# Can we have Rijndael s-boxes constructed using reducible polynomials?

Is it necessary for the Rijnael polynomial to be irreducible? Can we have s-boxes constructed using reducible polynomials? If not what is the mathematical property that is obstructing to do so?

• You could....but its meant to protect against shift frequency atks...by having an irreducible polynomial equation it is difficult to determine the shift interval – Light Bringer Mar 5 '17 at 6:07
• Can you please elaborate. If possible please provide an example where an secure S - box is constructed using a reducible polynomial. @light bringer – user38956 Mar 5 '17 at 10:24

As noted by poncho in the comments to your previous question, the Rijndael S-box is defined based on the multiplicative inverse of the input byte in a particular representation of the finite field $GF(2^8)$, where the representation is defined by the choice of the reduction polynomial.