k256: implement Scalar::sqrt
#400
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tarcieri
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tarcieri
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Aug 31, 2021
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| let w = { | ||
| let t0 = self; | ||
| let t1 = t0.square(); | ||
| let t2 = t1 * t0; | ||
| let t3 = t1.square(); | ||
| let t4 = t3.square(); |
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Curiously I wasn't able to replace this with pow_vartime as copied/pasted from p256, which is how sqrt is implemented there.
I tried the following:
let w = self.pow_vartime(&[
0x77fa4bd19a06c82,
0xd755db9cd5e91407,
0xffffffffffffffff,
0x1fffffffffffffff,
]);
It's possible I miscomputed the exponent. Here's my Sage:
sage: n = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141
sage: n
115792089237316195423570985008687907852837564279074904382605163141518161494337
sage: s = 6
sage: t = (n - 1) >> s
sage: t
1809251394333065553493296640760748560200586941860545380978205674086221273349
sage: (t - 1) / 2
904625697166532776746648320380374280100293470930272690489102837043110636674
sage: hex((t - 1) / 2)
'0x1fffffffffffffffffffffffffffffffd755db9cd5e9140777fa4bd19a06c82'
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Scalar::sqrtScalar::sqrt
Implements Tonelli-Shank's algorithm for q mod 16 = 1 as synthesized using the `ff_derive` crate, similar to #392 which implements it for the `p256`. Like `p256`, as part of implementing this it was discovered that `root_of_unity()` was incorrect. Here it is (re)calculated with sage: sage: n = 0xFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFEBAAEDCE6AF48A03BBFD25E8CD0364141 sage: GF(n).primitive_element() 7 sage: s = 6 sage: t = (n - 1) >> s sage: power_mod(7,t,n) 5480320495727936603795231718619559942670027629901634955707709633242980176626 Note that the value was computed correctly originally, but the hex digits were shifted such that the resulting value was left shifted by 4-bits. This has now been corrected.
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Codecov Report
@@ Coverage Diff @@
## master #400 +/- ##
==========================================
+ Coverage 54.88% 58.59% +3.71%
==========================================
Files 29 29
Lines 4207 4565 +358
==========================================
+ Hits 2309 2675 +366
+ Misses 1898 1890 -8
Continue to review full report at Codecov.
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tarcieri
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I had previously attempted to do this in #400 but had trouble tracking down why it wasn't working. It seems the lower two limbs were shifted shifted 8-bits, i.e. a copy-paste error from Sage. I computed the correct limbs using `ff_derive`, and the tests now pass.
tarcieri
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I had previously attempted to do this in #400 but had trouble tracking down why it wasn't working. It seems the lower two limbs were shifted shifted 8-bits, i.e. a copy-paste error from Sage. I computed the correct limbs using `ff_derive`, and the tests now pass.
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Implements Tonelli-Shank's algorithm for q mod 16 = 1 as synthesized using the
ff_derivecrate, similar to #392 which implements it for thep256.Like in
p256, as part of implementing this it was discovered thatroot_of_unity()was incorrect. Here it is (re)calculated with sage:Note that the value was computed correctly originally, but the hex digits were shifted such that the resulting value was left shifted by 4-bits. This has now been corrected.