**Post-Quantum Security of the Even-Mansour Cipher**

*Gorjan Alagic and Chen Bai and Jonathan Katz and Christian Majenz*

**Abstract: **The Even-Mansour cipher is a simple method for constructing a (keyed) pseudorandom permutation $E$ from a public random permutation $P:\{0,1\}^n \rightarrow \{0,1\}^n$. It is a core ingredient in a wide array of symmetric-key constructions, including several lightweight cryptosystems presently under consideration for standardization by NIST. It is secure against classical attacks, with optimal attacks requiring $q_E$ queries to $E$ and $q_P$ queries to $P$ such that $q_E \cdot q_P \approx 2^n$. If the attacker is given *quantum* access to both $E$ and $P$, however, the cipher is completely insecure, with attacks using $q_E, q_P = O(n)$ queries known.

In any plausible real-world setting, however, a quantum attacker would have only *classical* access to the keyed permutation $E$ implemented by honest parties, while retaining quantum access to $P$. Attacks in this setting with $q_E \cdot q_P^2 \approx 2^n$ are known, showing that security degrades as compared to the purely classical case, but leaving open the question as to whether the Even-Mansour cipher can still be proven secure in this natural ``post-quantum'' setting.

We resolve this question, showing that any attack in that setting requires $q_E \cdot q^2_P + q_P \cdot q_E^2 \approx 2^n$. Our results apply to both the two-key and single-key variants of Even-Mansour. Along the way, we establish several generalizations of results from prior work on quantum-query lower bounds that may be of independent interest.

**Category / Keywords: **secret-key cryptography /

**Date: **received 7 Dec 2021, last revised 15 Dec 2021

**Contact author: **galagic at gmail com, jkatz2 at gmail com, chmaj at dtu dk, cbai1 at terpmail umd edu

**Available format(s): **PDF | BibTeX Citation

**Version: **20211215:103807 (All versions of this report)

**Short URL: **ia.cr/2021/1601

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