Konstruksi Dan Implementasi Transformasi Wavelet Kuantum Berbasis Integer Atas Aljabar Min-max-plus

Firmansyah, Mohamad Ilham Dwi (2026) Konstruksi Dan Implementasi Transformasi Wavelet Kuantum Berbasis Integer Atas Aljabar Min-max-plus. Doctoral thesis, Institut Teknologi Sepuluh Nopember.

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Abstract

Transformasi wavelet merupakan alat matematis penting dalam pengolahan sinyal karena kemampuannya merepresentasikan sinyal secara efisien pada domain waktu dan frekuensi secara simultan. Namun, implementasi konvensional berbasis aritmetika floating point rentan terhadap kesalahan dan ketidakstabilan numerik, khususnya pada sistem dengan presisi terbatas seperti perangkat keras digital. Pendekatan berbasis aljabar max-plus dan turunannya, yang menggantikan operasi aritmetika konvensional dengan operasi maksimum dan penjumlahan, menawarkan sistem yang lebih stabil, bebas kesalahan numerik, serta efisien untuk implementasi perangkat keras. Transformasi wavelet banyak diterapkan dalam berbagai bidang, termasuk pengolahan citra dan kriptografi. Seiring berkembangnya komputasi kuantum, muncul tantangan baru dalam mengonstruksi transformasi wavelet kuantum yang mampu beroperasi pada domain kuantum sekaligus mempertahankan stabilitas dan ketepatan numerik. Pada disertasi ini diusulkan algoritma kriptografi baru yang memanfaatkan transformasi wavelet berbasis integer melalui skema lifting atas aljabar min-max-plus (MMPLS-IWavelet). Algoritma ini dirancang menggunakan pasangan operator predict dan update berbasis aljabar min-max-plus untuk mencapai kualitas enkripsi tinggi, sensitivitas kunci yang kuat, serta efisiensi komputasi yang optimal. Selain itu, dikembangkan pula transformasi wavelet integer kuantum satu dimensi, yaitu QMP-Wavelet Tipe IVa, yang dikonstruksi berdasarkan struktur MP-Wavelet Tipe IVa dan memanfaatkan operasi komparasi bilangan bulat kuantum melalui Quantum Fourier Transform (QFT). Pendekatan ini memungkinkan manipulasi langsung terhadap sinyal kuantum dengan kompleksitas komputasi linier terhadap panjang data. Seluruh rancangan algoritma dan transformasi diimplementasikan serta divalidasi menggunakan platform IBM Qiskit, dengan analisis meliputi jumlah gerbang kuantum, waktu tunda, dan kebutuhan qubit tambahan. Hasil penelitian ini memberikan kontribusi penting bagi pengembangan teori dan aplikasi transformasi wavelet integer, khususnya dalam pengolahan sinyal dan sistem keamanan informasi pada komputasi klasik maupun kuantum.
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Wavelet transforms are essential mathematical tools in signal processing due to their ability to represent signals efficiently in both the time and frequency domains simultaneously. However, conventional implementations based on floating-point arithmetic are prone to numerical errors and instability, particularly in systems with limited precision such as digital hardware. Algebraic approaches based on max-plus and its variants, which replace conventional arithmetic operations with maximization and addition, offer more stable, numerically error-free, and hardware-efficient systems. Wavelet transforms have been widely applied in various fields, including image processing and cryptography. With the rapid advancement of quantum computing, new challenges arise in constructing quantum wavelet transforms capable of operating in the quantum domain while preserving numerical stability and accuracy. This dissertation proposes a novel cryptographic algorithm that exploits integer-based wavelet transforms constructed via a lifting scheme over the min-max-plus algebra (MMPLS-IWavelet). The algorithm is designed using pairs of predict and update operators defined within the min-max-plus framework to achieve high encryption quality, strong key sensitivity, and optimal computational efficiency. Furthermore, a one-dimensional quantum integer wavelet transform, referred to as the QMP-Wavelet Type IVa, is developed based on the structure of the MP-Wavelet Type IVa and employs quantum integer comparison operations implemented through the Quantum Fourier Transform (QFT). This approach enables direct manipulation of quantum signals with computational complexity that is linear in the data length. All proposed algorithms and transforms are implemented and validated using the IBM Qiskit platform, with performance analyses covering quantum gate count, circuit latency, and additional qubit requirements. The results of this study provide significant contributions to the theoretical development and practical applications of integer wavelet transforms, particularly in signal processing and information security systems for both classical and quantum computing paradigms.

Item Type:	Thesis (Doctoral)
Uncontrolled Keywords:	komputasi kuantum, wavelet kuantum integer, wavelet, aljabar min-max-plus, lifting scheme
Subjects:	Q Science > QA Mathematics > QA159 Algebra Q Science > QA Mathematics > QA184 Algebra, Linear Q Science > QA Mathematics > QA403.3 Wavelets (Mathematics)
Divisions:	Faculty of Science and Data Analytics (SCIENTICS) > Mathematics > 44002-(S3) PhD Thesis
Depositing User:	Mohamad Ilham Dwi Firmansyah
Date Deposited:	04 Feb 2026 08:49
Last Modified:	04 Feb 2026 08:49
URI:	http://repository.its.ac.id/id/eprint/132142

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