PreprintCC BY 4.0arXiv
Fast tensor transforms and ring-valued orthogonal matrices: an application to cryptography
Abstract
We present a generalization of the Fast Fourier and fast Walsh transform algorithms to tensor products of n arbitrary q× q matrices over a commutative ring, reducing the cost of applying such a tensor product from q2n to nqn ring operations. We then give an explicit construction of square orthogonal matrices over a commutative unitary ring, starting from a prescribed first row, and specialize this construction to ℤ/256ℤ. Combining these two ingredients, we propose a symmetric cryptosystem in which the secret key determines~n orthogonal matrices over ℤ/256ℤ whose tensor product encrypts a block of qn bytes; encryption and decryption both exploit the fast tensor product algorithm. We give three examples of implementations.
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BibTeX
@misc{peyriere2026fast,
title = {{Fast tensor transforms and ring-valued orthogonal matrices: an application to cryptography}},
author = {Jacques Peyrière},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.14782},
eprint = {2609.14782},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.14782},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Fast tensor transforms and ring-valued orthogonal matrices: an application to cryptography
AU - Jacques Peyrière
PY - 2026
DA - 2026/09/13
PB - arXiv
DO - 10.48550/arxiv.2609.14782
UR - https://arxiv.org/abs/2609.14782
AB - We present a generalization of the Fast Fourier and fast Walsh transform algorithms to tensor products of n arbitrary q× q matrices over a commutative ring, reducing the cost of applying such a tensor product from q^2n to nq^n ring operations. We then give an explicit construction of square orthogonal matrices over a commutative unitary ring, starting from a prescribed first row, and specialize this construction to ℤ/256ℤ. Combining these two ingredients, we propose a symmetric cryptosystem in which the secret key determines~n orthogonal matrices over ℤ/256ℤ whose tensor product encrypts a block of q^n bytes; encryption and decryption both exploit the fast tensor product algorithm. We give three examples of implementations.
ER -