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All Journal Epsilon: Jurnal Matematika Murni dan Terapan CAUCHY: Jurnal Matematika Murni dan Aplikasi Specta Journal of Technology Proximal: Jurnal Penelitian Matematika dan Pendidikan Matematika Euler : Jurnal Ilmiah Matematika, Sains dan Teknologi LOSARI: Jurnal Pengabdian Kepada Masyarakat Journal of Fundamental Mathematics and Applications (JFMA) Jurnal Statistika dan Komputasi (STATKOM) Parameter: Jurnal Matematika, Statistika dan Terapannya
Yanuar Bhakti Wira Tama
Department Of Mathematics, Institut Teknologi Kalimantan, Balikpapan, East Kalimantan
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Algebraic Cryptanalysis of an Elliptic-Curve-Based Polyalphabetic Hill Cipher over Z/26Z Yanuar Bhakti Wira Tama; Sindi Lukmaini; Muhammad Firdhausi Fahmi
CAUCHY: Jurnal Matematika Murni dan Aplikasi Vol 11, No 2 (2026): CAUCHY: JURNAL MATEMATIKA MURNI DAN APLIKASI
Publisher : Mathematics Department, Maulana Malik Ibrahim State Islamic University of Malang

Show Abstract | Download Original | Original Source | Check in Google Scholar | DOI: 10.18860/cauchy.v11i2.45911

Abstract

This study revisits a Hill-type polyalphabetic cipher in which the 26 uppercase letters are represented by points of the elliptic curve y2 = x3 + 9x + 1 over F29. We prove that the selected curve group contains 26 elements and that B = (0,1) has order 26; hence the group is cyclic and isomorphic to Z26. Under the generator-coordinate map, the point-valued matrix transformation is therefore conjugate to the classical Hill transformation over Z26. This characterization yields explicit plaintext-based attacks. Under the stated fixed-key and public-mapping assumptions, m chosen plaintext blocks recover an m m key exactly. For the reported 4 4 key, a known-plaintext attack also recovers the key from the first four plaintext-ciphertext block pairs because the resulting plaintext matrix is invertible modulo 26. For the Indonesian Constitution benchmark, a reproducible experiment using 100 random invertible keys for each block size m = 2, 3, 4, 5 shows that mean ciphertext entropy increases from 4.5592 to 4.6769 bits per symbol and the index of coincidence approaches 1/26, while chosen-plaintext key recovery remains 100%. Thus, flatter symbol frequencies do not remove the linear weakness of the construction or provide elliptic-curve discrete-logarithm security.
Co-Authors Aditya Putra Pratama Amalia Kartika Ardila, Dwiki Jaya Desilya Lau Diva Rizqir Aurellia Fahmi, Muhammad Firdhausi Harli Andrika Lomo Kartika Nugraheni Levina Michella Muhammad Firdhausi Fahmi Muhammad Noor Ihsan Fadhilah Muhammad Rifqiandi Paulus Anthony Zejlstra Rangga Hermawa Retno Wahyu Dewanti Robby Robby Robby, Robby Simatupang, Syalam Ali Wira Dinata Sindi Lukmaini Sindi Lukmaini Syamsul Mujahidin Syamsul Mujahidin, Syamsul Winarni Winarni