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CTU Journal of Innovation and Sustainable Development

Demonstration of Grover’s algorithm for retrieving secret keys in a basic SPN block cipher

Minh Thang Vu, Van Nghi Nguyen, Quoc Dat Le, Quang Trung Do

Published Oct 25, 2024• 1 citation
Publisher

Abstract

In this study, we present in detail the application of Grover's quantum algorithm to the searching problem of the secret key of a simple SPN (Substitution–permutation network) block cipher called Yo-yo. The main goal of the paper is to clarify the construction of the quantum circuit and the operation phases of Grover's algorithm to find the secret key with the condition of knowing at least 1 pair of plaintext-ciphertext. To achieve this goal, we consider 2 cases: the case where there is a unique key that satisfies and the case where there are 2 keys that satisfy at the same time. As a result, our implementation technique, implemented in the Qiskit programming language, requires only 17 qubits to find the key of the Yo-yo block cipher correctly. This technique can be effectively applied on IBM quantum computers for large-scale SPN block ciphers, such as AES and GOST R.34.10.2015, which are widely used today.

DOI
10.22144/ctujoisd.2024.317

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In this study, we present in detail the application of Grover's quantum algorithm to the searching problem of the secret key of a simple SPN (Substitution–permutation network) block cipher called Yo-yo. The main goal of the paper is to clarify the construction of the quantum circuit and the operation phases of Grover's algorithm to find the secret key with the condition of knowing at least 1 pair of plaintext-ciphertext. To achieve this goal, we consider 2 cases: the case where there is a unique key that satisfies and the case where there are 2 keys that satisfy at the same time. As a result, our implementation technique, implemented in the Qiskit programming language, requires only 17 qubits to find the key of the Yo-yo block cipher correctly. This technique can be effectively applied on IBM quantum computers for large-scale SPN block ciphers, such as AES and GOST R.34.10.2015, which are widely used today.
The emergence of quantum computers is threatening the security of cryptography through various quantum algorithms. Among them, the Grover search algorithm is known to be efficient in accelerating brute force attacks on block cipher algorithms. To utilize the Grover’s algorithm for brute force attacks, block ciphers must be implemented in quantum circuits. In this paper, we present optimized quantum circuits of the SPN (Substitution Permutation Network) structured lightweight block cipher, namely the PIPO block cipher. In particular, the compact design of quantum circuits for the 8-bit Sbox is investigated. These optimization techniques are used to implement other cryptographic operations as quantum circuits. Finally, we evaluate quantum resources of Grover search algorithm for the PIPO block cipher in ProejctQ, a quantum simulator provided by IBM.
Key recovery attack on SPN ciphers such as AES is NP hard in complexity. Previous studies have estimated the quantum resources required for implementing Grover’s key search for AES, but these are largely theoretical in nature. The key challenges to implement Grover’s algorithm on current quantum hardware include the number of available qubits, noise, and decoherence. However, the rapid advancement of hardware has brought us closer to practical realization for the cryptanalysis of light-weight SPN ciphers. This study presents an efficient quantum circuit implementation of Mini-AES, a 16-bit variant of AES, employing standard CNOT and Toffoli gates. To the best of our knowledge, our work provides the first complete quantum implementations of all core Mini-AES components for the quantum oracle. The effects of noise are emulated for each module of the quantum oracle and quantified. Furthermore, Grover’s algorithm is applied to Mini-AES to perform successful key recovery through a known-plaintext attack, demonstrating significant amplitude amplification while using near-optimal Grover’s iterations, thereby reducing the total quantum volume. The optimal Grover’s iterations have been obtained empirically as well as validated with finite-shot analysis. Notably, the application of Grover’s algorithm reveals key clustering behavior in Mini-AES, which has been corroborated through classical simulation.
Grover search algorithm is the most representative quantum attack method that threatens the security of symmetric key cryptography. If the Grover search algorithm is applied to symmetric key cryptography, the security level of target symmetric key cryptography can be lowered from n-bit to n2-bit. When applying Grover’s search algorithm to the block cipher that is the target of potential quantum attacks, the target block cipher must be implemented as quantum circuits. Starting with the AES block cipher, a number of works have been conducted to optimize and implement target block ciphers into quantum circuits. Recently, many studies have been published to implement lightweight block ciphers as quantum circuits. In this paper, we present optimal quantum circuit designs of symmetric key cryptography, including PRESENT and GIFT block ciphers. The proposed method optimized PRESENT and GIFT block ciphers by minimizing qubits, quantum gates, and circuit depth. We compare proposed PRESENT and GIFT quantum circuits with other results of lightweight block cipher implementations in quantum circuits. Finally, quantum resources of PRESENT and GIFT block ciphers required for the oracle of the Grover search algorithm were estimated.
In this work, a quantum design for the Simplified-Advanced Encryption Standard (S-AES) algorithm is presented. Also, a quantum Grover attack is modeled on the proposed quantum S-AES. First, quantum circuits for the main components of S-AES in the finite field F2[x]/(x4 + x + 1), are constructed. Then, the constructed circuits are put together to form a quantum version of S-AES. A C-NOT synthesis is used to decompose some of the functions to reduce the number of the needed qubits. The quantum S-AES is integrated into a black-box queried by Grover's algorithm. A new approach is proposed to uniquely recover the secret key when Grover attack is applied. The entire work is simulated and tested on a quantum mechanics simulator. The complexity analysis shows that a block cipher can be designed as a quantum circuit with a polynomial cost. In addition, the secret key is recovered in quadratic speedup as promised by Grover's algorithm.