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Topic: Almost perfect nonlinear permutations
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jdm

Posts: 16
Registered: 11/22/10
Almost perfect nonlinear permutations
Posted: Jun 14, 2011 2:32 PM
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I've been wrestling with a bit of a thorny problem lately, and if
anyone knows whether a particular result relating to APN permutations
has been proven it would help enormously.

Consider an APN bijection S operating on GF(2^n) (n >= 5). It is easy
to show that the first 2^{n-1} entries of the truth table of S cannot
all be < 2^{n-1}, since if they were, all input differences < 2^{n-1}
would result in output differences < 2^{n-1}, and this would mean
that, for instance, input difference 1 would require 2^{n-1} 2s to be
placed in row 1 of the difference distribution table of S in the
columns corresponding to nonzero output differences < 2^{n-1}. As
there are only 2^{n-1}-1 such columns, this would lead to a
contradiction.

Now, consider a generalisation of this problem. Where S is an APN
bijection operating on GF(2^n), is there any value m (3 <= m < n) such
that it is possible for the first 2^m entries of the truth table of S
all to be < 2^{m} (and hence for the first 2^m entries of the truth
table of S to define an APN bijection over GF(2^m))? If not, can
anyone point me towards a proof?

I have tried to prove that this is not possible for the case m=(n-2),
but was unable to do so. More generally, I believe that no m exists
such that this is possible, but have not been able to prove this.

Many thanks,

James McLaughlin.



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