Two novel symmetric multidimensional affine nested variations of the Hill Cipher are presented. The Hill Cipher is a block polygraphic substitution encryption scheme based on a linear transformation of plaintext characters into ciphertext characters. In the time since Hill first published his encryption scheme, variations, modifications, and improvements of theoretical and practical importance have been published every year indicating that the Hill Cipher is an active area of cryptography research. The first variation presented in this paper incorporated invertible key matrices of orders 2, 4, and 8 such that the matrix values of the 2×2 matrix rotate positions with each block of characters in a similar manner to the rotating letter wheels of a German Enigma Encoder, then results of the 2×2 key matrices output are passed to 4×4 key matrices, and 8x8 key matrix, 4×4 key matrices, and rotative-value 2×2 key matrices. The second variation is configured with invertible key matrices of orders 4, 8, and 16 without rotation of matrix values in a similar manner to the first variation. In both variations, plaintext characters of each block are operated on by exclusive-or (XOR) vectors prior to multiplication with the matrices to create the affine ciphers. Strengths, weaknesses, and other considerations are provided in the discussion. Two proposals are also argued with rationale for a more robust character set for encryption and the increase in modulus that the character set allows, and the possible advantages and disadvantages of affine XOR vectors.
Published in | Mathematics and Computer Science (Volume 9, Issue 3) |
DOI | 10.11648/j.mcs.20240903.11 |
Page(s) | 46-56 |
Creative Commons |
This is an Open Access article, distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution and reproduction in any medium or format, provided the original work is properly cited. |
Copyright |
Copyright © The Author(s), 2024. Published by Science Publishing Group |
Cryptography, Hill Cipher, Matrix Theory, Invertible Matrices
Object | Dimension | Number Required |
---|---|---|
Pseudo-Key XOR Row Vector | 1x2 | 12 |
Pseudo-Key XOR Row Vector | 1x4 | 2 |
Pseudo-Key XOR Row Vector | 1x8 | 3 |
Key Matrix | 2x2 | 8 |
Key Matrix | 4x4 | 4 |
Key Matrix | 8x8 | 1 |
State 0 | State 1 | State 2 | State 3 |
---|---|---|---|
Object | Dimension | Number Required |
---|---|---|
Pseudo-Key XOR Row Vector | 1x4 | 12 |
Pseudo-Key XOR Row Vector | 1x8 | 2 |
Pseudo-Key XOR Row Vector | 1x16 | 3 |
Key Matrix | 4x4 | 8 |
Key Matrix | 8x8 | 4 |
Key Matrix | 8x8 | 1 |
Modulus | nxn Matrix | Number of Possible Permutations in Matrix slots |
---|---|---|
26 | 2x2 Matrix | 26^{4}=456,976 |
29 | 2x2 Matrix | 29^{4}=707,281 |
191 | 2x2 Matrix | 191^{4}=1,330,863,361 |
191 | 4x4 Matrix | 191^{6}>3.137E36 |
191 | 8x8 Matrix | 191^{64}>9.685E145 |
191 | 16x16 Matrix | 191^{256}>8.801E583 |
Modulo p | |
---|---|
26 | 157,248 |
29 | 682,080 |
191 | 1,323,859,200 |
j^{th} Block of the i^{th} Matrix of order n | |
j^{th} Block of the i^{th} Character Raised to the j^{th}^{ }Power | |
XOR | Exclusive-or Function |
General Linear Group of degree 2 (2x2) Invertible Matrices Over the Integers, of Prime p. |
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APA Style
Coggins, P. E. (2024). Two Novel Multidimensional Affine Variations of the Hill Cipher. Mathematics and Computer Science, 9(3), 46-56. https://doi.org/10.11648/j.mcs.20240903.11
ACS Style
Coggins, P. E. Two Novel Multidimensional Affine Variations of the Hill Cipher. Math. Comput. Sci. 2024, 9(3), 46-56. doi: 10.11648/j.mcs.20240903.11
AMA Style
Coggins PE. Two Novel Multidimensional Affine Variations of the Hill Cipher. Math Comput Sci. 2024;9(3):46-56. doi: 10.11648/j.mcs.20240903.11
@article{10.11648/j.mcs.20240903.11, author = {Porter Eldridge Coggins}, title = {Two Novel Multidimensional Affine Variations of the Hill Cipher }, journal = {Mathematics and Computer Science}, volume = {9}, number = {3}, pages = {46-56}, doi = {10.11648/j.mcs.20240903.11}, url = {https://doi.org/10.11648/j.mcs.20240903.11}, eprint = {https://article.sciencepublishinggroup.com/pdf/10.11648.j.mcs.20240903.11}, abstract = {Two novel symmetric multidimensional affine nested variations of the Hill Cipher are presented. The Hill Cipher is a block polygraphic substitution encryption scheme based on a linear transformation of plaintext characters into ciphertext characters. In the time since Hill first published his encryption scheme, variations, modifications, and improvements of theoretical and practical importance have been published every year indicating that the Hill Cipher is an active area of cryptography research. The first variation presented in this paper incorporated invertible key matrices of orders 2, 4, and 8 such that the matrix values of the 2×2 matrix rotate positions with each block of characters in a similar manner to the rotating letter wheels of a German Enigma Encoder, then results of the 2×2 key matrices output are passed to 4×4 key matrices, and 8x8 key matrix, 4×4 key matrices, and rotative-value 2×2 key matrices. The second variation is configured with invertible key matrices of orders 4, 8, and 16 without rotation of matrix values in a similar manner to the first variation. In both variations, plaintext characters of each block are operated on by exclusive-or (XOR) vectors prior to multiplication with the matrices to create the affine ciphers. Strengths, weaknesses, and other considerations are provided in the discussion. Two proposals are also argued with rationale for a more robust character set for encryption and the increase in modulus that the character set allows, and the possible advantages and disadvantages of affine XOR vectors. }, year = {2024} }
TY - JOUR T1 - Two Novel Multidimensional Affine Variations of the Hill Cipher AU - Porter Eldridge Coggins Y1 - 2024/07/23 PY - 2024 N1 - https://doi.org/10.11648/j.mcs.20240903.11 DO - 10.11648/j.mcs.20240903.11 T2 - Mathematics and Computer Science JF - Mathematics and Computer Science JO - Mathematics and Computer Science SP - 46 EP - 56 PB - Science Publishing Group SN - 2575-6028 UR - https://doi.org/10.11648/j.mcs.20240903.11 AB - Two novel symmetric multidimensional affine nested variations of the Hill Cipher are presented. The Hill Cipher is a block polygraphic substitution encryption scheme based on a linear transformation of plaintext characters into ciphertext characters. In the time since Hill first published his encryption scheme, variations, modifications, and improvements of theoretical and practical importance have been published every year indicating that the Hill Cipher is an active area of cryptography research. The first variation presented in this paper incorporated invertible key matrices of orders 2, 4, and 8 such that the matrix values of the 2×2 matrix rotate positions with each block of characters in a similar manner to the rotating letter wheels of a German Enigma Encoder, then results of the 2×2 key matrices output are passed to 4×4 key matrices, and 8x8 key matrix, 4×4 key matrices, and rotative-value 2×2 key matrices. The second variation is configured with invertible key matrices of orders 4, 8, and 16 without rotation of matrix values in a similar manner to the first variation. In both variations, plaintext characters of each block are operated on by exclusive-or (XOR) vectors prior to multiplication with the matrices to create the affine ciphers. Strengths, weaknesses, and other considerations are provided in the discussion. Two proposals are also argued with rationale for a more robust character set for encryption and the increase in modulus that the character set allows, and the possible advantages and disadvantages of affine XOR vectors. VL - 9 IS - 3 ER -