MiMC:Efficient Encryption and Cryptographic Hashing with Minimal Multiplicative Complexity

We explore cryptographic primitives with low multiplicative complexity. This is motivated by recent progress in practical applications of secure multi-party computation (MPC), fully homomorphic encryption (FHE), and zero-knowledge proofs (ZK) where primitives from symmetric cryptography are needed and where linear computations are, compared to non-linear operations, essentially ``free\u27\u27. Starting with the cipher design strategy ``LowMC\u27\u27 from Eurocrypt 2015, a number of bit-oriented proposals have been put forward, focusing on applications where the multiplicative depth of the circuit describing the cipher is the most important optimization goal. Surprisingly, albeit many MPC/FHE/ZK-protocols natively support operations in \GF{p} for large $p$, very few primitives, even considering all of symmetric cryptography, natively work in such fields. To that end, our proposal for both block ciphers and cryptographic hash functions is to reconsider and simplify the round function of the Knudsen-Nyberg cipher from 1995. The mapping $F(x) := x^3$ is used as the main component there and is also the main component of our family of proposals called ``MiMC\u27\u27. We study various attack vectors for this construction and give a new attack vector that outperforms others in relevant settings. Due to its very low number of multiplications, the design lends itself well to a large class of new applications, especially when the depth does not matter but the total number of multiplications in the circuit dominates all aspects of the implementation. With a number of rounds which we deem secure based on our security analysis, we report on significant performance improvements in a representative use-case involving SNARKs

Cryptanalysis of Low-Data Instances of Full LowMCv2

LowMC is a family of block ciphers designed for a low multiplicative complexity. The specification allows a large variety of instantiations, differing in block size, key size, number of S-boxes applied per round and allowed data complexity. The number of rounds deemed secure is determined by evaluating a number of attack vectors and taking the number of rounds still secure against the best of these. In this paper, we demonstrate that the attacks considered by the designers of LowMC in the version 2 of the round-formular were not sufficient to fend off all possible attacks. In the case of instantiations of LowMC with one of the most useful settings, namely with few applied S-boxes per round and only low allowable data complexities, efficient attacks based on difference enumeration techniques can be constructed. We show that it is most effective to consider tuples of differences instead of simple differences, both to increase the range of the distinguishers and to enable key recovery attacks. All applications for LowMC we are aware of, including signature schemes like Picnic and more recent (ring/group) signature schemes have used version 3 of the roundformular for LowMC, which takes our attack already into account

A Formal Analysis of Boomerang Probabilities

In the past 20 years since their conception, boomerang attacks have become an important tool in the cryptanalysis of block ciphers. In the classical estimate of their success probability, assumptions are made about the independence of the underlying differential trails that are not well-founded. We underline the problems inherent in these independence assumptions by using them to prove that for any boomerang there exists a differential trail over the entire cipher with a higher probability than the boomerang.While cryptanalysts today have a clear understanding that the trails can be dependent, the focus of previous research has mostly gone into using these dependencies to improve attacks but little effort has been put into giving boomerangs and their success probabilities a stronger theoretical underpinning. With this publication, we provide such a formalization.We provide a framework which allows us to formulate and prove rigorous statements about the probabilities involved in boomerang attacks without relying on independence assumptions of the trails. Among these statements is a proof that two-round boomerangs on SPNs with differentially 4-uniform S-boxes always deviate from the classical probability estimate to the largest degree possible.We applied the results of this formalization to analyze the validity of some of the first boomerang attacks. We show that the boomerang constructed in the amplified boomerang attack on Serpent by Kelsey, Kohno, and Schneier has probability zero. For the rectangle attack on Serpent by Dunkelman, Biham, and Keller, we demonstrate that a minuscule fraction of only 2−43.4 of all differential trail combinations used in the original attack have a non-zero probability. In spite of this, the probability of the boomerang is in fact a little higher than the original estimate suggests as the non-zero trails have a vastly higher probability than the classical estimate predicts

Cryptanalysis of Low-Data Instances of Full LowMCv2

LowMC is a family of block ciphers designed for a low multiplicative complexity. The specification allows a large variety of instantiations, differing in block size, key size, number of S-boxes applied per round and allowed data complexity. The number of rounds deemed secure is determined by evaluating a number of attack vectors and taking the number of rounds still secure against the best of these. In this paper, we demonstrate that the attacks considered by the designers of LowMC in the version 2 of the round-formular were not sufficient to fend off all possible attacks. In the case of instantiations of LowMC with one of the most useful settings, namely with few applied S-boxes per round and only low allowable data complexities, efficient attacks based on difference enumeration techniques can be constructed. We show that it is most effective to consider tuples of differences instead of simple differences, both to increase the range of the distinguishers and to enable key recovery attacks. All applications for LowMC we are aware of, including signature schemes like Picnic and more recent (ring/group) signature schemes have used version 3 of the roundformular for LowMC, which takes our attack already into account

Finding Integral Distinguishers with Ease

The division property method is a technique to determine integral distinguishers on block ciphers. While the complexity of finding these distinguishers is higher, it has recently been shown that MILP and SAT solvers can efficiently find such distinguishers. In this paper, we provide a framework to automatically find those distinguishers which solely requires a description of the cryptographic primitive. We demonstrate that by finding integral distinguishers for 30 primitives with different design strategies. We provide several new or improved bit-based division property distinguishers for ChaCha, Chaskey, DES, GIFT, LBlock, Mantis, Qarma, RoadRunner, Salsa and SM4. Furthermore, we present an algorithm to find distinguishers with lower data complexity more efficiently

Security of the AES with a Secret S-Box

Abstract. How does the security of the AES change when the S-box is replaced by a secret S-box, about which the adversary has no knowledge? Would it be safe to reduce the number of encryption rounds? In this paper, we demonstrate attacks based on integral cryptanalysis which allow to recover both the secret key and the secret S-box for respectively four, five, and six rounds of the AES. Despite the signif-icantly larger amount of secret information which an adversary needs to recover, the attacks are very efficient with time/data complexities of 217/216, 238/240 and 290/264, respectively. Another interesting aspect of our attack is that it works both as chosen plaintext and as chosen ciphertext attack. Surprisingly, the chosen ci-phertext variant has a significantly lower time complexity in the attacks on four and five round, compared to the respective chosen plaintext at-tacks