Non-Boolean almost perfect nonlinear functions on non-Abelian groups

The purpose of this paper is to present the extended definitions and characterizations of the classical notions of APN and maximum nonlinear Boolean functions to deal with the case of mappings from a finite group K to another one N with the possibility that one or both groups are non-Abelian.Comment: 17 page

Using deep reinforcement learning to reveal how the brain encodes abstract state-space representations in high-dimensional environments

Humans possess an exceptional aptitude to efficiently make decisions from high-dimensional sensory observations. However, it is unknown how the brain compactly represents the current state of the environment to guide this process. The deep Q-network (DQN) achieves this by capturing highly nonlinear mappings from multivariate inputs to the values of potential actions. We deployed DQN as a model of brain activity and behavior in participants playing three Atari video games during fMRI. Hidden layers of DQN exhibited a striking resemblance to voxel activity in a distributed sensorimotor network, extending throughout the dorsal visual pathway into posterior parietal cortex. Neural state-space representations emerged from nonlinear transformations of the pixel space bridging perception to action and reward. These transformations reshape axes to reflect relevant high-level features and strip away information about task-irrelevant sensory features. Our findings shed light on the neural encoding of task representations for decision-making in real-world situations

Doubly Perfect Nonlinear Boolean Permutations

Due to implementation constraints the XOR operation is widely used in order to combine plaintext and key bit-strings in secret-key block ciphers. This choice directly induces the classical version of the differential attack by the use of XOR-kind differences. While very natural, there are many alternatives to the XOR. Each of them inducing a new form for its corresponding differential attack (using the appropriate notion of difference) and therefore block-ciphers need to use S-boxes that are resistant against these nonstandard differential cryptanalysis. In this contribution we study the functions that offer the best resistance against a differential attack based on a finite field multiplication. We also show that in some particular cases, there are robust permutations which offers the best resistant against both multiplication and exponentiation base differential attacks. We call them doubly perfect nonlinear permutations