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Quantum exhaustive key search with simplified-DES as a case study

Mishal Almazrooie, Azman Samsudin, Rosni Abdullah, Kussay Nugamesh Mutter

Published Sep 6, 2016• 10 citations
Publisher

Abstract

To evaluate the security of a symmetric cryptosystem against any quantum attack, the symmetric algorithm must be first implemented on a quantum platform. In this study, a quantum implementation of a classical block cipher is presented. A quantum circuit for a classical block cipher of a polynomial size of quantum gates is proposed. The entire work has been tested on a quantum mechanics simulator called libquantum. First, the functionality of the proposed quantum cipher is verified and the experimental results are compared with those of the original classical version. Then, quantum attacks are conducted by using Grover's algorithm to recover the secret key. The proposed quantum cipher is used as a black box for the quantum search. The quantum oracle is then queried over the produced ciphertext to mark the quantum state, which consists of plaintext and key qubits. The experimental results show that for a key of n-bit size and key space of N such that [Formula: see text], the key can be recovered in [Formula: see text] computational steps.

DOI
10.1186/s40064-016-3159-4

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To evaluate the security of a symmetric cryptosystem against any quantum attack, the symmetric algorithm must be first implemented on a quantum platform. In this study, a quantum implementation of a classical block cipher is presented. A quantum circuit for a classical block cipher of a polynomial size of quantum gates is proposed. The entire work has been tested on a quantum mechanics simulator called libquantum. First, the functionality of the proposed quantum cipher is verified and the experimental results are compared with those of the original classical version. Then, quantum attacks are conducted by using Grover's algorithm to recover the secret key. The proposed quantum cipher is used as a black box for the quantum search. The quantum oracle is then queried over the produced ciphertext to mark the quantum state, which consists of plaintext and key qubits. The experimental results show that for a key of n-bit size and key space of N such that [Formula: see text], the key can be recovered in [Formula: see text] computational steps.
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.
ABSTRACT Quantum computing environments make block ciphers susceptible to exhaustive key search attacks utilizing Grover's algorithm. However, such quantum‐based attacks remain impractical unless the targeted cipher is directly implemented on a quantum platform. Moreover, their efficiency significantly depends on the quantum circuit design and optimization of the block cipher in question. The cost of a quantum circuit implementation is typically measured by two main metrics: the number of qubits and the circuit depth ( T ‐depth). For most block ciphers, the S‐box is the principal factor contributing to increased T ‐depth and additional qubit requirements. This paper presents a method for generating quantum circuits directly from lookup tables of S‐boxes applicable to arbitrary block ciphers. We illustrate our approach using quantum circuit implementations of the PRESENT and DES ciphers as practical examples. Our proposed method is expected to efficiently implement arbitrary S‐boxes by employing polynomial evaluation, thus balancing time–space complexity.
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.
A quantum computer capable of running the Grover search algorithm, which reduces the complexity of brute-force attacks by a square root, has the potential to undermine the security strength of symmetric-key cryptography and hash functions. Recently, studies on quantum approaches have proposed analyzing potential quantum attacks using the Grover search algorithm in conjunction with optimized quantum circuit implementations for symmetric-key cryptography and hash functions. Analyzing quantum attacks on a cipher (i.e., quantum cryptanalysis) and estimating the necessary quantum resources are related to evaluating post-quantum security for the target cryptography algorithms. In this paper, we revisit quantum implementations of CHAM block cipher, an ultra lightweight cipher, with a focus on optimizing the linear operations in its key schedule. We optimized the linear equations of CHAM as matrices by applying novel optimized decomposition techniques. Using the improved CHAM quantum circuits, we estimate the cost of Grover’s key search and evaluate the post-quantum security strength with further reduced costs.