Page 353 - Hardware Implementation of Finite-Field Arithmetic

Binary Fields 333

www.arithmetic-circuits.org. The implementation results (Spartan3,
speed -5) are the following:

FFs LUTs Slices Period AverCycles AverTime
2,170 3,514 2,062 7.9 54,422.8 429,940

233
C.2 GF(2 )
233
Another NIST-recommended finite field is GF(2 ), that is, the set of
polynomials of degree smaller than 233 over GF(2), modulo the
233
74
irreducible polynomial f(x) = x + x + 1.
C.2.1 mod f(x) Multiplication
Only sequential circuits are considered. The entities interleaved_mult
and montgomery_mult are described in Chap. 7. The parameter values
are the following:
constant M: integer := 233;
constant F: std_logic_vector(M-1 downto 0):=
(0=> ‘1’, 74 => ‘1’, others => ‘0’);
In the case of the interleaved multiplier, several implementation
strategies, based on the number of bits G computed at each cycle,
have been considered. The implementation results (Spartan3, speed-5)
are given in Table C.2.

Total
N G FFs LUTs Slices Period Cycles time
Interleaved 1 763 723 417 6.4 223 1427
Interleaved 2 769 957 541 5.6 112 627
Interleaved 4 794 1,192 689 5.5 56 308
Interleaved 8 780 1,919 1,045 5.7 28 160
Interleaved 15 880 3,112 1,736 5.9 15 88
Interleaved 16 879 3,130 1,743 5.9 14 83
Interleaved 32 932 6,213 3,346 7.5 7 52
Interleaved 56 1321 10,112 5,718 11.5 4 46
Montgomery – 484 489 255 7.5 233 1,748
TABLE C.2 Cost and Delay of Multipliers over GF(2 233 )

All the source files are available at www.arithmetic-circuits.org.