FA-15601 / Numerics / Open access
Gf2 field multiplication: multiplier bit progression · case 01
The exact gf2 field multiplication result violates the stated contract at multiplier bit progression.
ROOT CAUSE
The multiplier bit progression step uses b<<1 instead of b>>1.
VERIFIED REPAIR
Use b>>1 at the multiplier bit progression step.
Unsuccessful approach: The partial repair b>>2 still violates the multiplier bit progression invariant.
Case contract
Input [a,b,irreducible polynomial bitmask], nonnegative field elements; return carryless product reduced by that monic polynomial.
Why this case matters
Exact discrete arithmetic with observable algorithmic state; no floating point approximation is used.
1 / The failure
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
a,b,poly=x;n=poly.bit_length()-1;out=0
for _ in range(n):
if b&1:out=out^a
b=b<<1
a<<=1
if a&(1<<n):a=a^poly
return out
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([1, 2, 7], 2), ([0, 0, 7], 0), ([15, 15, 31], 8), ([0, 1, 7], 0), ([0, 2, 7], 0), ([0, 3, 7], 0), ([1, 0, 7], 0), ([1, 1, 7], 1)], [([1, 3, 7], 3), ([2, 3, 7], 1), ([0, 0, 7], 0), ([15, 15, 31], 8), ([0, 1, 11], 0), ([0, 2, 11], 0), ([0, 3, 11], 0), ([0, 4, 11], 0)], [([2, 2, 7], 3), ([1, 2, 11], 2), ([0, 0, 7], 0), ([15, 15, 31], 8), ([2, 2, 11], 4), ([2, 3, 11], 6), ([2, 4, 11], 3), ([2, 5, 11], 1)], [([2, 3, 7], 1), ([1, 5, 11], 5), ([0, 0, 7], 0), ([15, 15, 31], 8), ([4, 3, 11], 7), ([4, 4, 11], 6), ([4, 5, 11], 2), ([4, 6, 11], 5)], [([3, 2, 7], 1), ([2, 2, 11], 4), ([0, 0, 7], 0), ([15, 15, 31], 8), ([6, 4, 11], 5), ([6, 5, 11], 3), ([6, 6, 11], 2), ([6, 7, 11], 4)]]
for i, (args, expected) in enumerate(fixtures[N-1]):
check("explicit oracle %d" % i, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| explicit oracle 0 | 0 | 2 | Failed |
| explicit oracle 1 | 0 | 0 | Passed |
| explicit oracle 2 | 15 | 8 | Failed |
| explicit oracle 3 | 0 | 0 | Passed |
| explicit oracle 4 | 0 | 0 | Passed |
| explicit oracle 5 | 0 | 0 | Passed |
| explicit oracle 6 | 0 | 0 | Passed |
| explicit oracle 7 | 1 | 1 | Passed |
SHA-256 / c544dcc59fe91d0c32d7aa12bdd28fdfb3ff1eb772045f58994ac6808f598da7
2 / The unsuccessful fix
Exit 1"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
a,b,poly=x;n=poly.bit_length()-1;out=0
for _ in range(n):
if b&1:out=out^a
b=b>>2
a<<=1
if a&(1<<n):a=a^poly
return out
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([1, 2, 7], 2), ([0, 0, 7], 0), ([15, 15, 31], 8), ([0, 1, 7], 0), ([0, 2, 7], 0), ([0, 3, 7], 0), ([1, 0, 7], 0), ([1, 1, 7], 1)], [([1, 3, 7], 3), ([2, 3, 7], 1), ([0, 0, 7], 0), ([15, 15, 31], 8), ([0, 1, 11], 0), ([0, 2, 11], 0), ([0, 3, 11], 0), ([0, 4, 11], 0)], [([2, 2, 7], 3), ([1, 2, 11], 2), ([0, 0, 7], 0), ([15, 15, 31], 8), ([2, 2, 11], 4), ([2, 3, 11], 6), ([2, 4, 11], 3), ([2, 5, 11], 1)], [([2, 3, 7], 1), ([1, 5, 11], 5), ([0, 0, 7], 0), ([15, 15, 31], 8), ([4, 3, 11], 7), ([4, 4, 11], 6), ([4, 5, 11], 2), ([4, 6, 11], 5)], [([3, 2, 7], 1), ([2, 2, 11], 4), ([0, 0, 7], 0), ([15, 15, 31], 8), ([6, 4, 11], 5), ([6, 5, 11], 3), ([6, 6, 11], 2), ([6, 7, 11], 4)]]
for i, (args, expected) in enumerate(fixtures[N-1]):
check("explicit oracle %d" % i, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| explicit oracle 0 | 0 | 2 | Failed |
| explicit oracle 1 | 0 | 0 | Passed |
| explicit oracle 2 | 14 | 8 | Failed |
| explicit oracle 3 | 0 | 0 | Passed |
| explicit oracle 4 | 0 | 0 | Passed |
| explicit oracle 5 | 0 | 0 | Passed |
| explicit oracle 6 | 0 | 0 | Passed |
| explicit oracle 7 | 1 | 1 | Passed |
SHA-256 / bfc6d91376f8713b81dc3181a8ddd98ccd7eaff7042e496ca5511b6dce233280
3 / The verified repair
Exit 0"""Failure Map reference implementation. Python standard library only."""
import json
import math
import itertools
from fractions import Fraction
N = 1
observations = []
def solve(x):
a,b,poly=x;n=poly.bit_length()-1;out=0
for _ in range(n):
if b&1:out=out^a
b=b>>1
a<<=1
if a&(1<<n):a=a^poly
return out
def check(label, actual, expected):
observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[([1, 2, 7], 2), ([0, 0, 7], 0), ([15, 15, 31], 8), ([0, 1, 7], 0), ([0, 2, 7], 0), ([0, 3, 7], 0), ([1, 0, 7], 0), ([1, 1, 7], 1)], [([1, 3, 7], 3), ([2, 3, 7], 1), ([0, 0, 7], 0), ([15, 15, 31], 8), ([0, 1, 11], 0), ([0, 2, 11], 0), ([0, 3, 11], 0), ([0, 4, 11], 0)], [([2, 2, 7], 3), ([1, 2, 11], 2), ([0, 0, 7], 0), ([15, 15, 31], 8), ([2, 2, 11], 4), ([2, 3, 11], 6), ([2, 4, 11], 3), ([2, 5, 11], 1)], [([2, 3, 7], 1), ([1, 5, 11], 5), ([0, 0, 7], 0), ([15, 15, 31], 8), ([4, 3, 11], 7), ([4, 4, 11], 6), ([4, 5, 11], 2), ([4, 6, 11], 5)], [([3, 2, 7], 1), ([2, 2, 11], 4), ([0, 0, 7], 0), ([15, 15, 31], 8), ([6, 4, 11], 5), ([6, 5, 11], 3), ([6, 6, 11], 2), ([6, 7, 11], 4)]]
for i, (args, expected) in enumerate(fixtures[N-1]):
check("explicit oracle %d" % i, solve(args), expected)
print(json.dumps({"observations": observations, "passed": all(x["passed"] for x in observations)}, ensure_ascii=False))
raise SystemExit(0 if all(x["passed"] for x in observations) else 1)
| Boundary fixture | Actual | Expected | Outcome |
|---|---|---|---|
| explicit oracle 0 | 2 | 2 | Passed |
| explicit oracle 1 | 0 | 0 | Passed |
| explicit oracle 2 | 8 | 8 | Passed |
| explicit oracle 3 | 0 | 0 | Passed |
| explicit oracle 4 | 0 | 0 | Passed |
| explicit oracle 5 | 0 | 0 | Passed |
| explicit oracle 6 | 0 | 0 | Passed |
| explicit oracle 7 | 1 | 1 | Passed |
SHA-256 / e9ea5ef882cddedd8d416128d23ddf4ae319ce2558b94298c1bf2223b1a861bc
Verification & scope
A deterministic bounded teaching model. Inputs are restricted to the explicit contract; this is not a production algebra library. This reproducer isolates one failure mechanism. Results cover the supplied fixtures. Variants within a family share a test contract and should remain grouped when constructing evaluation splits. Related mechanisms with a shared evaluation_group must also remain together; these controlled models are not independent production incidents.
Observations recorded using Python 3.12.14 at 2026-09-29T14:39:28.370455+00:00.
Case digest / eb2031ccfe68882ab6c0cab33a2bbb7fb58f5a4bf7a47a54ec60d90ec4da9594