Invariants for EA- and CCZ-equivalence of APN and AB functions

An (n,m)-function is a mapping from ${\mathbb {F}_{2}^{n}}$ to ${\mathbb {F}_{2}^{m}}$. Such functions have numerous applications across mathematics and computer science, and in particular are used as building blocks of block ciphers in symmetric cryptography. The classes of APN and AB functions have been identified as cryptographically optimal with respect to the resistance against two of the most powerful known cryptanalytic attacks, namely differential and linear cryptanalysis. The classes of APN and AB functions are directly related to optimal objects in many other branches of mathematics, and have been a subject of intense study since at least the early 90’s. Finding new constructions of these functions is hard; one of the most significant practical issues is that any tentatively new function must be proven inequivalent to all the known ones. Testing equivalence can be significantly simplified by computing invariants, i.e. properties that are preserved by the respective equivalence relation. In this paper, we survey the known invariants for CCZ- and EA-equivalence, with a particular focus on their utility in distinguishing between inequivalent instances of APN and AB functions. We evaluate each invariant with respect to how easy it is to implement in practice, how efficiently it can be calculated on a computer, and how well it can distinguish between distinct EA- and CCZ-equivalence classes.publishedVersio

ACCUMULATED PREDICTION ERRORS, INFORMATION CRITERIA AND OPTIMAL FORECASTING FOR AUTOREGRESSIVE TIME SERIES

The predictive capability of a modification of Rissanen's accumulated prediction error (APE) criterion, APE$_{\delta_{n}}$,is investigated in infinite-order autoregressive (AR($\infty$)) models. Instead of accumulating squares of sequential prediction errors from the beginning, APE$_{\delta_{n}}$ is obtained by summing these squared errors from stage $n\delta_{n}$, where $n$ is the sample size and $0Accumulated prediction errors, Asymptotic equivalence, Asymptotic efficiency, Information criterion, Order selection, Optimal forecasting