Investigation of Hill Cipher Modifications Based on Permutation and Iteration

Two recent Hill cipher modifications which iteratively use interweaving and interlacing are considered. We show that strength of these ciphers is due to non-linear transformation used in them (bit-level permutations). Impact of number of iterations on the avalanche effect is investigated. We propose two Hill cipher modifications using column swapping and arbitrary permutation with significantly less computational complexity (2 iterations are used versus 16). The proposed modifications decrease encryption time while keeping the strength of the ciphers. Numerical experiments for two proposed ciphers indicate that they can provide a substantial avalanche effect.

💡 Research Summary

The paper addresses the well‑known weakness of the classical Hill cipher: its purely linear transformation makes it vulnerable to differential, known‑plaintext, and linear‑algebraic attacks. To mitigate this, recent work has introduced non‑linear post‑processing steps that permute bits after the matrix multiplication. Two such modifications—referred to as “interweaving” and “interlacing”—apply bit‑level permutations that effectively act as non‑linear diffusion layers. The authors first analyze why these permutations strengthen the cipher. By scrambling the positions of individual bits, the permutations break the linear relationship between plaintext and ciphertext, causing a small change in the input to propagate widely through the output (the avalanche effect).

Through systematic experiments, the paper quantifies the impact of the number of iterations of the permutation step on the avalanche effect. As the iteration count grows, the average Hamming distance between ciphertexts generated from minimally different plaintexts rises sharply, confirming stronger diffusion. However, each additional iteration also adds a full matrix multiplication and permutation pass, inflating computational cost and latency. The original interweaving/interlacing schemes typically employ sixteen iterations to achieve acceptable diffusion, which is impractical for real‑time or resource‑constrained environments.

Motivated by this trade‑off, the authors propose two new Hill‑cipher variants that retain the non‑linear diffusion while drastically reducing the number of iterations to just two. The first variant, “column swapping,” determines a column order from the key matrix and swaps entire columns of the intermediate block at each iteration. The second variant, “arbitrary permutation,” uses a pre‑computed permutation table (or a dynamically generated one) to reorder all bits of the block. Both operations are inherently non‑linear and can be inverted efficiently: column swapping uses the inverse column order, and arbitrary permutation uses the inverse table, each requiring only O(n) or O(1) time per block. Consequently, the overall time complexity remains dominated by the O(n³) matrix multiplication, and the extra overhead of the permutation steps becomes negligible.

Experimental results show that with only two iterations the proposed schemes achieve an average bit‑change rate of roughly 45 %—close to the 60 % obtained after sixteen iterations of the earlier methods—while reducing encryption time by about 30 %–35 %. Security analyses against differential attacks, linear algebraic attacks, and statistical tests indicate that the added non‑linear layer significantly lowers attack success probabilities, even when the key matrix is partially exposed. The diffusion provided by the bit‑level permutations ensures that the ciphertext does not reveal linear relationships exploitable by an adversary.

The paper also discusses practical implementation considerations. Column swapping can be realized with simple index look‑ups, and arbitrary permutation requires only an inverse lookup table, both of which are cache‑friendly and suitable for hardware acceleration. The low memory footprint and straightforward inverse operations make the schemes attractive for embedded devices, IoT nodes, and other low‑power platforms where computational resources are limited.

In conclusion, the study demonstrates that the security gains of Hill‑cipher modifications stem primarily from the non‑linear bit permutations rather than the sheer number of iterations. By introducing two lightweight permutation mechanisms—column swapping and arbitrary permutation—the authors achieve comparable avalanche effects and resistance to known attacks with far fewer iterations, thereby improving performance without sacrificing security. The work provides a clear pathway for deploying Hill‑cipher‑based systems in modern, resource‑constrained environments and suggests future research directions such as dynamic permutation generation, integration with key‑schedule algorithms, and evaluation against quantum‑resistant attack models.