FA-71966 / Error-correcting codes / Open access

GF(256) multiply tests the carry after shifting · case 01

Operands with the top bit set are not reduced.

Verified by executionVariant 1 · 8 checks per implementationDownload source bundle ↓JSON ↗

ROOT CAUSE

The carry bit is sampled after the shift and mask, when it has already been discarded.

THE FAILURE

The carry bit is sampled after the shift and mask, when it has already been discarded.

Unsuccessful approach: Sampling bit 6 before the shift reduces one step early.

Case contract

Multiply two elements of GF(2^8) defined by the primitive polynomial x^8+x^4+x^3+x^2+1 (0x11D) using shift-and-add. Operands outside 0..255 return None.

Why this case matters

Reed-Solomon codecs for storage and QR codes are built on GF(2^8) arithmetic with polynomial 0x11D.

1 / The failure

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(a, b):
    if not (0 <= a < 256 and 0 <= b < 256):
        return None
    p = 0
    for _ in range(8):
        if b & 1:
            p ^= a
        b >>= 1
        a = (a << 1) & 0xFF
        carry = a & 0x80
        if carry:
            a ^= 0x1D
    return p
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression [158, 185]', [158, 185], 141], ['regression [66, 7]', [66, 7], 211], ['control [0, 5]', [0, 5], 0], ['control [3, 7]', [3, 7], 9], ['control [256, 1]', [256, 1], None], ['control [198, 115]', [198, 115], 153], ['control [136, 144]', [136, 144], 15], ['control [184, 92]', [184, 92], 29]], [['regression [198, 115]', [198, 115], 153], ['regression [136, 144]', [136, 144], 15], ['control [256, 1]', [256, 1], None], ['control [0, 5]', [0, 5], 0], ['control [3, 7]', [3, 7], 9], ['control [95, 65]', [95, 65], 1], ['control [44, 122]', [44, 122], 249], ['control [232, 214]', [232, 214], 217]], [['regression [243, 34]', [243, 34], 240], ['regression [9, 184]', [9, 184], 17], ['control [3, 7]', [3, 7], 9], ['control [256, 1]', [256, 1], None], ['control [0, 5]', [0, 5], 0], ['control [231, 130]', [231, 130], 45], ['control [28, 61]', [28, 61], 246], ['control [253, 149]', [253, 149], 58]], [['regression [44, 122]', [44, 122], 249], ['regression [232, 214]', [232, 214], 217], ['control [0, 5]', [0, 5], 0], ['control [3, 7]', [3, 7], 9], ['control [256, 1]', [256, 1], None], ['control [117, 177]', [117, 177], 104], ['control [70, 227]', [70, 227], 18], ['control [114, 166]', [114, 166], 104]], [['regression [51, 91]', [51, 91], 155], ['regression [231, 130]', [231, 130], 45], ['control [256, 1]', [256, 1], None], ['control [0, 5]', [0, 5], 0], ['control [3, 7]', [3, 7], 9], ['control [1, 200]', [1, 200], 200], ['control [2, 128]', [2, 128], 29], ['control [128, 2]', [128, 2], 29]]]
for label, args, expected in fixtures[N - 1]:
    check(label, 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
regression [158, 185]	`107`	`141`	Failed
regression [66, 7]	`233`	`211`	Failed
control [0, 5]	`0`	`0`	Passed
control [3, 7]	`9`	`9`	Passed
control [256, 1]	`None`	`None`	Passed
control [198, 115]	`188`	`153`	Failed
control [136, 144]	`104`	`15`	Failed
control [184, 92]	`191`	`29`	Failed

SHA-256 / 57ecb8ed83a58ea02b25ea965f56cec3d7c5ced08db2cdf124652fcd003fa7a3

2 / The unsuccessful fix

Exit 1

"""Failure Map reference implementation. Python standard library only."""
import json

N = 1
observations = []
def solve(a, b):
    if not (0 <= a < 256 and 0 <= b < 256):
        return None
    p = 0
    for _ in range(8):
        if b & 1:
            p ^= a
        b >>= 1
        carry = a & 0x40
        a = (a << 1) & 0xFF
        if carry:
            a ^= 0x1D
    return p
def check(label, actual, expected):
    observations.append({"check": label, "actual": actual, "expected": expected, "passed": actual == expected})
fixtures = [[['regression [158, 185]', [158, 185], 141], ['regression [66, 7]', [66, 7], 211], ['control [0, 5]', [0, 5], 0], ['control [3, 7]', [3, 7], 9], ['control [256, 1]', [256, 1], None], ['control [198, 115]', [198, 115], 153], ['control [136, 144]', [136, 144], 15], ['control [184, 92]', [184, 92], 29]], [['regression [198, 115]', [198, 115], 153], ['regression [136, 144]', [136, 144], 15], ['control [256, 1]', [256, 1], None], ['control [0, 5]', [0, 5], 0], ['control [3, 7]', [3, 7], 9], ['control [95, 65]', [95, 65], 1], ['control [44, 122]', [44, 122], 249], ['control [232, 214]', [232, 214], 217]], [['regression [243, 34]', [243, 34], 240], ['regression [9, 184]', [9, 184], 17], ['control [3, 7]', [3, 7], 9], ['control [256, 1]', [256, 1], None], ['control [0, 5]', [0, 5], 0], ['control [231, 130]', [231, 130], 45], ['control [28, 61]', [28, 61], 246], ['control [253, 149]', [253, 149], 58]], [['regression [44, 122]', [44, 122], 249], ['regression [232, 214]', [232, 214], 217], ['control [0, 5]', [0, 5], 0], ['control [3, 7]', [3, 7], 9], ['control [256, 1]', [256, 1], None], ['control [117, 177]', [117, 177], 104], ['control [70, 227]', [70, 227], 18], ['control [114, 166]', [114, 166], 104]], [['regression [51, 91]', [51, 91], 155], ['regression [231, 130]', [231, 130], 45], ['control [256, 1]', [256, 1], None], ['control [0, 5]', [0, 5], 0], ['control [3, 7]', [3, 7], 9], ['control [1, 200]', [1, 200], 200], ['control [2, 128]', [2, 128], 29], ['control [128, 2]', [128, 2], 29]]]
for label, args, expected in fixtures[N - 1]:
    check(label, 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
regression [158, 185]	`107`	`141`	Failed
regression [66, 7]	`233`	`211`	Failed
control [0, 5]	`0`	`0`	Passed
control [3, 7]	`9`	`9`	Passed
control [256, 1]	`None`	`None`	Passed
control [198, 115]	`188`	`153`	Failed
control [136, 144]	`104`	`15`	Failed
control [184, 92]	`191`	`29`	Failed

SHA-256 / 285673dd2c6c72c69543c19f1a6f885359fcba930f684eaf5423a0392fd41091

HELD IN THE MEMBER ARCHIVE

The verified repair and its recorded checks are member-only.

This mechanism has 8 recorded checks per implementation. The open-access tier publishes the failure and the unsuccessful fix; the repaired source that passes every check, and the observations that prove it, are available to members.

Every case sharing this mechanism uses the same contract and the same repair, so this one record is held back for all of them.

Member access is invitation-based. Sign in with your invited account to inspect the repair.

Verification & scope

A deterministic, bounded teaching model of the named code under the stated contract; not a production codec. 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 2026-09-29T14:48:34.267656+00:00.

Case digest / bac7e8177c25c7314cc7affb5c9c7a7dc39dd9a26ba88e5ce1ad5df85dc688da