Linear cryptanalysis resistance of AES Sbox

Question

If you look at the AES Linear Approximation Table (computed for example with Sage) you will see there are many entries with what looks like a high bias of -16 ("absolute bias" scale).

I know AES is designed to be resistant to Linear cryptanalysis. If you agree that -16 is a high bias, then there are 2 (3) options:

• either the AES Sbox is weak to linear cryptanalysis, but the overall cipher is not thanks to the properties of ShiftRow and MixColumn
• or it is difficult/impossible to concatenate these entries with high bias to form a linear characteristic with high bias for more than 1 round (imagine we ignore the ShiftRow and MixColum and try to concatenate the linear approximation for more than one consecutive Sbox)
• or both

Which one is it? I read that AES Sbox is based on multiplicative inverse in Galois Field which is supposed to be "highly nonlinear", but I'm not sure this applies here.

Answer 1

The Sbox on its own is not optimally nonlinear but highly nonlinear.

What this means is that the ideal criteria against linear and differential cryptanalysis would be to have an Sbox that is Almost Bent and Almost Perfect Nonlinear. And APN is not achievable for an even number of bits. So already we are suboptimal.

But there are other security properties to think of, such as Strict Avalanche Criterion, Sum of Squares Indicator etc.

The design philosophy balances all these. And the ShiftRows and MixColumns help as well.

You should read The Design of Rijndael by the designers to understand all these tradeoffs. It's available here at Daemen's homepage.

The paper here (see example 6) mentions that AES achieves the lowest possible sum of squares indicator for a power permutation (which it is, $S:x\mapsto x^{2^n-2}$ usually wrongly written as $S:x\mapsto x^{-1}$ since zero does not have an inverse.