- secp256k1_scalar_cadd_bit
- secp256k1_modinvXX_normalize_YY
- secp256k1_modinvXX_divsteps_ZZ
- ECMULT_CONST_TABLE_GET_GE
Even though those code loations are not problematic right now
(with current compilers).
This introduces variants of the divsteps-based GCD algorithm used for
modular inverses to compute Jacobi symbols. Changes compared to
the normal vartime divsteps:
* Only positive matrices are used, guaranteeing that f and g remain
positive.
* An additional jac variable is updated to track sign changes during
matrix computation.
* There is (so far) no proof that this algorithm terminates within
reasonable amount of time for every input, but experimentally it
appears to almost always need less than 900 iterations. To account
for that, only a bounded number of iterations is performed (1500),
after which failure is returned. In VERIFY mode a lower iteration
count is used to make sure that callers exercise their fallback.
* The algorithm converges to f=g=gcd(f0,g0) rather than g=0. To keep
this test simple, the end condition is f=1, which won't be reached
if started with non-coprime or g=0 inputs. Because of that we only
support coprime non-zero inputs.
Instead of using eta=-delta, use zeta=-(delta+1/2) to represent
delta. This variant only needs at most 590 iterations for 256-bit
inputs rather than 724 (by convex hull bounds analysis).
The magnitude of the f and g variables generally goes down as the algorithm
progresses. Make use of this by keeping tracking how many limbs are used, and
when the number becomes small enough, make use of this to reduce the complexity
of arithmetic on them.
Refactored by: Pieter Wuille <pieter@wuille.net>
This commit adds functions to verify and compare numbers in signed{30,62} notation,
and uses that to do more extensive bounds checking on various variables in the modinv
code.
This adds a long comment explaining the algorithm and implementation choices by building
it up step by step in Python.
Comments in the code are also reworked/added, with references to the long explanation.
