Proof of Mirror Theory for a Wide Range of $$\xi _{\max }$$

In CRYPTO’03, Patarin conjectured a lower bound on the number of distinct solutions $$(P_1, \ldots , P_{q}) \in (\{0,1\}^{n})^{q}$$ satisfying system equations form $$X_i \oplus X_j = \lambda _{i,j}$$ such that $$P_1, P_2, $$ $$P_{q}$$ are pairwise distinct. This result is known as “ $$P_i P_j$$ Theorem for any $$\xi _{\max }$$ ” or alternatively Mirror Theory general which was later proved by in ICISC’05. theory stands powerful tool to provide high-security guarantee many blockcipher-(or even ideal permutation-) based designs. Unfortunately, proof contains gaps non-trivial fix. this work, we present first complete theorem wide range typically up order $$O(2^{n/4}/\sqrt{n})$$ . Furthermore, our approach made simpler using new type equation, dubbed link-deletion roughly corresponds half so-called orange from earlier works. As an illustration result, also revisit security proofs two optimally secure blockcipher-based pseudorandom functions, and n-bit six round Feistel cipher, updated bounds.