Symmetric Cryptography via Dirichlet Convolution
Abstract
We discuss a new symmetric-key cipher for digital data encryption. Its implementation is fast, memory efficient, and resilient against classical, AI-assisted, and quantum attacks. Let x, k, and c be elements of a finite abelian group G with operation + and neutral element $0$. Suppose one is given ciphertext c=x+k. Retrieving the plaintext x=c−k from the ciphertext is trivial when the key k is known. Without the key, however, recovering x amounts to a blind search. Even if an efficient criterion were available for identifying the plaintext among all group elements, the average number of trials would remain prohibitively large when the group is sufficiently large. The challenge in constructing a practical cryptosystem is therefore to identify a sufficiently rich group G, together with efficient algorithms that enable an immersion of \emph{real digital data} into G and efficient evaluation of the group operations. The proposed cipher is based on generalized Dirichlet convolution over finite fields. We investigate its security against plaintext attacks from an information-theoretic perspective. The nonlocal structure of Dirichlet convolution naturally permits plaintext salting, providing protection against plaintext attacks while allowing controlled key reuse. We establish perfect masking properties over finite fields and prove that the resulting cipher achieves information-theoretic security under key reuse when the underlying arithmetic is carried out in 𝔽p with p>2. For arithmetic over 𝔽2, some information leakage remains, leading to a finite key-reuse budget.
Related
- Existence of Admissible Subsolutions to the Dirichlet Problem for Symmetric Augmented k-Hessian Type Equations in Bounded Domains
- Multivariate Cryptography-Based Anonymous Certificate Scheme
- Perfect Combinatorial Structures in Coding Theory and Cryptography
- Fast tensor transforms and ring-valued orthogonal matrices: an application to cryptography
- The convolution product is generally not an open map
BibTeX
@misc{sowa2026symmetric,
title = {{Symmetric Cryptography via Dirichlet Convolution}},
author = {Artur Sowa and Elias Hassani Sangani and Francis Bui and Grant Harris and Jonathan Norton},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.14040},
eprint = {2610.14040},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.14040},
}
RIS
TY - UNPB
TI - Symmetric Cryptography via Dirichlet Convolution
AU - Artur Sowa
AU - Elias Hassani Sangani
AU - Francis Bui
AU - Grant Harris
AU - Jonathan Norton
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.14040
UR - https://arxiv.org/abs/2610.14040
AB - We discuss a new symmetric-key cipher for digital data encryption. Its implementation is fast, memory efficient, and resilient against classical, AI-assisted, and quantum attacks. Let x, k, and c be elements of a finite abelian group G with operation + and neutral element $0$. Suppose one is given ciphertext c=x+k. Retrieving the plaintext x=c−k from the ciphertext is trivial when the key k is known. Without the key, however, recovering x amounts to a blind search. Even if an efficient criterion were available for identifying the plaintext among all group elements, the average number of trials would remain prohibitively large when the group is sufficiently large. The challenge in constructing a practical cryptosystem is therefore to identify a sufficiently rich group G, together with efficient algorithms that enable an immersion of \emph{real digital data} into G and efficient evaluation of the group operations. The proposed cipher is based on generalized Dirichlet convolution over finite fields. We investigate its security against plaintext attacks from an information-theoretic perspective. The nonlocal structure of Dirichlet convolution naturally permits plaintext salting, providing protection against plaintext attacks while allowing controlled key reuse. We establish perfect masking properties over finite fields and prove that the resulting cipher achieves information-theoretic security under key reuse when the underlying arithmetic is carried out in 𝔽_p with p>2. For arithmetic over 𝔽_2, some information leakage remains, leading to a finite key-reuse budget.
ER -