In this paper, we classify$(q,q)$-biprojective almost perfect nonlinear (APN) functions over$\mathbb {L}\times \mathbb {L}$under the natural left and right action of$\mathop {\mathrm {GL}}\nolimits (2, \mathbb {L})$where$\mathbb {L}$is a finite field of characteristic 2. This shows in particular that the only quadratic APN functions (up to${\mathsf {CCZ}}$-equivalence) over$\mathbb {L}\times \mathbb {L}$that satisfy the so-called subfield property are the Gold functions and the function$\kappa: \mathbb {F}_{64} \to \mathbb {F}_{64}$which is the only known APN function that is equivalent to a permutation over$\mathbb {L}\times \mathbb {L}$up to${\mathsf {CCZ}}$-equivalence as shown in Browning et al. (2010). Deciding whether there exist other quadratic APN functions${\mathsf {CCZ}}$-equivalent to permutations that satisfy subfield property or equivalently, generalizing$\kappa $to higher dimensions was an open problem listed for instance in Carlet (2015) as one of the interesting open problems on cryptographic functions.

In this paper, we introduce the concept of biprojectivity in studying the cryptographically important almost perfect nonlinear (APN) functions. Although several known families of biprojective APN functions exist in the literature, the concept itself has not been explicitly observed and studied in detail. We give a survey of known APN families and functions that fall into the biprojective setting, then provide a method for finding such families and functions. We finally give two new infinite families of biprojective APN functions${\mathsf {CCZ}}$-inequivalent to any known APN function and study their properties.

New bounds on the cardinality of permutation codes equipped with the Ulam distance are presented. First, an integer-programming upper bound is derived, which improves on the Singleton-type upper bound in the literature for some lengths. Second, several probabilistic lower bounds are developed, which improve on the known lower bounds for large minimum distances. The results of a computer search for permutation codes are also presented.

Blondeau and colleagues recently conjectured a characterization of when$x^{2^{t}-1}$is an almost perfect nonlinear function on$ \BBF _{ 2^{n}}$. In this paper, we will prove the conjecture when$n$is even. The short proof uses a theorem of Hou and colleagues concerning reversed Dickson polynomials and a theorem of Payne.

Kloosterman sums are exponential sums on finite fields that have important applications in cryptography and coding theory. We use Stickelberger's theorem and the Gross-Koblitz formula to determine the value of the binary Kloosterman sum at$a$modulo 64, modulo 128, and modulo 256 in terms of coefficients of the characteristic polynomial of$a$.