International Association
for Cryptologic Research

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