Polynomial functions on the units of Z_{2^n}

TL;DR

This paper characterizes polynomial functions on the units of Z_{2^n}, introduces a unique reduced polynomial representation, and applies these results to construct large k-ary quasigroups for cryptography, also providing a new proof of Rivest's result.

Contribution

It identifies a finite set of reduced polynomials that induce all polynomial functions on Q_n and demonstrates their use in cryptographic constructions, offering a simpler proof of Rivest's characterization.

Findings

Finite set of reduced polynomials for Q_n

Unique polynomial representation for each function

Construction of large k-ary quasigroups for cryptography

Abstract

Polynomial functions on the group of units Q_n of the ring Z_{2^n} are considered. A finite set of reduced polynomials RP_n in Z[x] that induces the polynomial functions on Q_n is determined. Each polynomial function on Q_n is induced by a unique reduced polynomial - the reduction being made using a suitable ideal in Z[x]. The set of reduced polynomials forms a multiplicative 2-group. The obtained results are used to efficiently construct families of exponential cardinality of, so called, huge k-ary quasigroups, which are useful in the design of various types of cryptographic primitives. Along the way we provide a new (and simpler) proof of a result of Rivest characterizing the permutational polynomials on Z_{2^n}.