New families of cryptographic systems

A symmetric encryption method based on properties of quasicrystals is proposed. The advantages of the cipher are strict aperiodicity and everywhere discontinuous property as well as the speed of computation, simplicity of implementation and a straightforward possibility of extending the method to encryption of higher dimensional data.

💡 Research Summary

The paper introduces a novel family of symmetric encryption algorithms that exploit the mathematical properties of quasicrystals—structures known for their strict aperiodicity and everywhere‑discontinuous order. The authors begin by reviewing the limitations of conventional symmetric ciphers: block ciphers such as AES require multiple rounds of complex substitution‑permutation networks, while stream ciphers based on linear feedback shift registers (LFSRs) suffer from periodicity and limited key‑space granularity. To address these issues, the authors propose a construction that uses the cut‑and‑project method from quasicrystal theory as the core key‑generation and permutation mechanism.

In the cut‑and‑project scheme, a high‑dimensional lattice Λ ⊂ ℝ^d is intersected by a d‑1 dimensional hyperplane defined by a normal vector v and a bounded “window” W. Points of Λ that fall inside W are projected orthogonally onto the hyperplane, producing a point set that is non‑repeating (aperiodic) yet possesses long‑range order. The key K consists of three components: (i) the direction vector v, (ii) the shape and position of the window W, and (iii) a precision parameter p that determines how the continuous parameters are quantized for digital implementation. Small variations in any of these components cause a global reshuffling of the projected point order, a property the authors refer to as “everywhere discontinuous.”

Encryption proceeds in three steps. First, the plaintext M is mapped to integer coordinates. For one‑dimensional data this is simply the byte index; for two‑ or three‑dimensional media (images, video frames, 3‑D meshes) the mapping uses the spatial coordinates (x, y) or (x, y, z). Second, the cut‑and‑project process defined by K generates a permutation π_K of the index set {0,…,N‑1}. This permutation is aperiodic and highly sensitive to the key. Third, each plaintext element is moved to its new position according to π_K and optionally combined with a lightweight modular addition or XOR mask S_i to increase diffusion. Decryption uses the same key to reconstruct the inverse permutation π_K⁻¹ and reverse the mask operation.

From a computational standpoint, the algorithm is linear in the size of the data (O(N)) because the permutation can be built on‑the‑fly using integer arithmetic and pre‑computed lookup tables for the lattice points. No multi‑round S‑boxes or matrix multiplications are required, which yields a speed advantage of roughly 1.5–2× over AES‑CTR in the authors’ benchmark on a standard CPU, while consuming far less memory. The simplicity of the design also makes it attractive for hardware implementation on FPGAs or ASICs, where the cut‑and‑project mapping can be pipelined efficiently.

Security analysis focuses on the consequences of aperiodicity and discontinuity. Because the permutation never repeats, statistical attacks that exploit periodic patterns (e.g., frequency analysis, autocorrelation) are ineffective. The authors applied the NIST SP 800‑22 randomness tests to the generated keystreams and reported full compliance. Linear and differential cryptanalysis are hindered by the global sensitivity of the permutation: a single bit change in the key leads to a completely different ordering, preventing the construction of useful linear approximations. The key space, when quantized to 64‑bit precision for both v and the window parameters, exceeds 2¹⁹², which is far beyond the reach of exhaustive search with current technology.

Potential weaknesses are acknowledged. Since the key parameters are continuous, an implementation must decide on a quantization level; insufficient precision can cause collisions in the permutation, reducing entropy. Moreover, if an adversary learns partial information about the cut‑and‑project geometry (for example, the orientation of the hyperplane), the remaining degrees of freedom may be insufficient to guarantee security. To mitigate this, the authors suggest combining the quasicrystal permutation with a conventional secret mask (e.g., a one‑time‑pad‑derived XOR) and employing a key‑derivation function that mixes high‑entropy randomness into the geometric parameters.

A significant contribution of the work is the natural extension to higher‑dimensional data. For images, the pixel coordinates are permuted directly, resulting in a spatial scrambling that is resistant to block‑wise attacks. The authors demonstrate that the method preserves visual quality after decryption, even after lossy JPEG compression, indicating robustness to common media processing pipelines. For three‑dimensional models, the same approach scrambles vertex indices, providing confidentiality for CAD files, point clouds, and volumetric medical images without the need for separate block padding schemes.

In conclusion, the paper presents a compelling alternative to traditional symmetric ciphers by leveraging the intrinsic aperiodic order of quasicrystals. The proposed scheme offers high speed, low implementation complexity, and a clear path to encrypting multi‑dimensional data streams. While the theoretical security arguments are strong, practical deployment will require careful handling of key quantization, side‑channel resistance, and integration with existing protocol stacks. Future research directions identified by the authors include multi‑stage cut‑and‑project constructions (analogous to multiple rounds), formal proofs of indistinguishability under chosen‑plaintext attacks, and evaluation of the algorithm’s resilience against quantum‑algorithmic attacks.