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All Journal CAUCHY: Jurnal Matematika Murni dan Aplikasi
Muhammad Firdhausi Fahmi
Department of Mathematics, Institut Teknologi Kalimantan, Balikpapan, East Kalimantan
Mathematics

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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 Sindi Lukmaini Yanuar Bhakti Wira Tama