Round functions used as building blocks for iterated block ciphers, both in the case of Substitution-Permutation Networks (SPN) and Feistel Networks (FN), are often obtained as the composition of different layers. The bijectivity of any encryption function is guaranteed by the use of invertible layers or by the Feistel structure. In this work a new family of ciphers, called wave ciphers, is introduced. In wave ciphers, round functions feature wave functions, which are vectorial Boolean functions obtained as the composition of non-invertible layers, where the confusion layer enlarges the message which returns to its original size after the diffusion layer is applied. Efficient decryption is guaranteed by the use of wave functions in FNs. It is shown how to avoid that the group generated by the round functions acts imprimitively, a serious flaw for the cipher. The primitivity is a consequence of a more general result, which reduce the problem of proving that a given FN generates a primitive group to proving that an SPN, directly related to the given FN, generates a primitive group. Finally, a concrete instance of real-world size wave cipher is proposed as an example, and its resistance against differential and linear cryptanalyses is also established.

While technological advancements and their deep integration in connected and automated vehicles is a central aspect in the evolving trend of automotive industry, they also depict a growing size attack surface for malicious actors: the latter ones typically aim at exploiting known and unknown security vulnerabilities, with potentially disastrous consequences on the safety of vehicles, people, and infrastructures. In recent years, remarkable efforts have been spent to mitigate security vulnerabilities in intelligent and connected vehicles, in particular in the inside of vehicles, the so-called intra-vehicle networks. Despite those efforts, securing intra-vehicle networks remains a non-trivial task due to their heterogeneous and increasingly complex context. Starting from the above remarks and motivated by the industrial research and innovation project EMERGE, in this paper we report on a novel cryptographic hardware-software solution that we have designed and developed for securing the intra-vehicle network of intelligent connected vehicles: the Crypto-Engine. The Crypto-Engine relies on a lightweight hybrid-key cryptographic scheme to provide confidentiality and authentication without compromising the normal communication performance. We tested the Crypto-Engine and demonstrated that, once configured according to application-defined performance requirements, it can authenticate parties and secure the communications with a negligible overhead.

A partition into distinct parts is refinable if one of its parts a can be replaced by two different integers which do not belong to the partition and whose sum is a, and it is unrefinable otherwise. Clearly, the condition of being unrefinable imposes on the partition a non-trivial limitation on the size of the largest part and on the possible distributions of the parts. We prove a $$O(n^{1/2})$$O ( n 1 / 2 ) -upper bound for the largest part in an unrefinable partition of n, and we call maximal those which reach the bound. We show a complete classification of maximal unrefinable partitions for triangular numbers, proving that if n is even there exists only one maximal unrefinable partition of $$n(n+1)/2$$n ( n + 1 ) / 2 , and that if n is odd the number of such partitions equals the number of partitions of $$\lceil n/2\rceil $$⌈ n / 2 ⌉ into distinct parts. In the second case, an explicit bijection is provided.