International audienceThe left-regular multiplication is explicitly embedded in the notion of perfect nonlinearity. But there exist many other group actions. By replacing translations by another group action the new concept of group action-based perfect nonlinearity has been introduced. In this paper we show that this generalized concept of nonlinearity is actually equivalent to a new bentness notion that deals with functions defined on a finite Abelian group G that acts on a finite set X and with values in the finite-dimensional vector space of complex-valued functions defined on X

Perfect nonlinear functions are used to construct DES-like cryptosystems that are resistant to differential attacks. We present generalized DES-like cryptosystems where the XOR operation is replaced by a general group action. The new cryptosystems, when combined with G-perfect nonlinear functions (similar to classical perfect nonlinear functions with one XOR replaced by a general group action), allow us to construct systems resistant to modified differential attacks. The more general setting enables robust cryptosystems with parameters that would not be possible in the classical setting. We construct several examples of G-perfect nonlinear functions, both Z 2 -valued and Z a 2 -valued. Our final constructions demonstrate G-perfect nonlinear planar permutations (from Z a 2 to itself), thus providing an alternative implementation to current uses of almost perfect nonlinear functions.Keywords G-perfect nonlinear functions · Difference sets AMS Classifications 05B10 · 11T71 Background on cryptosystems and group action modificationsIn an r-round iterative block cipher such as the Data Encryption Standard (DES) [17] or the Advanced Encryption Standard (AES) [9,18] the ciphertext x r is obtained from a plaintext x 0 by r iterations of the round function f

Perfect nonlinear functions from a finite group G to another one H are those functions f : G → H such that for all nonzero α ∈ G, the derivative dαf : x → f (αx)f(x) −1 is balanced. In the case where both G and H are Abelian groups, f : G → H is perfect nonlinear if and only if f is bent i.e. for all nonprincipal character χ of H, the (discrete) Fourier transform of χ • f has a constant magnitude equals to |G|. In this paper, using the theory of linear representations, we exhibit similar bentness-like characterizations in the cases where G and/or H are (finite) non Abelian groups. Thus we extend the concept of bent functions to the framework of non Abelian groups.

Commutative Hilbertian Frobenius algebras are those commutative semigroup objects in the monoidal category of Hilbert spaces, for which the Hilbert adjoint of the multiplication satisfies the Frobenius compatibility relation, that is, this adjoint is a bimodule map. In this note we prove that they split as an orthogonal direct sum of two closed ideals, their Jacobson radical which in fact is nothing but their annihilator, and the closure of the linear span of their group-like elements. As a consequence such an algebra is semisimple if, and only if, its multiplication has a dense range. In particular every commutative special Hilbertian algebra, that is, with a coisometric multiplication, is semisimple. Extending a known result in the finite-dimensional situation, we prove that the structures of such Frobenius algebras on a given Hilbert space are in one-one correspondence with its bounded above orthogonal sets. We show, moreover, that the category of commutative Hilbertian Frobenius algebras is dually equivalent to a category of pointed sets. Thus, each semigroup morphism between commutative Hilbertian Frobenius semigroups arises from a unique base-point preserving map (of some specific kind), from the set of minimal ideals of its codomain to the set of minimal ideals of its domain, both with zero added.MSC 2010: Primary 46J40, Secondary 16T15. over a field is Morita equivalent to a Frobenius algebra [23, Corollary 3.11, p. 351], but also in representation theory because of their similarity with group algebras [9], in mathematical physics as their category is equivalent to that of 2-dimensional topological quantum field theories [1], in (categorical) quantum computing since they provide an algebraic characterization of orthonormal bases and observables [7,8] or also in the theory of Hopf algebras in particular because their relation with weak Hopf algebras [6].As is well-known the Hilbert space ℓ 2 (X) of square-summable functions on X is by no means free over X. However in this note we prove that X -or more precisely X plus a new point added -freely generates a (non-unital, when X is not finite) commutative Frobenius algebra, whose underlying space is (unitarily isomorphic to) ℓ 2 (X). More generally we show that every commutative Frobenius algebra (H, µ), where (H, µ) is a semigroup in the category of Hilbert space and for which µ † satisfies the Frobenius compatibility relation, splits into an orthogonal direct sum of two ideals ℓ 2 (X) ⊕ 2 A(H, µ), where A(H, µ) is the annihilator of (H, µ) and X is a set equipotent to that of minimal ideals of (H, µ). The free Frobenius algebras, that is, those of the form ℓ 2 (X), are precisely the semisimple ones. Before describing in more detail the content of this note, let us put it into perspective.A Frobenius algebra may be described in several equivalent ways ([9, Theorem 61.3, p. 414]), for instance it is a unital algebra over some base field which as a left module over itself is isomorphic to its algebraic (right) dual. This definition implies directly that the alg...