On the cryptographic properties of weightwise affine and weightwise quadratic functions

Weightwise degree-d functions are Boolean functions that take the values of a function of degree at most d on each set of fixed Hamming weight. The class of weightwise affine functions encompasses both the symmetric functions and the Hidden Weight Bit Function (HWBF). The good cryptographic properties of the HWBF, except for the nonlinearity, motivates to investigate a larger class with functions that share the good properties and have a better nonlinearity. Additionally, the homomorphic friendliness of symmetric functions exhibited in the context of hybrid homomorphic encryption and the recent results on homomorphic evaluation of Boolean functions make this class of functions appealing for efficient privacy-preserving protocols. In this article we realize the first study on weightwise degree-d functions, focusing on weightwise affine and weightwise quadratic functions. We show some properties on these new classes of functions, in particular on the subclass of cyclic weightwise functions. We provide balanced constructions and prove nonlinearity lower bounds for all cyclic weightwise affine functions and for a family of weightwise quadratic functions. We complement our work with experimental results, they show that other cyclic weightwise linear functions than the HWBF have better cryptographic parameters, and considering weightwise quadratic functions allows to reach higher algebraic immunity and substantially better nonlinearity

On the cryptographic properties of weightwise affine and weightwise quadratic functions

Weightwise degree-d functions are Boolean functions that take the values of a function of degree at most d on each set of fixed Hamming weight. The class of weightwise affine functions encompasses both the symmetric functions and the Hidden Weight Bit Function (HWBF). The good cryptographic properties of the HWBF, except for the nonlinearity, motivates to investigate a larger class with functions that share the good properties and have a better nonlinearity. Additionally, the homomorphic friendliness of symmetric functions exhibited in the context of hybrid homomorphic encryption and the recent results on homomorphic evaluation of Boolean functions make this class of functions appealing for efficient privacy-preserving protocols. In this article we realize the first study on weightwise degree-d functions, focusing on weightwise affine and weightwise quadratic functions. We show some properties on these new classes of functions, in particular on the subclass of cyclic weightwise functions. We provide balanced constructions and prove nonlinearity upper bounds for all cyclic weightwise affine functions and for a family of weightwise quadratic functions. We complement our work with experimental results, they show that other cyclic weightwise linear functions than the HWBF have better cryptographic parameters, and considering weightwise quadratic functions allows to reach higher algebraic immunity and substantially better nonlinearity

PPP-Completeness with Connections to Cryptography

Polynomial Pigeonhole Principle (PPP) is an important subclass of TFNP with profound connections to the complexity of the fundamental cryptographic primitives: collision-resistant hash functions and one-way permutations. In contrast to most of the other subclasses of TFNP, no complete problem is known for PPP. Our work identifies the first PPP-complete problem without any circuit or Turing Machine given explicitly in the input, and thus we answer a longstanding open question from [Papadimitriou1994]. Specifically, we show that constrained-SIS (cSIS), a generalized version of the well-known Short Integer Solution problem (SIS) from lattice-based cryptography, is PPP-complete. In order to give intuition behind our reduction for constrained-SIS, we identify another PPP-complete problem with a circuit in the input but closely related to lattice problems. We call this problem BLICHFELDT and it is the computational problem associated with Blichfeldt's fundamental theorem in the theory of lattices. Building on the inherent connection of PPP with collision-resistant hash functions, we use our completeness result to construct the first natural hash function family that captures the hardness of all collision-resistant hash functions in a worst-case sense, i.e. it is natural and universal in the worst-case. The close resemblance of our hash function family with SIS, leads us to the first candidate collision-resistant hash function that is both natural and universal in an average-case sense. Finally, our results enrich our understanding of the connections between PPP, lattice problems and other concrete cryptographic assumptions, such as the discrete logarithm problem over general groups

Constructions of Almost Optimal Resilient Boolean Functions on Large Even Number of Variables

In this paper, a technique on constructing nonlinear resilient Boolean functions is described. By using several sets of disjoint spectra functions on a small number of variables, an almost optimal resilient function on a large even number of variables can be constructed. It is shown that given any $m$, one can construct infinitely many $n$-variable ($n$ even), $m$-resilient functions with nonlinearity $>2^{n-1}-2^{n/2}$. A large class of highly nonlinear resilient functions which were not known are obtained. Then one method to optimize the degree of the constructed functions is proposed. Last, an improved version of the main construction is given.Comment: 14 pages, 2 table