G-Perfect Nonlinear Functions

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 Z2 -valued and Za2 -valued. Our final constructions demonstrate G-perfect nonlinear planar permutations (from Za2 to itself), thus providing an alternative implementation to current uses of almost perfect nonlinear functions

A new characterization of group action-based perfect nonlinearity

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

Construction of ASIC-POVMs via 2-to-1 PN functions and the Li bound

Symmetric informationally complete positive operator-valued measures (SIC-POVMs) in finite dimension $d$ are a particularly attractive case of informationally complete POVMs (IC-POVMs) which consist of $d^{2}$ subnormalized projectors with equal pairwise fidelity. However, it is difficult to construct SIC-POVMs and it is not even clear whether there exists an infinite family of SIC-POVMs. To realize some possible applications in quantum information processing, Klappenecker et al. [33] introduced an approximate version of SIC-POVMs called approximately symmetric informationally complete POVMs (ASIC-POVMs). In this paper, we present two new constructions of ASIC-POVMs in dimensions $q$ and $q+1$ by $2$-to-$1$ PN functions and the Li bound, respectively, where $q$ is a prime power. In the first construction, we show that all $2$-to-$1$ PN functions can be used for constructing ASIC-POVMs of dimension $q$, which not only generalizes the construction in [33, Theorem 5], but also generalizes the general construction in [11, Theorem III.3]. We show that some $2$-to-$1$ PN functions that do not satisfy the condition in [11, Theorem III.3] can be also utilized for constructing ASIC-POVMs of dimension $q$. We also give a class of biangular frames related to our ASIC-POVMs. The second construction gives a new method to obtain ASIC-POVMs in dimension $q+1$ via a multiplicative character sum estimate called the Li bound