Page 339 - Hardware Implementation of Finite-Field Arithmetic

APPENDIX B

Optimal Extension

Fields

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B.1 GF(239 )

B.1.1 VHDL Models and Constant Definitions
Several VHDL models have been generated for circuits executing
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operations over GF(239 ), that is, the set of polynomials of a degree
smaller than 17 over the field Z , modulo the irreducible polynomial
239
f(x) = x − 2. The MSE_first_mod_f_multiplier.vhd file includes the
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following entities: mod_239_reducer, adder_subtractor, mod_ 239_
multiplier, MSE_first_mod_ f_multiplier, LSE_first_mod_ f_multiplier. The
parameter values are the following:
constant K: natural := 8;
constant P: std_logic_vector(k-1 downto 0)
:= conv_std_logic_vector(239, k);
constant M: natural := 17;
type long_polynomial is array(M downto 0) of
std_logic_vector(K-1 downto 0);
type polynomial is array(M-1 downto 0) of
std_logic_vector(K-1 downto 0);
--LOGM is the number of bits of m-1
constant LOGM: natural := 5;
constant f: long_polynomial :=
(“00000001”, “00000000”, “00000000”, “00000000”,
“00000000”, “00000000”, “00000000”, “00000000”, “00000000”,
“00000000”, “00000000”, “00000000”, “00000000”,
“00000000”, “00000000”, “00000000”, “00000000”, “11101101” );

The oef.vhd file includes the following additional entities: mod_
239 _inverter and oef. The latter is a divider over the particular optimal
extension field under consideration. The mod 239 inversion is
performed with a table storing the 238 inverses. The table is modeled

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