# Permutable Polynomials and Rational Functions

### From NZJM

**New Zealand Journal of Mathematics**

**Vol.** 41, (2011), **Pages** 83-122

**Garry J. Tee**

Department of Mathematics

The University of Auckland

Auckland, New Zealand

mailto:tee@math.auckland.ac.nz

**Abstract** Permutable functions have recently been applied to cryptography. It has been known since 1951 that every sequence of permutable polynomials, which contains at least one polynomial of every positive degree, is either the sequence of simple positive powers or the sequence of Chebyshev polynomials of the first kind, or else it is related to those by a similarity transform by a linear function. The only known infinite sequences of permutable rational functions were the simple powers and Stirling's functions, which express tan *n**x* as rational functions of tan *x*: otherwise only 2 pairs of permutable rational functions have been published. Many infinite sequences of permutable rational functions are constructed on the basis of trigonometric functions and elliptic functions. Many identities connect 24 infinite sequences of permutable rational functions based on Jacobi's 12 elliptic functions.

**Keywords** Permutable polynomials, Chebyshev polynomials,
permutable chains, permutable rational functions, Jacobi elliptic functions,
real multiplication, trigonometric functions, rational function
identities, cryptography.

**Classification** (MSC2000) Primary 33E05, 26C05, 26C15; Secondary 20K99, 14G50.