GCR Checksum Algorithm
The GCR checksum algorithm is applied to the 524 bytes of sector data on GCR-encoded floppy disks. While it calculates a three-byte value which is appended to the sector, it is more than simply a checksum as it causes the bytes of the sector to be mangled as it is calculated, a process which must be reversed to recover the original data.
It is a cursed algorithm which is unclearly or erroneously detailed in several documents. This document shall attempt to set the record straight.
Mangling
Mangling is the process by which the data bytes of the sector are encoded before being recorded onto a GCR disk.
Pseudocode
- Set checksum_A to 0.
- Set checksum_B to 0.
- Set checksum_C to 0.
- Set carry to 0.
- Loop:
- Set byte to the next byte in the input data or exit loop now if there are no more bytes.
- Set checksumA to the sum of itself, byte, and carry.
- If checksumA is greater than 255:
- Subtract 256 from checksumA.
- Set carry to 1.
- Else:
- Set carry to 0.
- Set byte to the bitwise exclusive-OR (XOR) of itself and checksumC.
- Write byte to output.
- Set byte to the next byte in the input data or exit loop now if there are no more bytes.
- Set checksumB to the sum of itself, byte, and carry.
- If checksumB is greater than 255:
- Subtract 256 from checksumB.
- Set carry to 1.
- Else:
- Set carry to 0.
- Set byte to the bitwise exclusive-OR (XOR) of itself and checksumA.
- Write byte to output.
- Set byte to the next byte in the input data or exit loop now if there are no more bytes.
- Set checksumC to the sum of itself, byte, and carry.
- If checksumC is greater than 255:
- Subtract 256 from checksumC.
- Set carry to 1.
- Else:
- Set carry to 0.
- Set byte to the bitwise exclusive-OR (XOR) of itself and checksumB.
- Write byte to output.
- If checksumC is greater than 127:
- Subtract 128 from checksumC.
- Set carry to 1.
- Multiply checksumC
- Add 1 to checksumC.
- Else:
- Set carry to 0.
- Multiply checksum_C by 2.
Python 3.x Code
def mangle(data, checksum_a=0, checksum_b=0, checksum_c=0):
def _mangle_gen(data):
nonlocal checksum_a, checksum_b, checksum_c
data = iter(data)
carry = 0
while True:
try:
byte = next(data)
except StopIteration:
break
carry, checksum_a = divmod(checksum_a + byte + carry, 256)
byte ^= checksum_c
yield byte
try:
byte = next(data)
except StopIteration:
break
carry, checksum_b = divmod(checksum_b + byte + carry, 256)
byte ^= checksum_a
yield byte
try:
byte = next(data)
except StopIteration:
break
carry, checksum_c = divmod(checksum_c + byte + carry, 256)
byte ^= checksum_b
yield byte
carry = 1 if (checksum_c & 0x80) else 0
checksum_c = ((checksum_c << 1) & 0xFF) | carry
return bytes(_mangle_gen(data)), bytes((checksum_a, checksum_b, checksum_c))
Unmangling
Unmangling is the process by which the data bytes of the sector are decoded after being read from a GCR disk.
Pseudocode
- Set checksum_A to 0.
- Set checksum_B to 0.
- Set checksum_C to 0.
- Loop:
- If checksumC is greater than 127:
- Subtract 128 from checksumC.
- Set carry to 1.
- Multiply checksumC
- Add 1 to checksumC.
- If checksumC is greater than 127:
- Else:
- Set carry to 0.
- Multiply checksumC by 2.
- Set byte to the next byte in the input data or exit loop now if there are no more bytes.
- Set byte to the bitwise exclusive-OR (XOR) of itself and checksumC.
- Write byte to output.
- Set checksumA to the sum of itself, byte, and carry.
- If checksumA is greater than 255:
- Subtract 256 from checksumA.
- Set carry to 1.
- Else:
- Set carry to 0.
- Set byte to the next byte in the input data or exit loop now if there are no more bytes.
- Set byte to the bitwise exclusive-OR (XOR) of itself and checksumA.
- Write byte to output.
- Set checksumB to the sum of itself, byte, and carry.
- If checksumB is greater than 255:
- Subtract 256 from checksumB.
- Set carry to 1.
- Else:
- Set carry to 0.
- Set byte to the next byte in the input data or exit loop now if there are no more bytes.
- Set byte to the bitwise exclusive-OR (XOR) of itself and checksumB.
- Write byte to output.
- Set checksumC to the sum of itself, byte, and carry.
- If checksumC is greater than 255:
- Subtract 256 from checksum_C.
- Set carry to 1.
- Else:
- Set carry to 0.
Python 3.x Code
def unmangle(data):
checksum_a = checksum_b = checksum_c = 0
def _unmangle_gen(data):
nonlocal checksum_a, checksum_b, checksum_c
data = iter(data)
while True:
carry = 1 if (checksum_c & 0x80) else 0
checksum_c = ((checksum_c << 1) & 0xFF) | carry
try:
byte = next(data)
except StopIteration:
break
byte ^= checksum_c
yield byte
carry, checksum_a = divmod(checksum_a + byte + carry, 256)
try:
byte = next(data)
except StopIteration:
break
byte ^= checksum_a
yield byte
carry, checksum_b = divmod(checksum_b + byte + carry, 256)
try:
byte = next(data)
except StopIteration:
break
byte ^= checksum_b
yield byte
carry, checksum_c = divmod(checksum_c + byte + carry, 256)
return bytes(_demangle_gen(data)), bytes((checksum_a, checksum_b, checksum_c))
Example
Unmangled Data
000 54 41 47 53 54 41 47 53 54 41 47 53 44 41 54 41 TAGSTAGSTAGSDATA
010 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
020 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
030 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
040 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
050 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
060 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
070 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
080 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
090 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
0A0 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
0B0 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
0C0 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
0D0 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
0E0 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
0F0 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
100 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
110 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
120 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
130 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
140 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
150 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
160 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
170 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
180 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
190 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
1A0 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
1B0 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
1C0 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
1D0 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
1E0 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
1F0 44 41 54 41 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATADATA
200 44 41 54 41 44 41 54 41 44 41 54 41 DATADATADATA
Mangled Data
000 54 15 06 DD F3 D4 D8 BC BC A6 76 63 32 34 25 D4 T.........vc24%.
010 F3 F4 F9 4D B3 A2 1A 0A 0E D3 D8 7C 90 91 A8 69 ...M.......|...i
020 56 C3 3D 27 C3 EF F3 F6 B4 AA A5 04 69 2D D2 C0 V.='........i-..
030 1F 8A 96 1E 49 46 17 23 0C 4E F0 DD E4 A9 89 A8 ....IF.#.N......
040 6E 63 02 3F 27 74 94 F6 A8 45 AD C1 01 7C D1 DB nc.?'t...E...|..
050 3A D2 88 93 ED 60 40 BC 27 19 28 F6 DD C0 AD AC :....`@.'.(.....
060 C1 7F 7B F4 C4 32 B9 94 F0 23 42 48 0D 1A 1E 7A ..{..2...#BH...z
070 D9 CF AC B0 8B 39 66 65 31 36 31 D2 FD E8 FD 4F .....9fe161....O
080 AF 9A 04 7E 7E D5 D4 5C 92 85 E8 6B 42 43 3F 1B ...~~..\...kBC?.
090 C2 EE CF F4 B5 9E B9 07 65 25 D3 34 0F 8D E2 7E ........e%.4...~
0A0 4A BA 55 22 78 CD F0 D1 DF A8 85 9E 6F 57 16 3E J.U"x.......oW.>
0B0 13 4C 95 EA D8 44 B9 A0 03 70 12 DA 36 54 89 87 .L...D...p..6T..
0C0 E8 60 5C 43 26 0D D2 F7 C9 D4 AC A0 E9 7C 77 45 .`\C&........|wE
0D0 3A 26 DE 93 EC DC 40 BC F9 18 6A 65 DE C3 9E AF :&....@...je....
0E0 87 5D 7B 59 F8 32 0D 61 F0 DC 96 49 9B 4D 1D 72 .]{Y.2.a...I.M.r
0F0 D8 CC 20 A0 8B F9 10 62 BE 50 31 0F E2 E8 DB B4 .. ....b.P1.....
100 AF 92 38 7E 51 22 D5 28 0D 87 FE 7A 4C AE AC 1D ..8~Q".(...zL...
110 74 3B C9 C5 33 A3 91 D6 60 4B E7 38 0F AE EF DE t;..3...`K.8....
120 24 BE B4 28 05 64 03 D7 22 76 8C FB B4 5D A8 C9 $..(.d.."v...]..
130 19 01 C1 F3 C5 F2 A0 94 AD 78 43 3D 3E 1A 2F 8E .........xC=>./.
140 D8 3E 45 B0 C5 17 66 EC DC 37 89 AC F3 03 7A 4D .>E...f..7....zM
150 4D 32 19 FB F0 D0 A3 48 97 37 1E 66 35 CE 3C CA M2.....H.7.f5.<.
160 95 ED CD 67 AA FA 3C 03 BF EC D7 DF AA 86 EB 02 ...g..<.........
170 4D 49 D0 1C D6 89 CA 8C 4E A2 48 1F 60 83 CB 39 MI......N.H.`..9
180 42 A2 ED F5 60 BF BF 38 7B 21 EE D2 07 BE 80 6E B...`..8{!.....n
190 05 58 B6 D4 1E 0D 82 EF 5F 5A A4 9F 18 75 14 F1 .X......_Z...u..
1A0 31 48 A6 88 D0 77 5F 92 3C 0E 7C 8D D4 99 59 A4 1H...w_.<.|...Y.
1B0 76 10 72 4E D7 2B F4 A6 EF 88 7D 41 52 2C 15 95 v.rN.+....}AR,..
1C0 EA C4 7E 43 83 9D 17 5A 78 C6 08 60 8D E1 90 5C ..~C...Zx..`...\
1D0 A6 51 0A 77 E0 E3 23 B0 A0 FA 30 79 B9 10 2E 10 .Q.w..#...0y....
1E0 63 FE C6 B7 44 96 3A 15 5C 26 C2 2D 15 9A F9 0A c...D.:.\&.-....
1F0 59 B3 4D 33 77 FD E7 C6 BF B9 9C DF 7D 4C 95 CF Y.M3w.......}L..
200 0A 4B 9B E3 D2 52 B0 94 11 69 7A E8 .K...R...iz.
| Checksum Byte | Value in Hexadecimal | | ------------- | -------------------- | | checksum_A | A9 | | checksum_B | 69 | | checksum_C | 2E |
References
- [[http://bitsavers.org/pdf/apple/disk/sony/uPD72070Specification1991.pdf|uPD72070 Floppy Disk Controller Specification]], Page 6
- [[http://bitsavers.org/pdf/apple/disk/sony/SWIMChipUsersRef198801.pdf|SWIM Chip User's Reference, Revision 1.5 (11-Jan-1988)]], Page 6