more rounds, especially in the backward direction. In this work, we present a number of techniques that allowed us to make extension much more efficient and as such to increase the bounds significantly. Notably, we prove that the minimum weight of any 4-round trail is 80, the minimum weight of any 6-round trail is at least 132 and the minimum weight of any 12-round trail is at least 264, both for differential and linear trails. As a byproduct we found families of trails that have predictable weight once extended to more rounds and use them to compute upper bounds for the minimum weight of trails for arbitrary numbers of rounds.

Proving upper bounds for the expected differential probability (DP) of differential trails is a standard requirement when proposing a new symmetric primitive. In the case of cryptographic primitives with a bit-oriented round function, such as Keccak, Xoodoo and Subterranean, computer assistance is required in order to prove strong upper bounds on the probability of differential trails. The techniques described in the literature make use of the fact that the non-linear step of the round function is an S-box layer. In the case of Keccak and Xoodoo, the S-boxes are instances of the chi mapping operating on l-bit circles with l equal to 5 and 3 respectively. In that case the differential propagation properties of the non-linear layer can be evaluated efficiently by the use of pre-computed difference distribution tables.Subterranean 2.0 is a recently proposed cipher suite that has exceptionally good energy-efficiency when implemented in hardware (ASIC and FPGA). The non-linear step of its round function is also based on the chi mapping, but operating on an l = 257-bit circle, comprising all the state bits. This making the brute-force approach proposed and used for Keccak and Xoodoo infeasible to apply. Difference propagation through the chi mapping from input to output can be treated using linear algebra thanks to the fact that chi has algebraic degree 2. However, difference propagation from output to input is problematic for big-circle chi. In this paper, we tackle this problem, and present new techniques for the analysis of difference propagation for big-circle chi.We implemented these techniques in a dedicated program to perform differential trail search in Subterranean. Thanks to this, we confirm the maximum DP of 3-round trails found by the designers, we determine the maximum DP of 4-round trails and we improve the upper bounds for the DP of trails over 5, 6, 7 and 8 rounds.

Ascon is a family of cryptographic primitives for authenticated encryption and hashing introduced in 2015. It is selected as one of the ten finalists in the NIST Lightweight Cryptography competition. Since its introduction, Ascon has been extensively cryptanalyzed, and the results of these analyses can indicate the good resistance of this family of cryptographic primitives against known attacks, like differential and linear cryptanalysis.Proving upper bounds for the differential probability of differential trails and for the squared correlation of linear trails is a standard requirement to evaluate the security of cryptographic primitives. It can be done analytically for some primitives like AES. For other primitives, computer assistance is required to prove strong upper bounds for differential and linear trails. Computer-aided tools can be classified into two categories: tools based on general-purpose solvers and dedicated tools. General-purpose solvers such as SAT and MILP are widely used to prove these bounds, however they seem to have lower capabilities and thus yield less powerful bounds compared to dedicated tools.In this work, we present a dedicated tool for trail search in Ascon. We arrange 2-round trails in a tree and traverse this tree in an efficient way using a number of new techniques we introduce. Then we extend these trails to more rounds, where we also use the tree traversal technique to do it efficiently. This allows us to scan much larger spaces of trails faster than the previous methods using general-purpose solvers. As a result, we prove tight bounds for 3-rounds linear trails, and for both differential and linear trails, we improve the existing upper bounds for other number of rounds. In particular, for the first time, we prove bounds beyond 2−128 for 6 rounds and beyond 2−256 for 12 rounds of both differential and linear trails.

The sequential structure of some side-channel attacks makes them subject to error propagation, i.e. when an error occurs during the recovery of some part of a secret key, all the following guesses might as well be chosen randomly. We propose a methodology that strengthens sequential attacks by automatically identifying and correcting errors. The core ingredient of our methodology is a change-detection test that monitors the distribution of the distinguisher values used to reconstruct the secret key. Our methodology includes an error-correction procedure that can cope both with false positives of the change-detection test, and inaccuracies of the estimated location of the wrong key guess. The proposed methodology is general and can be included in several attacks. As meaningful examples, we conduct two different side-channel attacks against RSA-2048: an horizontal power-analysis attack based on correlation and a vertical timing attack. Our experiments show that, in all the considered cases, strengthened attacks outperforms their original counterparts and alternative solutions that are based on thresholds. In particular, strengthened attacks achieve high success rates even when the side-channel measurements are noisy or limited in number, without prohibitively increasing the computing time.

We present new techniques to efficiently scan the space of high-probability differential trails in bit-oriented ciphers. Differential trails consist in sequences of state patterns that we represent as ordered lists of basic components in order to arrange them in a tree. The task of generating trails with probability above some threshold starts with the traversal of the tree. Our choice of basic components allows us to efficiently prune the tree based on the fact that we can tightly bound the probability of all descendants for any node. Then we extend the state patterns resulting from the tree traversal into longer trails using similar bounding techniques. We apply these techniques to the 4 largest Keccak-f permutations, for which we are able to scan the space of trails with weight per round of 15. This space is orders of magnitude larger than previously best result published on Keccak-f[1600] that reached 12, which in turn is orders of magnitude larger than any published results achieved with standard tools, that reached at most 9. As a result we provide new and improved bounds for the minimum weight of differential trails on 3, 4, 5 and 6 rounds. We also report on new trails that are, to the best of our knowledge, the ones with the highest known probability.