Quadratic Equations from APN Power Functions

We develop several tools to derive quadratic equations from algebraic S-boxes and to prove their linear independence. By applying them to all known almost perfect nonlinear (APN) power functions and the inverse function, we can estimate the resistance against algebraic attacks. As a result, we can show that APN functions have different resistance against algebraic attacks, and especially S-boxes with Gold or Kasami exponents have very weak resistance. key words: algebraic attack, quadratic equations, almost perfect nonlinear (APN), linear independence, nonlinearity.