## Cryptology ePrint Archive: Report 2013/252

On the Primitivity of some Trinomials over Finite Fields

LI Yujuan, WANG Huaifu, ZHAO Jinhua

Abstract: In this paper, we give conditions under which the trinomials of the form $x^{n}+ax+b$ over finite field ${\mathbb{F}}_{p^{m}}$ are not primitive and conditions under which there are no primitive trinomials of the form $x^{n}+ax+b$ over finite field ${\mathbb{F}}_{p^{m}}$. For finite field ${\mathbb{F}}_{4}$, We show that there are no primitive trinomials of the form $x^{n}+x+\alpha$, if $n\equiv1\mod 3$ or $n\equiv0\mod 3$ or $n\equiv4\mod 5$.

Category / Keywords: secret-key cryptography / LFSR,primitive polynomial, finite field