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 $\mathbb{F}_p$ with $p>2$. For arithmetic over $\mathbb{F}_2$, some information leakage remains, leading to a finite key-reuse budget.