New math tools analyze input output patterns in finite field functions
Differential-linear profiles over finite fields of arbitrary characteristic
Information Theory
Summary
This work studies how certain mathematical functions change when their inputs shift, focusing on functions defined over finite fields with any prime number of elements. The authors extend a known tool (DLCT) that measures input-output relationships in binary functions to work in a broader setting. They find a way to capture detailed patterns of change and relate these to known properties of the functions. This helps characterize special functions used in cryptography and provides formulas for key examples in odd-prime cases. The results also show how these patterns behave under common transformations of the functions.
What this means in practice
- •For cryptography engineers: Use extended differential-linear profiles to analyze and design cryptographic functions in finite fields beyond binary cases.
- •For coding theory developers: Apply the characterization of planar functions and autocorrelation profiles to construct or study error-correcting codes over finite fields of odd characteristic.
A theory result. No direct application yet.
Authors
Kirpa Garg, Constanza Riera, Pantelimon Stănică
Abstract
The (binary) differential-linear connectivity table (DLCT) measures the dependence between an input difference and a linear mask applied to the corresponding output difference. For vectorial Boolean functions, each DLCT entry is one half of an additive autocorrelation value. We extend this relation to functions over finite fields of arbitrary prime characteristic by introducing a level-resolved p-ary differential-linear profile. Its entries are the centered numbers of inputs for which a derivative component has each prescribed trace value in $\mathbb{F}_p$. The discrete Fourier transform of this profile is the family of additive autocorrelations obtained by multiplying the output mask by the nonzero elements of $\mathbb{F}_p$; when p=2, the usual binary identity is recovered. For a fixed input difference, we show that the profiles over all nonzero output masks determine the corresponding DDT row exactly, and we give an explicit inversion formula. We establish a second-moment identity: the total profile energy in one derivative direction is a constant multiple of the squared Euclidean distance between that DDT row and the balanced row. Thus this energy is determined by the full row differential spectrum, not by differential uniformity alone. It follows that all profiles in a direction vanish exactly when the derivative is balanced; for square maps in odd characteristic, this gives a characterization of planarity. As concrete odd-characteristic examples, we determine the complete profile of the monomial $x^{p^k+1}$ and derive an exact Kloosterman-sum formula for the inverse monomial. Finally, we determine the behavior of the profiles under equivalence. EA-equivalence reindexes the input and output masks and translates the trace level, whereas a general CCZ equivalence may mix several derivative directions. Nevertheless, for square maps the global nontrivial profile energy is CCZ-invariant.