Equivalence for generalized Boolean functions

Equivalence plays a key-role for the classification of functions between elementary abelian groups Vₙ⁽ᵖ⁾ and Vₖ⁽ᵖ⁾. One distinguishes between affine equivalence, extended affine (EA) equivalence, and the most general CCZ-equivalence. Recently, there has been an increased interest in functions from elementary abelian groups Vₙ⁽ᵖ⁾ to cyclic groups Zpᵏ. We initiate the study of equivalence for functions from Vₙ⁽ᵖ⁾ to Zpᵏ. We show that CCZ-equivalence is more general than EA-equivalence. For some classes of functions, CCZ-equivalence reduces to EA-equivalence. We show that CCZ-equivalence between two functions from Vₙ⁽ᵖ⁾ to Zpᵏ implies CCZ-equivalence of two associated vectorial functions from Vₙ⁽ᵖ⁾ to Vₖ⁽ᵖ⁾.