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Trace formulae for irreducible polynomials over $$\mathbb F_P$$ with minimal order roots in $$\mathbb F_{P^q}$$. (English) Zbl 1178.11074
Let $$P$$ be a prime of the form $$P = q^ns+1$$ for a prime $$q$$. Then $$P^q = q^{n+1}sK+1$$, where $$\gcd(K,P-1) = 1$$. The author gives formulas involving values of the trace function $$\text{Tr}: \mathbb F_{P^q}\to\mathbb F_P$$ of elements $$\alpha \in \mathbb F_{P^q}$$ of order $$R$$ for a prime divisor $$R$$ of $$K$$. For instance $$\text{Tr}(\alpha)+\text{Tr}(\alpha^{-1}) = -1$$, $$\text{Tr}(\alpha)\text{Tr}(\alpha^{-1}) = (q+1)/2$$ if $$q > 2, R = 2q+1$$ (take e.g. $$P=401$$, $$q=5$$, $$R=11$$).
##### MSC:
 11T06 Polynomials over finite fields
##### Keywords:
Trace-function; minimal polynomial; reciprocal polynomial
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##### References:
 [1] Bach, Eric; Shallit, Jeffrey, Algorithmic number theory, vol. 1: efficient algorithms, (1997), MIT · Zbl 0873.11070 [2] Beaver, Cheryl; Gemmell, Peter; Johnston, Anna; Newmann, William, On the cryptographic value of the qth root problem, (), 135-142 · Zbl 1014.94553 [3] Cipolla, M., Un metodo per la risoluzione Della congruenza di secondo grado, Rend. accad. sci. fis. mat., 9, 154-163, (1903) · JFM 34.0219.02 [4] Johnston, Anna; Gemmell, Peter, Authenticated key exchange provably secure against the man-in-the-middle attack, J. cryptology, 15, 2, 139-148, (2002) · Zbl 0994.94027 [5] Anna M. Johnston, thesis, On the difficulty of the qth root problem in certain finite cyclic groups, University of London, Senate House, Malet Street, London, 2006 [6] Lild, Rudolf; Niederreiter, Harald, Finite fields, (1997), Cambridge University Press
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