Permutation Polynomials Over Finite Fields

Permutation polynomials over finite fields form a central theme in modern algebraic research, intertwining group theory, number theory and combinatorial design. A finite field is a set of elements closed under addition and multiplication with a fixed prime‐power cardinality. A permutation polynomial is a mapping defined by a polynomial which permutes the field’s elements. The classification of such polynomials seeks criteria that guarantee bijectivity and often leverages properties of exponents and field structure. Monomials of the form x^r can permute a field precisely when r is coprime to the field’s multiplicative order, while binomials and trinomials demand more intricate conditions, often expressed via greatest-common-divisor constraints or character sums. Complete permutation polynomials are those that remain permutation polynomials after addition of the identity map; these exhibit heightened symmetry and find particular use in coding theory. Underpinning many constructions are linearised polynomials, cyclotomic mapping techniques and trace functions from extension fields. The global significance is pronounced in cryptography, error-correcting codes and combinatorial designs, where explicit permutation polynomials deliver efficient substitution layers, pseudorandom generators and block designs with predetermined intersection properties. Over the past decade expanded criteria, such as those relying on Niho exponents or subfield decompositions, have enriched the catalogue of known families, while inverse maps and cycle-structure analyses have deepened understanding of their dynamical and algebraic behaviour.

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Permutation Polynomials Over Finite Fields publication trend

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Technical terms

Finite field: A field F_q with q elements, where q is a power of a prime, supporting arithmetic under addition and multiplication.

Permutation polynomial: A polynomial f in F_q[x] that induces a bijection on F_q, sending each field element to a unique image.

Complete permutation polynomial: A permutation polynomial f for which f(x) + x is also a permutation polynomial on the same field.

Niho exponent: An exponent of the form d(q–1)/e + 1, often employed to produce polynomials whose restriction to a subfield or subset yields permutation behaviour.

Inverse of a permutation polynomial: A polynomial g in F_q[x] satisfying f(g(x)) = g(f(x)) = x for all x in F_q, providing the algebraic inverse mapping.

References

1. On Inverses of Permutation Polynomials of Small Degree Over Finite Fields. IEEE Transactions on Information Theory (2019).
2. New classes of complete permutation polynomials. Finite Fields and Their Applications (2019).
3. Complete characterization of a class of permutation trinomials in characteristic five. Cryptography and Communications (2024).
4. On Polynomials of the Form xrf(x(q−1)/l). International Journal of Mathematics and Mathematical Sciences (2007).
5. Permutation binomials over finite fields. Transactions of the American Mathematical Society (2009).
6. Some estimate of character sums and its applications. Journal of Inequalities and Applications (2013).
7. Permutations on finite fields with invariant cycle structure on lines. Designs, Codes and Cryptography (2020).