In this paper, we present a high-speed, unified elliptic curve cryptography (ECC) processor for arbitrary Weierstrass curves over $GF\left(p\right)$, which to the best of our knowledge, outperforms other similar works in terms of execution time. Our approach employs the combination of the schoolbook long and Karatsuba multiplication algorithm for the elliptic curve point multiplication (ECPM) to achieve better parallelization while retaining low complexity. In the hardware implementation, the substantial gain in speed is also contributed by our n-bit pipelined Montgomery Modular Multiplier (pMMM), which is constructed from our n-bit pipelined multiplier-accumulators that utilizes digital signal processor (DSP) primitives as digit multipliers. Additionally, we also introduce our unified, pipelined modular adder/subtractor (pMAS) for the underlying field arithmetic, and leverage a more efficient yet compact scheduling of the Montgomery ladder algorithm. The implementation for 256-bit modulus size on the 7-series FPGA: Virtex-7, Kintex-7, and XC7Z020 yields 0.139, 0.138, and 0.206 ms of execution time, respectively. Furthermore, since our pMMM module is generic for any curve in Weierstrass form, we support multi-curve parameters, resulting in a unified ECC architecture. Lastly, our method also works in constant time, making it suitable for applications requiring high speed and SCA-resistant characteristics.

The use of quantum technology in NLP tasks, especially sentiment classification, has the potential to be developed. In this research, we investigate the best technique to represent sentiment sentences so that sentiment can be analyzed using the Quantum-Enhanced Support Vector Machine (QE-SVM) algorithm. Investigations were carried out using circuit parameter optimization methods and data transformation. The pipeline of the proposed method consists of sentence-to-circuit conversion, circuit parameter training, statevector formation, and finally the training and testing processes. As a result, we obtained the best classification results with an accuracy of 93.33% using the SPSA optimization method and PCA transformation data. These results have also outperformed the baseline SVM method.

This paper examines the asymptotic performance of multiplication and the cost of quantum implementation for the Naive schoolbook, Karatsuba, and Toom-Cook methods in the classical and quantum cases and provides insights into multiplication roles in the post-quantum cryptography (PQC) era. Further, considering that the lattice-based PQC algorithm is based on polynomial multiplication algorithms, including the Toom-Cook 4-way multiplier as its fundamental building block, we propose a higher-degree multiplier, the Toom-Cook 8-way multiplier, which has the lowest asymptotic performance and implementation cost. Additionally, the designed multiplication will include additional sub-operations to complete the multiplication of large integers in order to prevent side-channel attacks. To design our Toom-Cook 8-way in detail, we employ detailed step computations such as splitting, evaluation, point-wise multiplication, interpolation, and recomposition, as well as several strategies to reduce space and time requirements. Existing asymptotic performance and quantum implementation cost multipliers are compared with our 2-way, 4-way, and 8-way Toom-Cook multiplier designs. Our Toom-Cook 8-way quantum multiplier has the lowest asymptotic performance analysis or qubit count in terms of space efficiency, with $n\left({\frac {15}{8}}\right)^{\frac {\log 15}{(2\log 15 - \log 8)} \log _{8} n}$ or asymptotically $\mathcal {O} (n^{1.245})$ . The design has the lowest logical Toffoli counts bound at $112n^{\log _{8} 15}-128n $ and Toffoli depth of $n\left({\frac {15}{8}}\right)^{1-\frac {\log 15}{(2\log 15 - \log 8)} \log _{8} n }$ , asymptotically close to $\mathcal {O}(n^{1.0569})$ , which corresponds to a space- and time-efficient multiplication.

This paper presents quantum cryptanalysis for binary elliptic curves from a time-efficient implementation perspective (i.e., reducing the circuit depth), complementing the previous research that focuses on the space-efficiency perspective (i.e., reducing the circuit width). To achieve depth optimization, we propose an improvement to the existing circuit implementation of the Karatsuba multiplier and FLT-based inversion, then construct and analyze the resource in the Qiskit quantum computer simulator. The proposed multiplier architecture, which improves the quantum Karatsuba multiplier from the previous study, reduces the depth and yields a lower number of CNOT gates that bound to O ( n ^log₂(3) ) while maintaining a similar number of Toffoli gates and qubits. Furthermore, our improved FLT-based inversion reduces CNOT count and overall depth, with a tradeoff of higher qubit size. Finally, we employ the proposed multiplier and FLT-based inversion for performing quantum cryptanalysis of binary point addition as well as the complete Shor’s algorithm for the elliptic curve discrete logarithm problem (ECDLP). As a result, apart from depth reduction, we are also able to reduce up to 90% of the Toffoli gates required in a single-step point addition compared to prior work, leading to significant improvements and giving new insights on quantum cryptanalysis for a depth-optimized implementation.