Inversion in Finite Fields Using Logarithmic Depth

Litow & Davida (1988) show that inverses in large finite fields of small characteristicp, say p=2, can be computed by Boolean circuits of (order-optimal) logarithmic depth. We note that their numerical approach can also be implemented purely algebraically, and that the resulting much simpler algorithm yields, also for large p, both arithmetic and Boolean reductions of inversion in F"p"n to inversion in F"p.

[1]  Arjen K. Lenstra,et al.  Factoring polynominals over algebraic number fields , 1983, EUROCAL.

[2]  Joachim von zur Gathen.  Computing Powers in Parallel , 1987, SIAM J. Comput..

[3]  Joachim von zur Gathen,et al.  Parallel algorithms for algebraic problems , 1983, SIAM J. Comput..

[4]  Dario Bini,et al.  Parallel Solution of Certain Toeplitz Linear Systems , 1984, SIAM J. Comput..

[5]  Wayne Eberly.  Very Fast Parallel Polynomial Arithmetic , 1989, SIAM J. Comput..

[6]  Rudolf Lide,et al.  Finite fields , 1983 .

[7]  Susan Landau,et al.  Factoring Polynomials Over Algebraic Number Fields , 1985, SIAM J. Comput..

[8]  Victor Y. Pan,et al.  Polynomial division and its computational complexity , 1986, J. Complex..

[9]  F. R. Gantmakher.  The Theory of Matrices , 1984 .

[10]  Stephen A. Cook,et al.  Log Depth Circuits for Division and Related Problems , 1986, SIAM J. Comput..

[11]  Joachim von zur Gathen,et al.  Parallel Arithmetic Computations: A Survey , 1986, MFCS.

[12]  Donald E. Knuth,et al.  The art of computer programming. Vol.2: Seminumerical algorithms , 1981 .

[13]  Bruce E. Litow,et al.  O(log(n)) Parallel Time Finite Field Inversion , 1988, AWOC.

[14]  Joachim von zur Gathen,et al.  Boolean Circuits Versus Arithmetic Circuits , 1991, Inf. Comput..

[15]  J. Hopcroft,et al.  Fast parallel matrix and GCD computations , 1982, FOCS 1982.

[16]  Stephen A. Cook,et al.  A Taxonomy of Problems with Fast Parallel Algorithms , 1985, Inf. Control..

[17]  John H. Reif.  Logarithmic Depth Circuits for Algebraic Functions , 1986, SIAM J. Comput..

[18]  Allan Borodin,et al.  Parallel Computation for Well-Endowed Rings and Space-Bounded Probabilistic Machines , 1984, Inf. Control..