crypto: Specialize the Fq2 multiplication by ksi - #1648
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Multiplication by ksi = 9 + u, the non-residue defining the Fq6 extension, went through the generic Fq2 multiplication, which needs 4 field multiplications where 2 suffice: the u coefficient of ksi is 1, so a1 * 1 and a0 * 1 were computed for nothing. Add mul_by_ksi() computing (a0 + a1*u)*(9 + u) = (9a0 - a1) + (a0 + 9a1)*u directly and use it at the 13 sites, spread over the Fq6 and Fq12 multiplication, squaring and inversion, the cyclotomic squaring and the Miller loop's line multiplication. Cuts up to 7% off the ECPAIRING instruction count and a few percent off its cycles.
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Codecov Report✅ All modified and coverable lines are covered by tests. Additional details and impacted files@@ Coverage Diff @@
## master #1648 +/- ##
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- Coverage 97.72% 97.72% -0.01%
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Files 171 171
Lines 15631 15630 -1
Branches 3617 3617
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- Hits 15275 15274 -1
Misses 269 269
Partials 87 87
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Multiplication by ksi = 9 + u, the non-residue defining the Fq6 extension, went through the generic Fq2 multiplication. That costs 4 field multiplications where 2 suffice: the u coefficient of ksi is 1, so
a1 * 1anda0 * 1were computed for nothing.mul_by_ksi()computes(a0 + a1*u)*(9 + u) = (9a0 - a1) + (a0 + 9a1)*udirectly and is used at the 13 sites, spread over the Fq6 and Fq12 multiplication, squaring and inversion, the cyclotomic squaring and the Miller loop's line multiplication.The dependency depth is unchanged (multiplication then addition, exactly as before), so this is a strict removal of work rather than a trade-off.
ecpairing benchmark, interleaved A/B with two-point subtraction: instructions -6.6% (0.02% spread), cycles -2.6% by minimum and -4.6% by median over 7 repetitions, faster in 7/7. Only the cycles range is soft, the box was at load 50-64 throughout.
The obvious stronger version, computing 9a with additions only so that no multiplication is left at all, is a regression and the TODO in the code records it: -9.3% instructions but +5.2% cycles, IPC 1.78 to 1.54. The chain v, 2v, 4v, 8v, 9v is 4 dependent modular additions, each with its own carry chain and conditional subtraction, replacing multiplications that were independent and pipelined well. Fewer instructions, longer critical path.
Correctness:
mul_by_ksiwas differentially tested against the genericmultiply(a, ksi)over 200,100 inputs, the field edge cases (0, 1, 2, 8, 9, p-1, p-2, p/2, p/2+1, (p-1)/9 in both coordinates) and 200k random Fq2 values, with no mismatch. Thestatic_assertpins the coefficient that licenses dropping the two multiplications.🤖 Generated with Claude Code