OK, let's now look specifically at functions which are _permutations_. Suppose I have a finite field GF(p^k), which can be represented as k-element vector with elements [0..p-1]. Is there any characterization of these Lagrange interpolations that happen to generate only permutations? I.e., let P={x|x=[x0,x1,x2,...,x(k-1)] and 0<=xi<p and i/=j => xi/=xj}, then f(p)=q for p,q in P. Alternately, which algebraic constructions generate _only_ and _all_ permutations of [0..p-1]. Obviously additions and multiplications by constants generate some of the permutations, but not all of them. What else do we need to get all of the permutations? At 09:24 AM 8/22/2012, Neil Sloane wrote:
Sure, use a version of Lagrange interpolation. Make a list (a generalized truth table) of all the values the function takes at all possible arguments. Then write a monomial for each term and add them up. E.g. if you want the value 7 at x=3, y=4, z=5, the monomial is Prod_{a!=3}(x-a) Prod_{b!=4} (y-b) Prod_{c!=5) (z-c) 7/d, where d is the appropriate constant.
On Wed, Aug 22, 2012 at 12:06 PM, Henry Baker <hbaker1@pipeline.com> wrote:
It appears that all functions over GF(2) can be synthesized using only polynomials:
E.g.,
f()=0, f()=1
f(x)=0, f(x)=1, f(x)=x, f(x)=1+x
not(x)=1+x, and(x,y)=x*y, or(x,y)=x+y+x*y, etc.
Q: Can all functions over _any_ finite field by synthesized using only polynomials, and is there a simple synthesis procedure?
-- Dear Friends, I have now retired from AT&T. New coordinates:
Neil J. A. Sloane, President, OEIS Foundation 11 South Adelaide Avenue, Highland Park, NJ 08904, USA Phone: 732 828 6098; home page: http://NeilSloane.com Email: njasloane@gmail.com