New algorithms for finding irreducible polynomials over finite fields

An algorithm is presented for finding an irreducible polynomial of specified degree over a finite field. It is deterministic and runs in polynomial time for fields of small characteristics. A proof is given of the stronger result, that the problem of finding irreducible polynomials of specified degree over a finite field K is deterministic-polynomial-time reducible to the problem of factoring polynomials over the prime field of K.<<ETX>>

[1]  W. Burnside,et al.  Theory of equations , 1886 .

[2]  R. Tennant.  Algebra , 1941, Nature.

[3]  Elwyn R. Berlekamp,et al.  Algebraic coding theory , 1984, McGraw-Hill series in systems science.

[4]  E. Berlekamp.  Factoring polynomials over large finite fields* , 1970, SYMSAC '71.

[5]  E. Berlekamp.  Factoring polynomials over large finite fields* , 1971, SYMSAC '71.

[6]  Alfred V. Aho,et al.  The Design and Analysis of Computer Algorithms , 1974 .

[7]  W. Schmidt.  Equations over Finite Fields: An Elementary Approach , 1976 .

[8]  A. Odlyzko,et al.  A bound for the least prime ideal in the Chebotarev Density Theorem , 1979 .

[9]  Michael O. Rabin,et al.  Probabilistic Algorithms in Finite Fields , 1980, SIAM J. Comput..

[10]  Paul Camion.  Improving an algorithm for factoring polynomials over a finite field and constructing large irreducible polynomials , 1983, IEEE Trans. Inf. Theory.

[11]  Ronald L. Rivest,et al.  A Knapsack Type Public Key Cryptosystem Based On Arithmetic in Finite Fields , 1984, CRYPTO.

[12]  Wayne Eberly,et al.  Very Fast Parallel Matrix and Polynomial Arithmetic , 1984, FOCS.

[13]  J. Gathen,et al.  Factorization of multivariate polynomials over finite fields , 1985 .

[14]  Ming-Deh A. Huang.  Riemann hypothesis and finding roots over finite fields , 1985, STOC '85.

[15]  Leonard M. Adleman,et al.  Finding irreducible polynomials over finite fields , 1986, STOC '86.

[16]  Joachim von zur Gathen,et al.  Factoring Polynomials and Primitive Elements for Special Primes , 1987, Theor. Comput. Sci..

[17]  Eric Bach,et al.  Realistic analysis of some randomized algorithms , 1987, J. Comput. Syst. Sci..

[18]  Ronald L. Rivest,et al.  A knapsack-type public key cryptosystem based on arithmetic in finite fields , 1988, IEEE Trans. Inf. Theory.

[19]  Prabhakar Raghavan,et al.  Randomized algorithms and pseudorandom numbers , 1988, STOC '88.

[20]  Eli Upfal,et al.  A time-randomness tradeoff for oblivious routing , 1988, STOC '88.

[21]  J. Chahal.  Equations over Finite Fields , 1988 .

[22]  Eric Bach,et al.  Factoring Polynomials Using Fewer Random Bits , 1990, J. Symb. Comput..

[23]  Victor Shoup.  On the Deterministic Complexity of Factoring Polynomials over Finite Fields , 1990, Inf. Process. Lett..