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Galois rings and algebraic cryptography. (English) Zbl 0744.11063

Let \(R\) be a finite local commutative ring with maximal ideal \(M\) and residue field \(k=R/M\). The polynomial \(f\in R[x]\) is monic irreducible if \(\mu f\) is irreducible in \(k[x]\) where \(\mu\) is the natural projection from \(R[x]\) to \(k[x]\). For \(p\) prime and \(n\), \(m\) integers, the Galois ring \(GR(p^ n,m)\) can be viewed as \((\mathbb{Z}/\mathbb{Z}_{p^ n})[x]/(f)\).
The Dickson polynomial \(g_ d(x,a)\) of degree \(d\) over a commutative ring with identity \(R\) is \[ g_ d(x,a)=\sum^{[d/2]}_{t=0}{d\over d- t}{d-t\choose t}(-a)^ tx^{d-2t} \] where \([\cdot]\) is the greatest integer function. Conditions under which Dickson polynomials permute \(GR(p^ n,m)\) are studied. It is shown that for \(a\) a unit in \(GR(p^ n,m)\), the set \({\mathcal P}(a)=\{g_ d(x,a): g_ d(x,a)\) permutes \(GR(p^ n,m)\}\) is closed under composition iff \(a=\pm 1\). The application of these results to public key cryptosystems is studied.

MSC:

11T71 Algebraic coding theory; cryptography (number-theoretic aspects)
11T06 Polynomials over finite fields
13M10 Polynomials and finite commutative rings
94A60 Cryptography
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