Paper 2026/1104

The ABC of Symmetric Primitives over Integer Rings: Milk Before Meat

Abstract

Designing a secure symmetric-key cipher over a vector space over a field $\mathbb F_{p^n}^t$ is well known and understood by the cryptographic community. Even if the attacks are continuously improving, our current understanding regarding the design and security of the majority of the symmetric-key primitives has not fundamentally changed in the last 20 years. How does this picture change when we move to an integer ring $\mathbb Z_{p^n}^t$? Although the question is easy to state, it turns out to be far harder to answer. Indeed, there is a significant difference between the arithmetics of $\mathbb F_{p^n}^t$ and $\mathbb Z_{p^n}^t$ and attack vectors do not apply/translate directly between the two. As a case in point, a few ciphers have already been designed over integer rings, yet their initial versions have already been broken. In this paper, we lay the foundations for a more rigorous approach to designing ciphers over integer rings, noting that this is not only of theoretical interest, but also has concrete applications. We analyze how existing statistical and algebraic attacks will behave for these ciphers and also present new attacks that take into account that not all functions over integer rings admit a polynomial representation. Based on this, we discuss possible design strategies, in which we analyze the security effect of having/not having polynomial S-boxes. In particular, we introduce new properties for the non-polynomial S-boxes that measure their resistance against the attacks presented in this paper. Finally, we discuss how to design such non-polynomial S-boxes, presenting two concrete constructions, and one based on the "digit manipulation".