Analysis of a Public Key Approach Based on Polynomial Substitution

We set out to build a public key cryptosystem by repeatedly substituting for variables in multivariate polynomials and simplifying the results to conceal the substitution process. There seems, however, to be no way to build such a system that is both secure and has a public key of practical size when the devices used to limit the number of coefficeints are nilpotence and J-rings. We have only shown, however, that it is impossible to produce such a system if the total degree of the encryption polynomial determines the size of the public key. Perhaps, by properly choosing p₀ and p₁, we can employ the fundamental scheme to produce sparse encrypting polynomials. Then the public key could be kept small while the encrypting polynomial bas large total degree and is difficult to invert.