Certain cubic polynomials over finite fields
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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_{q^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_{q^2}.
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_{q^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_{q^2}.
참고문헌 (4)
- A. K. Lenstra, "The XTR public key system, Advances in cryptology—CRYPTO 2000 (Santa Barbara, CA)", Lecture Notes in Comput. Sci., pp.1~19 (2000), Springer
- A. K. Lenstra, "Key improvements to XTR, Advances in cryptology—ASIACRYPT", Lecture Notes in Comput. Sci., pp.220~233 (2000), Springer
- A. K. Lenstra, "Fast irreducibility and subgroup membership testing in XTR, Public key cryptography", Lecture Notes in Comput. Sci., pp.73~86 (2001), Springer
- A. J. Menezes., "Applications of Finite Fields" (1993), Kluwer Academic Publishers