Families of Vectorial Permutations Induced by Translations: Affine Structure, Differential Invariance, and the Block Size Bound

We study a family of vectorial permutations over F_2^8 generated by iterated composition of translations and a fixed permutation (S-box), motivated by a binary projection method based on the Gray code. We formalize the original family F(S) = {T_m(x) = S(x ⊕ m) : m ∈ F_2^8}, show that its elements are affine (not linear) maps, and characterize exactly the invariance of its differential distribution table (DDT) and the magnitudes of its linear approximation table (LAT). We extend the construction to a 256-bit key space via iterated key-alternating composition, verify injectivity computationally for r=1,2, and establish the combinatorial ceiling r ≤ 210 imposed by |S_256| = 256!. We show that the differential and linear metrics of the iterated family saturate at the level of a uniform random permutation from r=2 onward. Finally, we establish the block size theorem: the effective security of any block cipher under the chosen-plaintext model is bounded by min(2^κ, 2^n), where κ is the key length in bits, independently of the key space size. We apply the result to the construction of a 128-bit SPN with a 256-bit key, structurally equivalent to AES-256, and compare it with standard lightweight ciphers (PRESENT, GIFT, SKINNY) from the perspective of the area–rounds–security trade-off. We analyze the key schedule as a critical component, quantify the gap between the Even–Mansour model (independent round keys) and real ciphers (deterministically derived round keys), and propose a nonlinear key schedule with conditional security guarantees.