{"abstract":"Products that overflow the byte disagree with the Reed-Solomon field.","category":"Error-correcting codes","checks":8,"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.","contract_signature":"a, b","evaluation_group":"w2-error_correcting_codes-gf256-multiply","failed_approach":"XORing the full 0x11D after masking re-introduces bit 8 into the operand.","family":"w2-error_correcting_codes-gf256-multiply-reduction-polynomial","id":"FA-71956","implementations":{"attempt":{"sha256":"fc20360281b860de239171b74a47ff00b40bca66dd9193ac5bc1c6cdf378a374","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(a, b):\n    if not (0 <= a < 256 and 0 <= b < 256):\n        return None\n    p = 0\n    for _ in range(8):\n        if b & 1:\n            p ^= a\n        b >>= 1\n        carry = a & 0x80\n        a = (a << 1) & 0xFF\n        if carry:\n            a ^= 0x11D\n    return p\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['regression [158, 185]', [158, 185], 141], ['regression [66, 7]', [66, 7], 211], ['control [2, 96]', [2, 96], 192], ['control [0, 5]', [0, 5], 0], ['control [1, 200]', [1, 200], 200], ['control [3, 7]', [3, 7], 9], ['control [256, 1]', [256, 1], None], ['control [184, 92]', [184, 92], 29]], [['regression [198, 115]', [198, 115], 153], ['regression [136, 144]', [136, 144], 15], ['control [2, 96]', [2, 96], 192], ['control [0, 5]', [0, 5], 0], ['control [1, 200]', [1, 200], 200], ['control [3, 7]', [3, 7], 9], ['control [256, 1]', [256, 1], None], ['control [232, 214]', [232, 214], 217]], [['regression [243, 34]', [243, 34], 240], ['regression [9, 184]', [9, 184], 17], ['partial-repair [44, 122]', [44, 122], 249], ['control [2, 96]', [2, 96], 192], ['control [0, 5]', [0, 5], 0], ['control [1, 200]', [1, 200], 200], ['control [3, 7]', [3, 7], 9], ['control [256, 1]', [256, 1], None]], [['regression [44, 122]', [44, 122], 249], ['regression [232, 214]', [232, 214], 217], ['partial-repair [231, 130]', [231, 130], 45], ['partial-repair [253, 149]', [253, 149], 58], ['control [2, 96]', [2, 96], 192], ['control [0, 5]', [0, 5], 0], ['control [1, 200]', [1, 200], 200], ['control [3, 7]', [3, 7], 9]], [['regression [231, 130]', [231, 130], 45], ['regression [28, 61]', [28, 61], 246], ['partial-repair [117, 177]', [117, 177], 104], ['partial-repair [197, 117]', [197, 117], 168], ['control [2, 96]', [2, 96], 192], ['control [0, 5]', [0, 5], 0], ['control [1, 200]', [1, 200], 200], ['control [3, 7]', [3, 7], 9]]]\nfor label, args, expected in fixtures[N - 1]:\n    check(label, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"},"broken":{"sha256":"fc4015a04d28106043967a4fb0c1eee494adf2150ef20ab6ec034a5d32e22cda","source":"\"\"\"Failure Map reference implementation. Python standard library only.\"\"\"\nimport json\n\nN = 1\nobservations = []\ndef solve(a, b):\n    if not (0 <= a < 256 and 0 <= b < 256):\n        return None\n    p = 0\n    for _ in range(8):\n        if b & 1:\n            p ^= a\n        b >>= 1\n        carry = a & 0x80\n        a = (a << 1) & 0xFF\n        if carry:\n            a ^= 0x1B\n    return p\ndef check(label, actual, expected):\n    observations.append({\"check\": label, \"actual\": actual, \"expected\": expected, \"passed\": actual == expected})\nfixtures = [[['regression [158, 185]', [158, 185], 141], ['regression [66, 7]', [66, 7], 211], ['control [2, 96]', [2, 96], 192], ['control [0, 5]', [0, 5], 0], ['control [1, 200]', [1, 200], 200], ['control [3, 7]', [3, 7], 9], ['control [256, 1]', [256, 1], None], ['control [184, 92]', [184, 92], 29]], [['regression [198, 115]', [198, 115], 153], ['regression [136, 144]', [136, 144], 15], ['control [2, 96]', [2, 96], 192], ['control [0, 5]', [0, 5], 0], ['control [1, 200]', [1, 200], 200], ['control [3, 7]', [3, 7], 9], ['control [256, 1]', [256, 1], None], ['control [232, 214]', [232, 214], 217]], [['regression [243, 34]', [243, 34], 240], ['regression [9, 184]', [9, 184], 17], ['partial-repair [44, 122]', [44, 122], 249], ['control [2, 96]', [2, 96], 192], ['control [0, 5]', [0, 5], 0], ['control [1, 200]', [1, 200], 200], ['control [3, 7]', [3, 7], 9], ['control [256, 1]', [256, 1], None]], [['regression [44, 122]', [44, 122], 249], ['regression [232, 214]', [232, 214], 217], ['partial-repair [231, 130]', [231, 130], 45], ['partial-repair [253, 149]', [253, 149], 58], ['control [2, 96]', [2, 96], 192], ['control [0, 5]', [0, 5], 0], ['control [1, 200]', [1, 200], 200], ['control [3, 7]', [3, 7], 9]], [['regression [231, 130]', [231, 130], 45], ['regression [28, 61]', [28, 61], 246], ['partial-repair [117, 177]', [117, 177], 104], ['partial-repair [197, 117]', [197, 117], 168], ['control [2, 96]', [2, 96], 192], ['control [0, 5]', [0, 5], 0], ['control [1, 200]', [1, 200], 200], ['control [3, 7]', [3, 7], 9]]]\nfor label, args, expected in fixtures[N - 1]:\n    check(label, solve(*args), expected)\nprint(json.dumps({\"observations\": observations, \"passed\": all(x[\"passed\"] for x in observations)}, ensure_ascii=False))\nraise SystemExit(0 if all(x[\"passed\"] for x in observations) else 1)\n"}},"limitations":"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.","method":"Deterministic executable model with adversarial boundary fixtures.","provenance":{"created_by":"Failure Map","dependencies":"Python standard library","family":"w2-error_correcting_codes-gf256-multiply-reduction-polynomial","generated_at":"2026-09-29T14:48:34.184520+00:00","license":"CC0-1.0","python":"3.12.14","seed":1,"split":"open-access"},"relevance":"Reed-Solomon codecs for storage and QR codes are built on GF(2^8) arithmetic with polynomial 0x11D.","root_cause":"The reduction XORs 0x1B (x^8+x^4+x^3+x+1) instead of 0x1D.","sha256":"fdc7503bf08bbd98129727c51fc389a0d42cf256e689b5774ab46bd7f640bfb7","title":"GF(256) multiply reduces with the AES polynomial · case 01","variant":1,"variant_policy":"Five numbered records share a model and may reuse boundary fixtures.","verified":true,"visibility":"public","verification":{"attempt":{"elapsed_ms":38.348,"exit_code":1,"observations":[{"actual":397,"check":"regression [158, 185]","expected":141,"passed":false},{"actual":467,"check":"regression [66, 7]","expected":211,"passed":false},{"actual":192,"check":"control [2, 96]","expected":192,"passed":true},{"actual":0,"check":"control [0, 5]","expected":0,"passed":true},{"actual":200,"check":"control [1, 200]","expected":200,"passed":true},{"actual":9,"check":"control [3, 7]","expected":9,"passed":true},{"actual":null,"check":"control [256, 1]","expected":null,"passed":true},{"actual":285,"check":"control [184, 92]","expected":29,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression [158, 185]\", \"actual\": 397, \"expected\": 141, \"passed\": false}, {\"check\": \"regression [66, 7]\", \"actual\": 467, \"expected\": 211, \"passed\": false}, {\"check\": \"control [2, 96]\", \"actual\": 192, \"expected\": 192, \"passed\": true}, {\"check\": \"control [0, 5]\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"control [1, 200]\", \"actual\": 200, \"expected\": 200, \"passed\": true}, {\"check\": \"control [3, 7]\", \"actual\": 9, \"expected\": 9, \"passed\": true}, {\"check\": \"control [256, 1]\", \"actual\": null, \"expected\": null, \"passed\": true}, {\"check\": \"control [184, 92]\", \"actual\": 285, \"expected\": 29, \"passed\": false}], \"passed\": false}\n"},"broken":{"elapsed_ms":39.593,"exit_code":1,"observations":[{"actual":100,"check":"regression [158, 185]","expected":141,"passed":false},{"actual":213,"check":"regression [66, 7]","expected":211,"passed":false},{"actual":192,"check":"control [2, 96]","expected":192,"passed":true},{"actual":0,"check":"control [0, 5]","expected":0,"passed":true},{"actual":200,"check":"control [1, 200]","expected":200,"passed":true},{"actual":9,"check":"control [3, 7]","expected":9,"passed":true},{"actual":null,"check":"control [256, 1]","expected":null,"passed":true},{"actual":219,"check":"control [184, 92]","expected":29,"passed":false}],"passed":false,"stderr":"","stdout":"{\"observations\": [{\"check\": \"regression [158, 185]\", \"actual\": 100, \"expected\": 141, \"passed\": false}, {\"check\": \"regression [66, 7]\", \"actual\": 213, \"expected\": 211, \"passed\": false}, {\"check\": \"control [2, 96]\", \"actual\": 192, \"expected\": 192, \"passed\": true}, {\"check\": \"control [0, 5]\", \"actual\": 0, \"expected\": 0, \"passed\": true}, {\"check\": \"control [1, 200]\", \"actual\": 200, \"expected\": 200, \"passed\": true}, {\"check\": \"control [3, 7]\", \"actual\": 9, \"expected\": 9, \"passed\": true}, {\"check\": \"control [256, 1]\", \"actual\": null, \"expected\": null, \"passed\": true}, {\"check\": \"control [184, 92]\", \"actual\": 219, \"expected\": 29, \"passed\": false}], \"passed\": false}\n"}},"member_only":{"stages":["fixed"],"fields":["implementations.fixed","verification.fixed","harness","repair"],"note":"The verified repair, its recorded checks, the repair description, and the scoring harness are available to members."}}