Nonlinear Functions in Finite Fields

Nonlinear functions defined over finite fields lie at the heart of modern algebraic design, with wide-ranging influence from cryptography to coding theory. Such functions depart from linear or affine mappings, endowing systems with resistance against analytical attacks and enabling complex combinatorial structures. Key properties include nonlinearity—often gauged by the distance to the closest affine map—differential uniformity, which measures the predictability of output differences under input shifts, and algebraic degree, reflecting the highest exponent in a polynomial representation. Among these, Almost Perfect Nonlinear (APN) functions achieve the minimal differential uniformity possible in even characteristic and are prized as S-boxes in block cipher design. Recent extensions consider c-differential uniformity, capturing multiplicative difference criteria relevant to novel cryptographic modes. Implementation concerns further elevate decomposition techniques that split high-dimensional permutations into smaller keyed mappings, enabling efficient hardware realisations. Advancements in enumeration, classification and theoretical bounds for these mappings continue to shape the security and performance of encryption algorithms, error-correcting codes and pseudorandom generators. Interdisciplinary dialogue between pure algebra, combinatorics and system engineering ensures that evolving threats are met with robust finite-field constructions, balancing provable security with practical resource constraints.

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Nonlinear Functions in Finite Fields publication trend

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Technical terms

Finite field: A field with finitely many elements, denoted GF(q), where q is a prime power.

Nonlinear function: A mapping on a finite field whose polynomial representation contains monomials of degree greater than one, providing cryptographic strength.

Differential uniformity: The maximum number of solutions to f(x + a) – f(x) = b for nonzero a, indicating susceptibility to differential attacks.

Almost Perfect Nonlinear (APN) function: A function achieving the lowest possible differential uniformity of two in even characteristic fields.

S-box: A substitution box in block cipher architectures that applies a nonlinear finite-field mapping to introduce confusion.

Algebraic degree: The highest total degree of any monomial in the polynomial form of a function, influencing resistance to algebraic attacks.

c-differential uniformity: A generalisation of differential uniformity measuring output differences scaled by a constant multiplier c.

TU-decomposition: A method of expressing a high-dimensional permutation as the composition of two smaller keyed permutations (T and U) to reduce implementation complexity.

References

1. Low-Complexity Hardware Architecture of APN Permutations Using TU-Decomposition. IEEE Transactions on Circuits and Systems I Regular Papers (2024).
2. On the Pentanomial Power Mapping Classification of 8-bit to 8-bit S-Boxes. Mathematics (2024).
3. On construction and (non)existence of c-(almost) perfect nonlinear functions. Finite Fields and Their Applications (2021).