Implement error correction #205

There are two parameterizations of the bech32 checksum (see the "roots" unit test in src/primitives/polynomial.rs for what they are). In rust-bitcoin#203 we mixed them up, using the generator from one but the exponents from the other. We made the same mistake with codex32 apparently. When we implement error correction this will cause failures. Fix it.

Adds a CHARACTERISTIC constant to the Field trait, so this is yet another breaking change (though in practice I don't think anybody is implementing Field on their own types).

…near shift registers This provides a general-purpose implementation of the Berlekamp-Massey algorithm for finding a linear shift register that generates a given sequence prefix. If compiled without an allocator, it will run less efficiently (and be limited to a maximum size) but it will work. Also introduces a fuzz test to check that it works properly and does not crash.

This commit pulls everything together. The actual error correction code isn't too big: we interpret a residue as a polynomial, evaluate it at various powers of alpha to get a syndrome polynomial, call berlekeamp-massey on this to get a "connection polynomial", then use Forney's algorithm to get the actual error values. Each step in the above is encapsulated separately -- the "big" stuff, in particular Berlekamp-Massey and obtaining the relevant constants from the checksum definition, were in previous commits. This PR does need to add some more functionality to Polynomial. Specifically we need the ability to evaluate polynomials, take their formal derivatives, and multiply them modulo x^d for a given d. These are the bulk of this PR. The next commit will introduce a fuzztest which hammers on the correction logic to ensure that it's not crashing.

The codex32 test will more thoroughly exercise the algebra, since there we can correct up to 4 errors. The bech32 test on the other hand should work without an allocator (though to exercise this you need to manually edit fuzz/Cargo.toml to disable the alloc feature -- this is rust-lang/cargo#2980 which has been open for 10 years and counting..)

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Implement error correction #205

Implement error correction #205

Commits on Sep 30, 2024