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Why can we reverse the tempering function?
There are to kinds of operations in the tempering function of the MT:
1) Right shift
2) Left shift with an AND
To undo the first operation, a simple shift is straight forward. Let's consider
the following example of a 24-bit number
0010 1010 1011 1101 1111 0111 y
0000 0000 0000 0000 0000 1010 1010 1111 0111 1101 11 y >> 18 := N
^
orginal string
0010 1010 1011 1101 1111 1101 y ^ (y>>18) := M
On can see, that the first 18 bits of M are the same than y. That's clear,
because we have shifted 18 new 0 into it. But now we also now N completly, since
the last 6 bits are the first 6 of M (N is a shifted version of M).
If you shift less than 24/2 = 12 bits, you have to do the above steps in a
loop, because you cannot determin N completly one step.
To understand why one can undo the second operation was a bit harder for me.
For me it was the easiest to understand it with the feistel network.
y y << 15
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XOR<--------- F <-----------
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\_______________________ /
/\______________________
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