Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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CERTAIN CUBIC POLYNOMIALS OVER FINITE FIELDS

  • Published : 2009.01.31
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Abstract

Motivated by XTR cryptosystem which is based on an irreducible polynomial $x^3-cx^2+c^px-1$ over $F_{p^2}$, we study polynomials of the form $F(c,x)=x^3-cx^2+c^qx-1$ over $F_{p^2}$ with $q=p^m$. In this paper, we establish a one to one correspondence between the set of such polynomials and a certain set of cubic polynomials over $F_q$. Our approach is rather theoretical and provides an efficient method to generate irreducible polynomials over $F_{p^2}$.

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References

  1. A. K. Lenstra and E. R. Verheul, The XTR public key system, Advances in cryptology-CRYPTO 2000 (Santa Barbara, CA), 1–19, Lecture Notes in Comput. Sci., 1880, Springer, Berlin, 2000. https://doi.org/10.1007/3-540-44598-6_1
  2. A. K. Lenstra and E. R. Verheul, Key improvements to XTR, Advances in cryptology-ASIACRYPT 2000 (Kyoto), 220–233, Lecture Notes in Comput. Sci., 1976, Springer, Berlin, 2000. https://doi.org/10.1007/3-540-44448-3_17
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