ZK-SNARKs, a fundamental component of privacyoriented payment systems, identity protocols, or anonymous voting systems, are advanced cryptographic protocols for verifiable computation: modern SNARKs allow to encode the invariants of a program, expressed as an arithmetic circuit, in an appropriate constraint language from which short, zero-knowledge proofs for correct computations can be constructed. One of the most important computations that is run through SNARK systems is the verification of Merkle tree (MT) opening proofs, which relies on the evaluation of a fixed-input-length (FIL) cryptographic compression function over binary MTs. As classical, bit-oriented hash functions like SHA-2 are not compactly representable in SNARK frameworks, Arithmetization-Oriented (AO) cryptographic designs have emerged as an alternative, efficient solution. Today, the majority of AO compression functions are built from permutation-based hashing modes, such as Sponge. While this approach allows cost savings, compared to blockcipher-based modes, as it does not require key-scheduling, AO blockcipher schedulers are often cheap to compute. Furthermore, classical bitoriented cryptography has long studied how to construct provably secure compression functions from blockciphers, following the Preneel-Govaerts-Vandewalle (PGV) framework. The potential efficiency gains together with the strong provable security foundations in the classic setting, motivate the study of AO blockcipher-based compression functions. In this work, we propose AO PGV-LC and PGV-ELC, two AO blockcipher-based FIL compression modes inspired by and extending the classical PGV approach, offering flexible input and output sizes and coming with provable security guarantees in the AO setting. We prove the collision and preimage resistance in the ideal cipher model, and give bounds for collision and opening resistance over MTs of arbitrary arity. We compare experimentally the AO PGV-ELC mode over the HADES blockcipher with its popular and widely adopted Sponge instantiation, POSEIDON, and its improved variant POSEIDON2. Our resulting constructions are up to 3× faster than POSEIDONAND 2× faster than POSEIDON2 in native x86 execution, and up to 50% faster in the Groth16 SNARK framework. Finally, we study the benefits of using MTs of arity wider than two, proposing a new strategy to obtain a compact R1CS constraint system in such case. In fact, by combining an efficient parametrization of the HADES blockcipher over the PGV-ELC mode, together with an optimal choice of the MT arity, we measured an improvement of up to 9× in native MT construction time, and up to 2.5× in proof generation time, compared to POSEIDON over binary MTs.

Branching Algebra is the natural branching-time generalization of Allen’s Interval Algebra. As in the linear case, the consistency problem for Branching Algebra is NP-hard. Being relatively new, however, not much is known about the computational behaviour of the consistency problem of its sub-algebras, except in the case of the recently found subset of convex branching relations, for which the consistency of a network can be tested via path consistency and it is therefore deterministic polynomial. In this paper, following Nebel and Bürckert, we define the Horn fragment of Branching Algebra, and prove that it is a sub-algebra of the latter, being closed under inverse, intersection, and composition, that it strictly contains both the convex fragment of Branching Algebra and the Horn fragment of Interval Algebra, and that its consistency problem can be decided via path consistency. Finally, we experimentally prove that the Horn fragment of Branching Algebra can be used as an heuristic for checking the consistency of a generic network with a considerable improvement over the convex subset.