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Need help with reversing an old auth algo

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ozcar:
There being more than 0xffff seconds in a day, if your MCU could pull the PIC /MCLR down, or perhaps reset it by cutting power to it off, then you might be able to spin through all possibilities for a 16-bit challenge word in a day or so.

However has it been established how big lookup table results can be combined? Say for example, how do you go from knowing the first two of these to the third?


--- Code: ---ffff 0000 0000 0000  310f 3110 1315 1315
0000 0000 ffff 0000  1255 1255 495f 4960
ffff 0000 ffff 0000  f9d2 f9d3 9e0c 9e0d

--- End code ---

Nominal Animal:

--- Quote from: ozcar on November 05, 2023, 10:22:57 pm ---However has it been established how big lookup table results can be combined?
--- End quote ---
No.

I also discovered by examining challenges 'xx yy 00 00 00 00 00 00', '00 00 00 00 xx yy 00 00', and 'xx yy 00 00 xx yy 00 00' for yyxx in 8000 C000 E000 F000 F800 FC00 FE00 FF00 ... FFF0 FFF8 FFFC FFFE, that the hash function cannot described as an addition between the two functions; the mixing step is more complicated.

It is possible that a mask is constructed from each challenge word, using
    Mw = Cw1 | Cw2 | Cw3 | Cw4
    Mw = Mw | (Mw << 1) | (Mw << 2) | ... | (Mw << 14) | (Mw << 15)
so that all bits above the least significant set bits are always set in the mask.  The result words would be calculated using three cryptographic hash functions S(A,B), H(A,B), and G(A, B) via
    Rw1 = 0xC6EA + Cw1 + (Mw & S(H(Cw1, Cw2), G(Cw3, Cw4)))
    Rw2 = 0xC6EA + Cw2 + (Mw & S(H(Cw1, Cw2), G(Cw3, Cw4))) = Rw1 - Cw1 + Cw2
    Rw3 = 0xE64C + Cw3 + (Mw & S(H(Cw3, Cw4), G(Cw1, Cw2)))
    Rw3 = 0xE64C + Cw4 + (Mw & S(H(Cw3, Cw4), G(Cw1, Cw2))) = Rw3 - Cw3 + Cw4
The use of such a mask would be extremely odd, because it only weakens the security of the system.  It really is a hash function that takes 64 bit input and yields a 32-bit output.

We do know that H() is associative: H(A,B) = H(B,A).  We also know that H() is not a simple addition, subtraction, or exclusive-OR of its arguments.  We do not have enough challenge-response pairs to tell if H(A,B) is actually H(A+B) or H(A^B), i.e. really a 16-bit to 16-bit function.

Nominal Animal:
The information in each challenge-response pairs (as bytes)
    C1 C2 C3 C4 C5 C6 C7 → R1 R2 R3 R4 R5 R6 R7 R8
can be described as four 16-bit challenge and response words,
    C2C1 C4C3 C6C5 C8C7 → R2R1 R4R3 R6R5 R8R7
but the response itself only contains 32 bits of meaningful information,
    Rw12 = R2R1 - C2C1 - 0xC6EA = R4R3 - C4C3 - 0xC6EA
    Rw34 = R6R5 - C6C5 - 0xE64C = R8R7 - C8C7 - 0xE64C
and the other 32 bits (interleaved) is just superfluous information.

This is why I call this a 64-bit hash function with a 32-bit result.

Converting to this form, the current dataset we have is attached as dataset-12.txt (in hexadecimal and in binary).

We can derive key observations from the following patterns:

--- Code: ---C2C1 C4C3 C6C5 C8C7 │ Rw12 Rw34 ║    C2        C1          C4        C3          C6        C5          C8        C7     │        Rw12                  Rw34
════════════════════╪═══════════╬═══════════════════════════════════════════════════════════════════════════════════════╪═══════════════════════════════════════════
0000 0000 0000 0000 │ 0000 0000 ║ 0000'0000 0000'0000   0000'0000 0000'0000   0000'0000 0000'0000   0000'0000 0000'0000 │ 0000'0000 0000'0000   0000'0000 0000'0000
────────────────────┼───────────╫───────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────
FFFF 0000 FFFF 0000 │ 32E9 B7C1 ║ 1111'1111 1111'1111   0000'0000 0000'0000   1111'1111 1111'1111   0000'0000 0000'0000 │ 0011'0010 1110'1001   1011'0111 1100'0001
0000 FFFF 0000 FFFF │ 32E9 B7C1 ║ 0000'0000 0000'0000   1111'1111 1111'1111   0000'0000 0000'0000   1111'1111 1111'1111 │ 0011'0010 1110'1001   1011'0111 1100'0001
FFFF 0000 0000 FFFF │ 32E9 B7C1 ║ 1111'1111 1111'1111   0000'0000 0000'0000   0000'0000 0000'0000   1111'1111 1111'1111 │ 0011'0010 1110'1001   1011'0111 1100'0001
0000 FFFF FFFF 0000 │ 32E9 B7C1 ║ 0000'0000 0000'0000   1111'1111 1111'1111   1111'1111 1111'1111   0000'0000 0000'0000 │ 0011'0010 1110'1001   1011'0111 1100'0001
────────────────────┼───────────╫───────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────
8000 0000 8000 0000 │ 8000 8000 ║ 1000'0000 0000'0000   0000'0000 0000'0000   1000'0000 0000'0000   0000'0000 0000'0000 │ 1000'0000 0000'0000   1000'0000 0000'0000
0000 8000 0000 8000 │ 8000 8000 ║ 0000'0000 0000'0000   1000'0000 0000'0000   0000'0000 0000'0000   1000'0000 0000'0000 │ 1000'0000 0000'0000   1000'0000 0000'0000
8000 0000 0000 8000 │ 8000 8000 ║ 1000'0000 0000'0000   0000'0000 0000'0000   0000'0000 0000'0000   1000'0000 0000'0000 │ 1000'0000 0000'0000   1000'0000 0000'0000
0000 8000 8000 0000 │ 8000 8000 ║ 0000'0000 0000'0000   1000'0000 0000'0000   1000'0000 0000'0000   0000'0000 0000'0000 │ 1000'0000 0000'0000   1000'0000 0000'0000
────────────────────┼───────────╫───────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────
C000 0000 C000 0000 │ 4000 C000 ║ 1100'0000 0000'0000   0000'0000 0000'0000   1100'0000 0000'0000   0000'0000 0000'0000 │ 0100'0000 0000'0000   1100'0000 0000'0000
0000 C000 0000 C000 │ 4000 C000 ║ 0000'0000 0000'0000   1100'0000 0000'0000   0000'0000 0000'0000   1100'0000 0000'0000 │ 0100'0000 0000'0000   1100'0000 0000'0000
C000 0000 0000 C000 │ 4000 C000 ║ 1100'0000 0000'0000   0000'0000 0000'0000   0000'0000 0000'0000   1100'0000 0000'0000 │ 0100'0000 0000'0000   1100'0000 0000'0000
0000 C000 C000 0000 │ 4000 C000 ║ 0000'0000 0000'0000   1100'0000 0000'0000   1100'0000 0000'0000   0000'0000 0000'0000 │ 0100'0000 0000'0000   1100'0000 0000'0000
────────────────────┼───────────╫───────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────
E000 0000 E000 0000 │ 2000 E000 ║ 1110'0000 0000'0000   0000'0000 0000'0000   1110'0000 0000'0000   0000'0000 0000'0000 │ 0010'0000 0000'0000   1110'0000 0000'0000
0000 E000 0000 E000 │ 2000 E000 ║ 0000'0000 0000'0000   1110'0000 0000'0000   0000'0000 0000'0000   1110'0000 0000'0000 │ 0010'0000 0000'0000   1110'0000 0000'0000
E000 0000 0000 E000 │ 2000 E000 ║ 1110'0000 0000'0000   0000'0000 0000'0000   0000'0000 0000'0000   1110'0000 0000'0000 │ 0010'0000 0000'0000   1110'0000 0000'0000
0000 E000 E000 0000 │ 2000 E000 ║ 0000'0000 0000'0000   1110'0000 0000'0000   1110'0000 0000'0000   0000'0000 0000'0000 │ 0010'0000 0000'0000   1110'0000 0000'0000
────────────────────┼───────────╫───────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────
F000 0000 F000 0000 │ 5000 9000 ║ 1111'0000 0000'0000   0000'0000 0000'0000   1111'0000 0000'0000   0000'0000 0000'0000 │ 0101'0000 0000'0000   1001'0000 0000'0000
0000 F000 0000 F000 │ 5000 9000 ║ 0000'0000 0000'0000   1111'0000 0000'0000   0000'0000 0000'0000   1111'0000 0000'0000 │ 0101'0000 0000'0000   1001'0000 0000'0000
F000 0000 0000 F000 │ 5000 9000 ║ 1111'0000 0000'0000   0000'0000 0000'0000   0000'0000 0000'0000   1111'0000 0000'0000 │ 0101'0000 0000'0000   1001'0000 0000'0000
0000 F000 F000 0000 │ 5000 9000 ║ 0000'0000 0000'0000   1111'0000 0000'0000   1111'0000 0000'0000   0000'0000 0000'0000 │ 0101'0000 0000'0000   1001'0000 0000'0000
────────────────────┼───────────╫───────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────
F800 0000 F800 0000 │ B800 A800 ║ 1111'1000 0000'0000   0000'0000 0000'0000   1111'1000 0000'0000   0000'0000 0000'0000 │ 1011'1000 0000'0000   1010'1000 0000'0000
0000 F800 0000 F800 │ B800 A800 ║ 0000'0000 0000'0000   1111'1000 0000'0000   0000'0000 0000'0000   1111'1000 0000'0000 │ 1011'1000 0000'0000   1010'1000 0000'0000
F800 0000 0000 F800 │ B800 A800 ║ 1111'1000 0000'0000   0000'0000 0000'0000   0000'0000 0000'0000   1111'1000 0000'0000 │ 1011'1000 0000'0000   1010'1000 0000'0000
0000 F800 F800 0000 │ B800 A800 ║ 0000'0000 0000'0000   1111'1000 0000'0000   1111'1000 0000'0000   0000'0000 0000'0000 │ 1011'1000 0000'0000   1010'1000 0000'0000
────────────────────┼───────────╫───────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────
FC00 0000 FC00 0000 │ B400 4C00 ║ 1111'1100 0000'0000   0000'0000 0000'0000   1111'1100 0000'0000   0000'0000 0000'0000 │ 1011'0100 0000'0000   0100'1100 0000'0000
0000 FC00 0000 FC00 │ B400 4C00 ║ 0000'0000 0000'0000   1111'1100 0000'0000   0000'0000 0000'0000   1111'1100 0000'0000 │ 1011'0100 0000'0000   0100'1100 0000'0000
FC00 0000 0000 FC00 │ B400 4C00 ║ 1111'1100 0000'0000   0000'0000 0000'0000   0000'0000 0000'0000   1111'1100 0000'0000 │ 1011'0100 0000'0000   0100'1100 0000'0000
0000 FC00 FC00 0000 │ B400 4C00 ║ 0000'0000 0000'0000   1111'1100 0000'0000   1111'1100 0000'0000   0000'0000 0000'0000 │ 1011'0100 0000'0000   0100'1100 0000'0000
────────────────────┼───────────╫───────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────
FE00 0000 FE00 0000 │ BE00 DE00 ║ 1111'1110 0000'0000   0000'0000 0000'0000   1111'1110 0000'0000   0000'0000 0000'0000 │ 1011'1110 0000'0000   1101'1110 0000'0000
0000 FE00 0000 FE00 │ BE00 DE00 ║ 0000'0000 0000'0000   1111'1110 0000'0000   0000'0000 0000'0000   1111'1110 0000'0000 │ 1011'1110 0000'0000   1101'1110 0000'0000
FE00 0000 0000 FE00 │ BE00 DE00 ║ 1111'1110 0000'0000   0000'0000 0000'0000   0000'0000 0000'0000   1111'1110 0000'0000 │ 1011'1110 0000'0000   1101'1110 0000'0000
0000 FE00 FE00 0000 │ BE00 DE00 ║ 0000'0000 0000'0000   1111'1110 0000'0000   1111'1110 0000'0000   0000'0000 0000'0000 │ 1011'1110 0000'0000   1101'1110 0000'0000
────────────────────┼───────────╫───────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────
FF00 0000 FF00 0000 │ 0D00 6D00 ║ 1111'1111 0000'0000   0000'0000 0000'0000   1111'1111 0000'0000   0000'0000 0000'0000 │ 0000'1101 0000'0000   0110'1101 0000'0000
0000 FF00 0000 FF00 │ 0D00 6D00 ║ 0000'0000 0000'0000   1111'1111 0000'0000   0000'0000 0000'0000   1111'1111 0000'0000 │ 0000'1101 0000'0000   0110'1101 0000'0000
FF00 0000 0000 FF00 │ 0D00 6D00 ║ 1111'1111 0000'0000   0000'0000 0000'0000   0000'0000 0000'0000   1111'1111 0000'0000 │ 0000'1101 0000'0000   0110'1101 0000'0000
0000 FF00 FF00 0000 │ 0D00 6D00 ║ 0000'0000 0000'0000   1111'1111 0000'0000   1111'1111 0000'0000   0000'0000 0000'0000 │ 0000'1101 0000'0000   0110'1101 0000'0000
────────────────────┼───────────╫───────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────
FF80 0000 FF80 0000 │ 0480 F580 ║ 1111'1111 1000'0000   0000'0000 0000'0000   1111'1111 1000'0000   0000'0000 0000'0000 │ 0000'0100 1000'0000   1111'0101 1000'0000
0000 FF80 0000 FF80 │ 0480 F580 ║ 0000'0000 0000'0000   1111'1111 1000'0000   0000'0000 0000'0000   1111'1111 1000'0000 │ 0000'0100 1000'0000   1111'0101 1000'0000
FF80 0000 0000 FF80 │ 0480 F580 ║ 1111'1111 1000'0000   0000'0000 0000'0000   0000'0000 0000'0000   1111'1111 1000'0000 │ 0000'0100 1000'0000   1111'0101 1000'0000
0000 FF80 FF80 0000 │ 0480 F580 ║ 0000'0000 0000'0000   1111'1111 1000'0000   1111'1111 1000'0000   0000'0000 0000'0000 │ 0000'0100 1000'0000   1111'0101 1000'0000
────────────────────┼───────────╫───────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────
FFC0 0000 FFC0 0000 │ F8C0 6640 ║ 1111'1111 1100'0000   0000'0000 0000'0000   1111'1111 1100'0000   0000'0000 0000'0000 │ 1111'1000 1100'0000   0110'0110 0100'0000
0000 FFC0 0000 FFC0 │ F8C0 6640 ║ 0000'0000 0000'0000   1111'1111 1100'0000   0000'0000 0000'0000   1111'1111 1100'0000 │ 1111'1000 1100'0000   0110'0110 0100'0000
FFC0 0000 0000 FFC0 │ F8C0 6640 ║ 1111'1111 1100'0000   0000'0000 0000'0000   0000'0000 0000'0000   1111'1111 1100'0000 │ 1111'1000 1100'0000   0110'0110 0100'0000
0000 FFC0 FFC0 0000 │ F8C0 6640 ║ 0000'0000 0000'0000   1111'1111 1100'0000   1111'1111 1100'0000   0000'0000 0000'0000 │ 1111'1000 1100'0000   0110'0110 0100'0000
────────────────────┼───────────╫───────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────
FFE0 0000 FFE0 0000 │ D820 1620 ║ 1111'1111 1110'0000   0000'0000 0000'0000   1111'1111 1110'0000   0000'0000 0000'0000 │ 1101'1000 0010'0000   0001'0110 0010'0000
0000 FFE0 0000 FFE0 │ D820 1620 ║ 0000'0000 0000'0000   1111'1111 1110'0000   0000'0000 0000'0000   1111'1111 1110'0000 │ 1101'1000 0010'0000   0001'0110 0010'0000
FFE0 0000 0000 FFE0 │ D820 1620 ║ 1111'1111 1110'0000   0000'0000 0000'0000   0000'0000 0000'0000   1111'1111 1110'0000 │ 1101'1000 0010'0000   0001'0110 0010'0000
0000 FFE0 FFE0 0000 │ D820 1620 ║ 0000'0000 0000'0000   1111'1111 1110'0000   1111'1111 1110'0000   0000'0000 0000'0000 │ 1101'1000 0010'0000   0001'0110 0010'0000
────────────────────┼───────────╫───────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────
FFF0 0000 FFF0 0000 │ 7F70 9330 ║ 1111'1111 1111'0000   0000'0000 0000'0000   1111'1111 1111'0000   0000'0000 0000'0000 │ 0111'1111 0111'0000   1001'0011 0011'0000
0000 FFF0 0000 FFF0 │ 7F70 9330 ║ 0000'0000 0000'0000   1111'1111 1111'0000   0000'0000 0000'0000   1111'1111 1111'0000 │ 0111'1111 0111'0000   1001'0011 0011'0000
FFF0 0000 0000 FFF0 │ 7F70 9330 ║ 1111'1111 1111'0000   0000'0000 0000'0000   0000'0000 0000'0000   1111'1111 1111'0000 │ 0111'1111 0111'0000   1001'0011 0011'0000
0000 FFF0 FFF0 0000 │ 7F70 9330 ║ 0000'0000 0000'0000   1111'1111 1111'0000   1111'1111 1111'0000   0000'0000 0000'0000 │ 0111'1111 0111'0000   1001'0011 0011'0000
────────────────────┼───────────╫───────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────
FFF8 0000 FFF8 0000 │ 0BE8 E548 ║ 1111'1111 1111'1000   0000'0000 0000'0000   1111'1111 1111'1000   0000'0000 0000'0000 │ 0000'1011 1110'1000   1110'0101 0100'1000
0000 FFF8 0000 FFF8 │ 0BE8 E548 ║ 0000'0000 0000'0000   1111'1111 1111'1000   0000'0000 0000'0000   1111'1111 1111'1000 │ 0000'1011 1110'1000   1110'0101 0100'1000
FFF8 0000 0000 FFF8 │ 0BE8 E548 ║ 1111'1111 1111'1000   0000'0000 0000'0000   0000'0000 0000'0000   1111'1111 1111'1000 │ 0000'1011 1110'1000   1110'0101 0100'1000
0000 FFF8 FFF8 0000 │ 0BE8 E548 ║ 0000'0000 0000'0000   1111'1111 1111'1000   1111'1111 1111'1000   0000'0000 0000'0000 │ 0000'1011 1110'1000   1110'0101 0100'1000
────────────────────┼───────────╫───────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────
FFFC 0000 FFFC 0000 │ 756C 174C ║ 1111'1111 1111'1100   0000'0000 0000'0000   1111'1111 1111'1100   0000'0000 0000'0000 │ 0111'0101 0110'1100   0001'0111 0100'1100
0000 FFFC 0000 FFFC │ 756C 174C ║ 0000'0000 0000'0000   1111'1111 1111'1100   0000'0000 0000'0000   1111'1111 1111'1100 │ 0111'0101 0110'1100   0001'0111 0100'1100
FFFC 0000 0000 FFFC │ 756C 174C ║ 1111'1111 1111'1100   0000'0000 0000'0000   0000'0000 0000'0000   1111'1111 1111'1100 │ 0111'0101 0110'1100   0001'0111 0100'1100
0000 FFFC FFFC 0000 │ 756C 174C ║ 0000'0000 0000'0000   1111'1111 1111'1100   1111'1111 1111'1100   0000'0000 0000'0000 │ 0111'0101 0110'1100   0001'0111 0100'1100
────────────────────┼───────────╫───────────────────────────────────────────────────────────────────────────────────────┼───────────────────────────────────────────
FFFE 0000 FFFE 0000 │ 3746 EA42 ║ 1111'1111 1111'1110   0000'0000 0000'0000   1111'1111 1111'1110   0000'0000 0000'0000 │ 0011'0111 0100'0110   1110'1010 0100'0010
0000 FFFE 0000 FFFE │ 3746 EA42 ║ 0000'0000 0000'0000   1111'1111 1111'1110   0000'0000 0000'0000   1111'1111 1111'1110 │ 0011'0111 0100'0110   1110'1010 0100'0010
FFFE 0000 0000 FFFE │ 3746 EA42 ║ 1111'1111 1111'1110   0000'0000 0000'0000   0000'0000 0000'0000   1111'1111 1111'1110 │ 0011'0111 0100'0110   1110'1010 0100'0010
0000 FFFE FFFE 0000 │ 3746 EA42 ║ 0000'0000 0000'0000   1111'1111 1111'1110   1111'1111 1111'1110   0000'0000 0000'0000 │ 0011'0111 0100'0110   1110'1010 0100'0010
--- End code ---

Because Rw12 and Rw34 are the same regardless of swapping C2C1↔C4C3 and C6C5↔C8C7, we can define the hash algorithm as
    Rw12 = Mw & S1(H1(C2C1, C4C3), G1(C6C5, C8C7))
    Rw34 = Mw & S2(H2(C6C5, C8C7), G2(C2C1, C4C3))
where H1(), H2(), S1(), S2(), G1(), and G2() are functions that take two 16-bit arguments, and return one 16-bit result, and Mw is a mask where all bits above the least significant bit set in any challenge word are set, i.e.
    M' = C2C1 | C4C3 | C6C5 | C8C7
    Mw = M' | (M'<<1) | (M'<<2) | ... | (M'<<14) | (M'<<15)
which is easy to generate even on a small PIC.  The actual responses are then constructed from Rw12 and Rw34,
    R2R1 = Rw12 + C2C1 + 0xC6EA
    R4R3 = Rw12 + C4C3 + 0xC6EA
    R6R5 = Rw34 + C6C5 + 0xE64C
    R8R7 = Rw34 + C8C7 + 0xE64C
which fulfills the symmetries and patterns we have thus far observed.  I have also verified that addition (+) and not exclusive-or matches the current dataset.  (Adding 0xC6EA is equivalent to substracting 0x3916, and adding 0xE64C is equivalent to substracting 0x19B4, using 16-bit modular (wraparound) arithmetic.)

Similarities indicate that the functions are very likely similar, for example only differing by some 16-bit seed value; i.e. that they all are actually just F(A, B, seed).

We do not yet know enough challenge-response pairs to determine if some/any of the functions are actually just 16-bit in, 16-bit out: Fn(A, B) = Fn(A ⊕ B) for some arithmetic operator ⊕ like addition, subtraction, or exclusive-OR.  If it is, then patterns
    C1 C2 C3 C4 00 00 00 00
    C5 C6 00 00 00 00 00 00
will have the exact same response result Rw12 when
    C6C5 = C2C1 ⊕ C4C3
and similarly for Rw34 when the leading four bytes of each challenge are zeroes.
If we find any pair where the responses differ, that immediately excludes that particular arithmetic operator ⊕, so C2C1 and C4C3 should have relatively random bit patterns, so we don't accidentally assume false proof based on other quirks of the functions.

Here are a six such challenge sets for addition, subtraction, xor, and or each, in case twizzter could work these through:

--- Code: ---48 1D 09 B8 00 00 00 00 # 0x1D48 + 0xB809 = 0xD551
51 D5 00 00 00 00 00 00
00 00 00 00 48 1D 09 B8
00 00 00 00 51 D5 00 00
BB 24 B1 40 00 00 00 00 # 0x24BB + 0x40B1 = 0x656C
6C 65 00 00 00 00 00 00
00 00 00 00 BB 24 B1 40
00 00 00 00 6C 65 00 00
2D 91 3E DD 00 00 00 00 # 0x912D + 0xDD3E = 0x6E6B
6B 6E 00 00 00 00 00 00
00 00 00 00 2D 91 3E DD
00 00 00 00 6B 6E 00 00
D6 90 DE 46 00 00 00 00 # 0x90D6 + 0x46DE = 0xD7B4
B4 D7 00 00 00 00 00 00
00 00 00 00 D6 90 DE 46
00 00 00 00 B4 D7 00 00
38 85 74 D4 00 00 00 00 # 0x8538 + 0xD474 = 0x59AC
AC 59 00 00 00 00 00 00
00 00 00 00 38 85 74 D4
00 00 00 00 AC 59 00 00
8A BA 64 FD 00 00 00 00 # 0xBA8A + 0xFD64 = 0xB7EE
EE B7 00 00 00 00 00 00
00 00 00 00 8A BA 64 FD
00 00 00 00 EE B7 00 00
14 91 B3 D2 00 00 00 00 # 0x9114 - 0xD2B3 = 0xBE61
61 BE 00 00 00 00 00 00
00 00 00 00 14 91 B3 D2
00 00 00 00 61 BE 00 00
35 97 20 9C 00 00 00 00 # 0x9735 - 0x9C20 = 0xFB15
15 FB 00 00 00 00 00 00
00 00 00 00 35 97 20 9C
00 00 00 00 15 FB 00 00
AA 48 3C DB 00 00 00 00 # 0x48AA - 0xDB3C = 0x6D6E
6E 6D 00 00 00 00 00 00
00 00 00 00 AA 48 3C DB
00 00 00 00 6E 6D 00 00
75 5C AE 4B 00 00 00 00 # 0x5C75 - 0x4BAE = 0x10C7
C7 10 00 00 00 00 00 00
00 00 00 00 75 5C AE 4B
00 00 00 00 C7 10 00 00
25 29 85 CE 00 00 00 00 # 0x2925 - 0xCE85 = 0x5AA0
A0 5A 00 00 00 00 00 00
00 00 00 00 25 29 85 CE
00 00 00 00 A0 5A 00 00
DB 04 2F A8 00 00 00 00 # 0x04DB - 0xA82F = 0x5CAC
AC 5C 00 00 00 00 00 00
00 00 00 00 DB 04 2F A8
00 00 00 00 AC 5C 00 00
75 5F 54 E7 00 00 00 00 # 0x5F75 ^ 0xE754 = 0xB821
21 B8 00 00 00 00 00 00
00 00 00 00 75 5F 54 E7
00 00 00 00 21 B8 00 00
13 22 85 62 00 00 00 00 # 0x2213 ^ 0x6285 = 0x4096
96 40 00 00 00 00 00 00
00 00 00 00 13 22 85 62
00 00 00 00 96 40 00 00
9D 16 18 5F 00 00 00 00 # 0x169D ^ 0x5F18 = 0x4985
85 49 00 00 00 00 00 00
00 00 00 00 9D 16 18 5F
00 00 00 00 85 49 00 00
1A EC 65 91 00 00 00 00 # 0xEC1A ^ 0x9165 = 0x7D7F
7F 7D 00 00 00 00 00 00
00 00 00 00 1A EC 65 91
00 00 00 00 7F 7D 00 00
6D A2 CF 50 00 00 00 00 # 0xA26D ^ 0x50CF = 0xF2A2
A2 F2 00 00 00 00 00 00
00 00 00 00 6D A2 CF 50
00 00 00 00 A2 F2 00 00
F9 A2 45 FC 00 00 00 00 # 0xA2F9 ^ 0xFC45 = 0x5EBC
BC 5E 00 00 00 00 00 00
00 00 00 00 F9 A2 45 FC
00 00 00 00 BC 5E 00 00
AA 1F 52 23 00 00 00 00 # 0x1FAA | 0x2352 = 0x3FFA
FA 3F 00 00 00 00 00 00
00 00 00 00 AA 1F 52 23
00 00 00 00 FA 3F 00 00
B9 5F A1 21 00 00 00 00 # 0x5FB9 | 0x21A1 = 0x7FB9
B9 7F 00 00 00 00 00 00
00 00 00 00 B9 5F A1 21
00 00 00 00 B9 7F 00 00
E6 80 A5 66 00 00 00 00 # 0x80E6 | 0x66A5 = 0xE6E7
E7 E6 00 00 00 00 00 00
00 00 00 00 E6 80 A5 66
00 00 00 00 E7 E6 00 00
70 EA 2B B2 00 00 00 00 # 0xEA70 | 0xB22B = 0xFA7B
7B FA 00 00 00 00 00 00
00 00 00 00 70 EA 2B B2
00 00 00 00 7B FA 00 00
30 F0 3D A3 00 00 00 00 # 0xF030 | 0xA33D = 0xF33D
3D F3 00 00 00 00 00 00
00 00 00 00 30 F0 3D A3
00 00 00 00 3D F3 00 00
E5 82 50 7B 00 00 00 00 # 0x82E5 | 0x7B50 = 0xFBF5
F5 FB 00 00 00 00 00 00
00 00 00 00 E5 82 50 7B
00 00 00 00 F5 FB 00 00

--- End code ---

Basically, this set is a fishing expedition: if it succeeds, we've discovered something that hugely simplifies the algorithm – but it might say no, too.

twizzter:

--- Quote from: Nominal Animal on November 06, 2023, 08:27:59 am ---Here are a six such challenge sets for addition, subtraction, xor, and or each, in case twizzter could work these through:
[...]
Basically, this set is a fishing expedition: if it succeeds, we've discovered something that hugely simplifies the algorithm – but it might say no, too.

--- End quote ---
Little fishes (hopefully) below:

--- Code: ---0048 001D 0009 00B8 0000 0000 0000 0000     0084 00C0 0045 005B 00AB 0095 00AB 0095
0051 00D5 0000 0000 0000 0000 0000 0000     008D 0063 003C 008E 004B 0007 004B 0007
0000 0000 0000 0000 0048 001D 0009 00B8     0063 00F3 0063 00F3 0030 00AE 00F1 0048
0000 0000 0000 0000 0051 00D5 0000 0000     0083 00D8 0083 00D8 0039 0028 00E8 0052
00BB 0024 00B1 0040 0000 0000 0000 0000     0015 0086 000B 00A2 0058 0050 0058 0050
006C 0065 0000 0000 0000 0000 0000 0000     003E 00BF 00D2 0059 0068 0099 0068 0099
0000 0000 0000 0000 00BB 0024 00B1 0040     00E2 0020 00E2 0020 006F 00E6 0065 0002
0000 0000 0000 0000 006C 0065 0000 0000     007E 00AF 007E 00AF 0038 00CF 00CC 0069
002D 0091 003E 00DD 0000 0000 0000 0000     0005 0033 0016 007F 0081 00F0 0081 00F0
006B 006E 0000 0000 0000 0000 0000 0000     00E3 00A8 0078 003A 00B1 002F 00B1 002F
0000 0000 0000 0000 002D 0091 003E 00DD     0079 003C 0079 003C 006D 00AF 007E 00FB
0000 0000 0000 0000 006B 006E 0000 0000     0089 003D 0089 003D 008B 0073 0020 0005
00D6 0090 00DE 0046 0000 0000 0000 0000     0018 0068 0020 001E 00F8 0073 00F8 0073
00B4 00D7 0000 0000 0000 0000 0000 0000     0096 0001 00E2 0029 0020 002F 0020 002F
0000 0000 0000 0000 00D6 0090 00DE 0046     00E6 005D 00E6 005D 00E2 00FE 00EA 00B4
0000 0000 0000 0000 00B4 00D7 0000 0000     0066 0077 0066 0077 00A0 002E 00EC 0056
0038 0085 0074 00D4 0000 0000 0000 0000     004A 0072 0086 00C1 00E8 00E3 00E8 00E3
00AC 0059 0000 0000 0000 0000 0000 0000     00FE 0092 0052 0039 0028 0081 0028 0081
0000 0000 0000 0000 0038 0085 0074 00D4     00DE 0065 00DE 0065 00C4 00D5 0000 0025
0000 0000 0000 0000 00AC 0059 0000 0000     003E 00B6 003E 00B6 0078 0008 00CC 00AE
008A 00BA 0064 00FD 0000 0000 0000 0000     0038 0075 0012 00B8 006E 00E3 006E 00E3
00EE 00B7 0000 0000 0000 0000 0000 0000     001C 0060 002E 00A8 003E 0068 003E 0068
0000 0000 0000 0000 008A 00BA 0064 00FD     009C 0051 009C 0051 009E 0021 0078 0064
0000 0000 0000 0000 00EE 00B7 0000 0000     000C 0069 000C 0069 00C2 00F4 00D4 003C
0014 0091 00B3 00D2 0000 0000 0000 0000     0074 0049 0013 008B 006D 00ED 006D 00ED
0061 00BE 0000 0000 0000 0000 0000 0000     007D 0071 001C 00B3 009B 008C 009B 008C
0000 0000 0000 0000 0014 0091 00B3 00D2     00FD 008B 00FD 008B 00D4 0028 0073 006A
0000 0000 0000 0000 0061 00BE 0000 0000     00B3 008E 00B3 008E 00C9 0082 0068 00C4
0035 0097 0020 009C 0000 0000 0000 0000     0089 0075 0074 007A 000F 0089 000F 0089
0015 00FB 0000 0000 0000 0000 0000 0000     00E9 006D 00D4 0072 004F 002A 004F 002A
0000 0000 0000 0000 0035 0097 0020 009C     004F 00AC 004F 00AC 009D 006D 0088 0072
0000 0000 0000 0000 0015 00FB 0000 0000     008F 00F8 008F 00F8 007D 0099 0068 009E
00AA 0048 003C 00DB 0000 0000 0000 0000     0048 002B 00DA 00BD 0056 0070 0056 0070
006E 006D 0000 0000 0000 0000 0000 0000     009C 00D0 002E 0063 00BE 001D 00BE 001D
0000 0000 0000 0000 00AA 0048 003C 00DB     00D4 0003 00D4 0003 005E 0042 00F0 00D4
0000 0000 0000 0000 006E 006D 0000 0000     008C 0062 008C 0062 0042 00AE 00D4 0040
0075 005C 00AE 004B 0000 0000 0000 0000     007D 00DF 00B6 00CE 00C9 006A 00C9 006A
00C7 0010 0000 0000 0000 0000 0000 0000     00A7 0025 00E0 0014 00AD 001A 00AD 001A
0000 0000 0000 0000 0075 005C 00AE 004B     0051 003E 0051 003E 00D5 00D1 000E 00C1
0000 0000 0000 0000 00C7 0010 0000 0000     003D 0082 003D 0082 0087 00BA 00C0 00A9
0025 0029 0085 00CE 0000 0000 0000 0000     0003 00AD 0063 0052 0002 0026 0002 0026
00A0 005A 0000 0000 0000 0000 0000 0000     004A 0002 00AA 00A7 00AC 00E0 00AC 00E0
0000 0000 0000 0000 0025 0029 0085 00CE     00D4 0081 00D4 0081 00C1 00D9 0021 007F
0000 0000 0000 0000 00A0 005A 0000 0000     000A 0033 000A 0033 00EC 0037 004C 00DD
00DB 0004 002F 00A8 0000 0000 0000 0000     0039 005E 008D 0001 00CA 00AB 00CA 00AB
00AC 005C 0000 0000 0000 0000 0000 0000     00FE 002F 0052 00D3 0028 0076 0028 0076
0000 0000 0000 0000 00DB 0004 002F 00A8     00C4 0082 00C4 0082 00F7 0057 004B 00FB
0000 0000 0000 0000 00AC 005C 0000 0000     003E 0065 003E 0065 0078 00A3 00CC 0046
0075 005F 0054 00E7 0000 0000 0000 0000     0041 0083 0020 000B 0083 00C7 0083 00C7
0021 00B8 0000 0000 0000 0000 0000 0000     00BD 0001 009C 0049 00DB 003E 00DB 003E
0000 0000 0000 0000 0075 005F 0054 00E7     006B 00F1 006B 00F1 005D 00AB 003C 0033
0000 0000 0000 0000 0021 00B8 0000 0000     00F3 0089 00F3 0089 0089 0023 0068 006B
0013 0022 0085 0062 0000 0000 0000 0000     0055 009E 00C7 00DE 00B4 0011 00B4 0011
0096 0040 0000 0000 0000 0000 0000 0000     0094 0002 00FE 00C1 00D6 0070 00D6 0070
0000 0000 0000 0000 0013 0022 0085 0062     00D6 00B0 00D6 00B0 00E7 00B2 0059 00F3
0000 0000 0000 0000 0096 0040 0000 0000     0094 0045 0094 0045 00CA 00A2 0034 0062
009D 0016 0018 005F 0000 0000 0000 0000     00B1 0053 002C 009C 008F 008E 008F 008E
0085 0049 0000 0000 0000 0000 0000 0000     00F9 00A5 0074 005C 003F 00EA 003F 00EA
0000 0000 0000 0000 009D 0016 0018 005F     00EF 00DB 00EF 00DB 0085 0099 0000 00E2
0000 0000 0000 0000 0085 0049 0000 0000     005F 00C7 005F 00C7 006D 00C7 00E8 007D
001A 00EC 0065 0091 0000 0000 0000 0000     002A 00FB 0075 00A0 0095 00A5 0095 00A5
007F 007D 0000 0000 0000 0000 0000 0000     008F 00E1 0010 0064 0095 0076 0095 0076
0000 0000 0000 0000 001A 00EC 0065 0091     00D5 00D3 00D5 00D3 007A 0052 00C5 00F7
0000 0000 0000 0000 007F 007D 0000 0000     00D5 0079 00D5 0079 00DF 0046 0060 00C9
006D 00A2 00CF 0050 0000 0000 0000 0000     0087 0023 00E9 00D1 00C8 0078 00C8 0078
00A2 00F2 0000 0000 0000 0000 0000 0000     0048 00AF 00A6 00BC 0042 00E1 0042 00E1
0000 0000 0000 0000 006D 00A2 00CF 0050     00B2 00D0 00B2 00D0 00C1 00F2 0023 00A1
0000 0000 0000 0000 00A2 00F2 0000 0000     0018 0012 0018 0012 0076 0003 00D4 0010
00F9 00A2 0045 00FC 0000 0000 0000 0000     00CF 001B 001B 0075 0036 005D 0036 005D
00BC 005E 0000 0000 0000 0000 0000 0000     006E 006A 00B2 000B 0038 008A 0038 008A
0000 0000 0000 0000 00F9 00A2 0045 00FC     0010 0075 0010 0075 0035 0077 0081 00D0
0000 0000 0000 0000 00BC 005E 0000 0000     00AE 00C8 00AE 00C8 0008 0053 004C 00F4
00AA 001F 0052 0023 0000 0000 0000 0000     005C 0052 0004 0056 00E0 0029 00E0 0029
00FA 003F 0000 0000 0000 0000 0000 0000     0090 0059 0096 0019 00EA 00DF 00EA 00DF
0000 0000 0000 0000 00AA 001F 0052 0023     001E 001A 001E 001A 00F6 0073 009E 0077
0000 0000 0000 0000 00FA 003F 0000 0000     0010 0092 0010 0092 006E 0012 0074 00D2
00B9 005F 00A1 0021 0000 0000 0000 0000     00D7 00A2 00BF 0064 0042 001A 0042 001A
00B9 007F 0000 0000 0000 0000 0000 0000     0005 0084 004C 0004 0023 0075 0023 0075
0000 0000 0000 0000 00B9 005F 00A1 0021     0074 0001 0074 0001 0075 0069 005D 002B
0000 0000 0000 0000 00B9 007F 0000 0000     000B 00E0 000B 00E0 0041 00EC 0088 006C
00E6 0080 00A5 0066 0000 0000 0000 0000     00FE 000A 00BD 00F0 00C1 008F 00C1 008F
00E7 00E6 0000 0000 0000 0000 0000 0000     0087 009B 00A0 00B4 000D 00E5 000D 00E5
0000 0000 0000 0000 00E6 0080 00A5 0066     0079 00AB 0079 00AB 00A6 0047 0065 002D
0000 0000 0000 0000 00E7 00E6 0000 0000     005D 0069 005D 0069 00A7 00BD 00C0 00D6
0070 00EA 002B 00B2 0000 0000 0000 0000     0008 0042 00C3 0009 0021 00E7 0021 00E7
007B 00FA 0000 0000 0000 0000 0000 0000     0053 009F 00D8 00A4 0041 00F6 0041 00F6
0000 0000 0000 0000 0070 00EA 002B 00B2     0099 0027 0099 0027 0010 0032 00CB 00F9
0000 0000 0000 0000 007B 00FA 0000 0000     0079 00D7 0079 00D7 001B 00E4 00A0 00E9
0030 00F0 003D 00A3 0000 0000 0000 0000     00B4 00A6 00C1 0059 0017 000C 0017 000C
003D 00F3 0000 0000 0000 0000 0000 0000     0021 00FC 00E4 0008 00E7 00B0 00E7 00B0
0000 0000 0000 0000 0030 00F0 003D 00A3     0067 0025 0067 0025 00F8 00B5 0005 0069
0000 0000 0000 0000 003D 00F3 0000 0000     0097 00F9 0097 00F9 0085 003C 0048 0049
00E5 0082 0050 007B 0000 0000 0000 0000     0079 00DC 00E4 00D4 00EF 005B 00EF 005B
00F5 00FB 0000 0000 0000 0000 0000 0000     0009 0094 0014 0098 00EF 00A7 00EF 00A7
0000 0000 0000 0000 00E5 0082 0050 007B     00AF 002A 00AF 002A 004D 0030 00B8 0028
0000 0000 0000 0000 00F5 00FB 0000 0000     00EF 003B 00EF 003B 005D 00B7 0068 00BB


--- End code ---

Nominal Animal:
No catch, I'm afraid.  We ruled out Hn(A, B) = Hn(A ⊕ B) for ⊕ being addition, subtraction, inclusive-or, or exclusive-or.
It is still possible it is true for some other simple associative sequence of operations instead of just A ⊕ B, though; perhaps nibble-swapping and addition/substraction/xor?

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