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.
This change eases the use of alternate build systems by moving
the variables in `src/libsecp256k1-config.h` to compiler macros
for each invocation, preventing duplication of these variables
for each build system.
Co-authored-by: Ali Sherief <ali@notatether.com>
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.
