ethcore: minor optimization of modexp by using LR exponentiation

[andresilva]

Optimize modexp implementation by doing left-to-right binary exponentiation to keep multiplicands lower. The implementation can be further optimised by using a bigger limb size for the BigUint currently num is using u32.

benchcmp:

 name                          modexp_base ns/iter  modexp_final ns/iter  diff ns/iter   diff %  speedup
 modexp_eip_example1           501,834              304,093                   -197,741  -39.40%   x 1.65
 modexp_eip_example2           267                  200                            -67  -25.09%   x 1.33
 modexp_nagydani_1_pow0x10001  38,693               38,748                          55    0.14%   x 1.00
 modexp_nagydani_1_qube        7,477                6,087                       -1,390  -18.59%   x 1.23
 modexp_nagydani_1_square      5,180                3,528                       -1,652  -31.89%   x 1.47
 modexp_nagydani_2_pow0x10001  82,100               82,079                         -21   -0.03%   x 1.00
 modexp_nagydani_2_qube        14,532               10,895                      -3,637  -25.03%   x 1.33
 modexp_nagydani_2_square      10,033               6,095                       -3,938  -39.25%   x 1.65
 modexp_nagydani_3_pow0x10001  251,002              242,514                     -8,488   -3.38%   x 1.04
 modexp_nagydani_3_qube        43,365               30,771                     -12,594  -29.04%   x 1.41
 modexp_nagydani_3_square      29,139               16,189                     -12,950  -44.44%   x 1.80
 modexp_nagydani_4_pow0x10001  825,013              772,208                    -52,805   -6.40%   x 1.07
 modexp_nagydani_4_qube        137,210              94,985                     -42,225  -30.77%   x 1.44
 modexp_nagydani_4_square      92,899               47,773                     -45,126  -48.58%   x 1.94
 modexp_nagydani_5_pow0x10001  2,930,324            2,760,226                 -170,098   -5.80%   x 1.06
 modexp_nagydani_5_qube        493,096              330,857                   -162,239  -32.90%   x 1.49
 modexp_nagydani_5_square      330,625              166,783                   -163,842  -49.56%   x 1.98

[dvdplm]

For reviewers: http://www.cacr.math.uwaterloo.ca/hac/about/chap14.pdf

[dvdplm]

lgtm. Nice speedup!

Can't help wondering if SIMD could be brought to bear on this – did you look into that? :)

[andresilva]

I think if the num library starts using u64 for its "words" and u128 for "double words", LLVM will generate better code for the primitive operations like mul and mod, that will probably give a 2x speedup. I wouldn't know how to vectorize this code though 😅. I also implemented the LR k-ary algorithm (14.83) where you process more than one bit of the exponent at a time. But it wasn't worth it, probably only pays off for really big exponents (since you have to precompute a table).