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CRC Press eBooks

Grover's Algorithm: Quantum Search

Bhupendra Singh, Mohankumar Mylsamy, Thamaraimanalan Thangarajan

Published Nov 14, 2025• 0 citations
Publisher

Abstract

“Grover's Algorithm: Quantum Search” explores Grover's Algorithm in the context of quantum search, beginning with an overview of its mechanics. The section includes a literature survey that covers Grover's algorithm, stream ciphers, and the Grover attack, as well as quantum resource estimation. The section then introduces a simplified 4-bit Grain cipher (with a 4-bit key and 4-bit IV), detailing its design, initialization, and key stream generation. Following this, the chapter discusses about key recovery using Grover's algorithm, with an emphasis on quantum circuit development for the simplified Grain 4-bit cipher. It outlines key steps in the circuit, such as executing the Boolean function and XORing the NFSR's fourth bit. Further, the section details the development of a Grover attack quantum circuit, describing both single and double pair methods. Experimental results are provided across multiple cases, with a summary of test cases and classical computation times. Quantum resource estimation, including the costs associated with the simplified Grain 4-bit cipher and the Grover oracle, is analyzed. The section concludes with a discussion on the results, classical and quantum resource estimations, and the potential for future enhancements to the approach.

DOI
10.1201/9781003606338-4

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“Grover's Algorithm: Quantum Search” explores Grover's Algorithm in the context of quantum search, beginning with an overview of its mechanics. The section includes a literature survey that covers Grover's algorithm, stream ciphers, and the Grover attack, as well as quantum resource estimation. The section then introduces a simplified 4-bit Grain cipher (with a 4-bit key and 4-bit IV), detailing its design, initialization, and key stream generation. Following this, the chapter discusses about key recovery using Grover's algorithm, with an emphasis on quantum circuit development for the simplified Grain 4-bit cipher. It outlines key steps in the circuit, such as executing the Boolean function and XORing the NFSR's fourth bit. Further, the section details the development of a Grover attack quantum circuit, describing both single and double pair methods. Experimental results are provided across multiple cases, with a summary of test cases and classical computation times. Quantum resource estimation, including the costs associated with the simplified Grain 4-bit cipher and the Grover oracle, is analyzed. The section concludes with a discussion on the results, classical and quantum resource estimations, and the potential for future enhancements to the approach.
In this paper, we proposed a simplified version of the Grain-128 stream cipher and developed a corresponding quantum circuit. Leveraging Grover's algorithm, we demonstrate the ability to recover a key given plaintext, initialization vector (IV), and corresponding ciphertext through various methods, including single pair, double pair series, and double pair parallel approaches. Using the Qiskit Quantum Development Kit (QDK), we conduct these attacks and estimate the required quantum resources. Our research reveals that the simplified grain cipher is vulnerable to quantum attacks, showing how quantum computing could affect cryptographic systems.
“Simon's Algorithm: Collision Finding” explains Simon's Algorithm, focusing on how it helps in finding collisions. It starts with an overview of how the algorithm works. A literature review is included, covering both Simon's algorithm and its uses in cryptography. The section then introduces Simon's problem, comparing how classical and quantum methods solve it, and explains how to build a quantum circuit for a blackbox function. Different examples of blackbox functions (oracles) are shown, such as 1-to-1 and 2-to-1 mappings, with detailed implementations of Simon's algorithm for both 4-bit and 8-bit cases. The section also looks at a suggested attack method on the Grain-128a cipher, explaining how the key and initialization vector (IV) are set up, how the keystream is generated, and how the authenticated tag is created. It provides a quantum circuit design for the Simon attack on Grain-4a, along with steps for carrying out the attack. The section also presents experimental results from several test cases and quantum resource estimates. It wraps up with a discussion of the results and thoughts on future improvements and developments in using Simon's algorithm for cryptographic attacks.
Abstract A significant amount of study is being done to review the security promises made for the various ciphers now in use as a result of the development of quantum computing technology. A general attack against symmetric key cryptography primitives that can reduce search costs to the square root is Grover's search algorithm. To implement Grover's algorithm, it is necessary that the target cipher be implemented as a quantum circuit. Despite being relatively new, this area of study has received significant attention from the research community. The authors have estimated the cost of Grover's key search attack against the stream cipher Atom, for the first time, under circuit depth restrictions defined in National Institute of Standards and Technology (NIST) PQC standardisation process. The authors implement the quantum circuit of Atom in QISKIT, (open‐source software development kit for working with quantum computers running on IBM Quantum Experience). The results are also compared with other existing literature on LFSR‐based stream ciphers, such as Grain‐v1, Grain‐128‐AEAD, and Lizard. The authors also find that, to the best of their knowledge, in the existing literature on estimating the cost of Grover's attack on symmetric ciphers, Atom is the only 128‐bit key cipher that meets the threshold of ≈2 170 set by NIST for quantum security of 128‐bit key ciphers. The authors also analyse the security of Atom against quantum TMDTO attacks.
Quantum computing endangers classical cryptography via two main avenues: Shor’s al-gorithm reduces integer factorization to polynomial time, and Grover’s algorithm yields a quadratic speed-up for unstructured key search. We develop rigorous mathematical treatment of both algorithms, enrich the literature survey with detailed complexity and resource analyses, and extend the development section to include quantum er-ror-correction overhead, resource estimates for breaking 2048-bit RSA, and advanced as-ymptotic bounds. In Discussion, we propose lattice-, code-, multivariate-, and hash-based post-quantum schemes, analyze their security margins against quantum attacks, and out-line standardized migration strategies. We include three illustrative figures and 35+ se-quential IEEE citations.
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.
NIST(National Institute of Standards and Technology) estimates the Grover key search cost required for symmetric key cryptography as the postquantum security strength. The Grover search algorithm is applicable to key recovery in symmetric key cryptography, and for this, the target encryption process must be implemented as a quantum circuit. In this paper, we present an optimized implementation of Simplified AES as a quantum circuit. Substitution, Mix Columns, and Key Expansion are implemented efficiently, and as a result of resource analysis, qubits, quantum gates, and circuit depth are significantly reduced compared to the previous result.