Time-space complexity of quantum search algorithms in symmetric cryptanalysis: applying to AES and SHA-2

Performance of cryptanalytic quantum search algorithms is mainly inferred from query complexity which hides overhead induced by an implementation. To shed light on quantitative complexity analysis removing hidden factors, we provide a framework for estimating time-space complexity, with carefully accounting for characteristics of target cryptographic functions. Processor and circuit parallelization methods are taken into account, resulting in the time-space trade-off curves in terms of depth and qubit. The method guides howto rank different circuit designs in order of their efficiency. The framework is applied to representative cryptosystems NIST referred to as a guideline for security parameters, reassessing the security strengths of AES and SHA-2

Quantum Search for Scaled Hash Function Preimages

We present the implementation of Grover's algorithm in a quantum simulator to perform a quantum search for preimages of two scaled hash functions, whose design only uses modular addition, word rotation, and bitwise exclusive or. Our implementation provides the means to assess with precision the scaling of the number of gates and depth of a full-fledged quantum circuit designed to find the preimages of a given hash digest. The detailed construction of the quantum oracle shows that the presence of AND gates, OR gates, shifts of bits and the reuse of the initial state along the computation, require extra quantum resources as compared with other hash functions based on modular additions, XOR gates and rotations. We also track the entanglement entropy present in the quantum register at every step along the computation, showing that it becomes maximal at the inner core of the first action of the quantum oracle, which implies that no classical simulation based on Tensor Networks would be of relevance. Finally, we show that strategies that suggest a shortcut based on sampling the quantum register after a few steps of Grover's algorithm can only provide some marginal practical advantage in terms of error mitigation.Comment: 24 pages, 14 figure

Quantum Analysis of AES

Quantum computing is considered among the next big leaps in computer science. While a fully functional quantum computer is still in the future, there is an ever-growing need to evaluate the security of the symmetric key ciphers against a potent quantum adversary. Keeping this in mind, our work explores the key recovery attack using the Grover\u27s search on the three variants of AES (-128, -192, -256). In total, we develop a pool of 20 implementations per AES variant (thus totaling in 60), by taking the state-of-the-art advancements in the relevant fields into account. In a nutshell, we present the least Toffoli depth and full depth implementations of AES, thereby improving from Zou et al.\u27s Asiacrypt\u2720 paper by more than 97 percent for each variant of AES. We show that the qubit count - Toffoli depth product is reduced from theirs by more than 86 percent. Furthermore, we analyze the Jaques et al.\u27s Eurocrypt\u2720 implementations in details, fix the bugs (arising from some problem of the quantum computing tool used and not related to their coding) and report corrected benchmarks. To the best of our finding, our work improves from all the previous works (including the Asiacrypt\u2722 paper by Huang and Sun and the Asiacrypt\u2723 paper by Liu et al.) in terms of various quantum circuit complexity metrics (Toffoli depth, full depth, Toffoli/full depth - qubit count product, full depth - gate count product, etc.). Also, our bug-fixing of Jaques et al.\u27s Eurocrypt\u2720 implementations seem to improve from the authors\u27 own bug-fixing, thanks to our architecture consideration. Equipped with the basic AES implementations, we further investigate the prospect of the Grover\u27s search. We also propose three new implementations of the S-box, one new implementation of the MixColumn; as well as five new architecture (one is motivated by the architecture by Jaques et al. in Eurocrypt’20, and the rest four are entirely our innovation). Under the MAXDEPTH constraint (specified by NIST), the circuit depth metrics (Toffoli depth, T-depth and full depth) become crucial factors and parallelization for often becomes necessary. We provide the least depth implementation in this respect, that offers the best performance in terms of metrics for circuit complexity (like, depth-squared - qubit count product, depth - gate count product)

Quantum Search Algorithms for Constraint Satisfaction and Optimization Problems Using Grover\u27s Search and Quantum Walk Algorithms with Advanced Oracle Design

The field of quantum computing has emerged as a powerful tool for solving and optimizing combinatorial optimization problems. To solve many real-world problems with many variables and possible solutions for constraint satisfaction and optimization problems, the required number of qubits of scalable hardware for quantum computing is the bottleneck in the current generation of quantum computers. In this dissertation, we will demonstrate advanced, scalable building blocks for the quantum search algorithms that have been implemented in Grover\u27s search algorithm and the quantum walk algorithm. The scalable building blocks are used to reduce the required number of qubits in the design. The proposed architecture effectively scales and optimizes the number of qubits needed to solve large problems with a limited number of qubits. Thus, scaling and optimizing the number of qubits that can be accommodated in quantum algorithm design directly reflect on performance. Also, accuracy is a key performance metric related to how accurately one can measure quantum states. The search space of quantum search algorithms is traditionally created by using the Hadamard operator to create superposition. However, creating superpositions for problems that do not need all superposition states decreases the accuracy of the measured states. We present an efficient quantum circuit design that the user has control over to create the subspace superposition states for the search space as needed. Using only the subspace states as superposition states of the search space will increase the rate of correct solutions. In this dissertation, we will present the implementation of practical problems for Grover\u27s search algorithm and quantum walk algorithm in logic design, logic puzzles, and machine learning problems such as SAT, MAX-SAT, XOR-SAT, and like SAT problems in EDA, and mining frequent patterns for association rule mining