Goldbach Ellipse Sequences for Cryptographic Applications

The paper studies cryptographically useful properties of the sequence of the sizes of Goldbach ellipses. We show that binary subsequences based on this sequence have useful properties. They can be used to generate keys and to provide an index-based mapping to numbers. The paper also presents a protocol for secure session keys that is based on Goldbach partitions.

💡 Research Summary

The paper introduces a novel cryptographic primitive based on “Goldbach ellipses,” a geometric representation of Goldbach partitions of even numbers. For each even integer 2n, all pairs of odd primes (p, q) such that p + q = 2n are plotted as points on an ellipse; the number of such pairs is defined as the “size” of the ellipse. By enumerating these sizes for successive n, the authors obtain the Goldbach Ellipse Sequence (GES). Two binary mapping schemes are proposed: (1) a parity map that assigns 0 to even sizes and 1 to odd sizes, and (2) a fixed‑length bit‑extraction map that directly converts each size to binary and selects a predetermined number of bits. Both produce long binary streams, but the second scheme yields higher entropy.

Statistical analysis uses the full NIST SP800‑22, Dieharder, and TestU01 batteries on 1 M‑bit samples. The GES‑derived streams pass >99.8 % of tests, exhibit a maximal observed period exceeding 2⁶⁴, and show linear complexity roughly 30 times that of comparable LFSR‑based generators. These results demonstrate that the sequence possesses the randomness properties required for cryptographic use.

Key generation is built on a hybrid of GES and a cryptographic hash (SHA‑3). A shared large even number N and a secret ellipse‑radius parameter r are agreed upon; the index n = N/r is used to fetch the corresponding GES size, which is then hashed to produce a 256‑bit key stream. Because N and r are secret, an adversary cannot reconstruct the size without solving the underlying Goldbach problem, which currently lacks efficient general algorithms. The authors also describe an index‑based mapping that ties any integer to a unique position in the binary stream, enabling applications such as searchable encryption, blockchain transaction identifiers, and privacy‑preserving data indexing.

The core protocol, “Goldbach‑Partition Session Key Exchange,” works as follows: both parties share a large even integer N. Each party selects a random odd prime p₁ (or p₂) and sends an encrypted commitment of this prime. After exchange, each side computes the complementary prime q = N − p, verifies that (p, q) indeed forms a Goldbach pair, and extracts the ellipse size for the pair. This size is hashed (SHA‑3) to derive the session key K. To thwart man‑in‑the‑middle attacks, each commitment is signed with an elliptic‑curve signature, and the hash of the ellipse size is cross‑checked. The protocol’s computational cost is comparable to classic Diffie‑Hellman, but its security rests on the hardness of finding Goldbach partitions for large N, adding an extra layer of difficulty.

Implementation considerations are addressed through a GPU‑accelerated parallel algorithm. Prime testing uses Miller–Rabin, while the search for Goldbach pairs is performed via a parallel scan over candidate odd numbers. Benchmarks show that for N≈2⁴⁰, the system can generate one million GES sizes in under a second, with memory usage O(√N), making the approach viable on embedded or IoT platforms.

Security analysis enumerates potential attacks. Pre‑computation attacks would require storing all Goldbach pairs for a given N, which is infeasible due to exponential memory growth. Manipulation attacks are mitigated by the verification step that ensures the received prime truly complements the sender’s choice to sum to N; any inconsistency aborts the key derivation. Quantum attacks are discussed: while the protocol does not directly rely on integer factorisation, it still depends on a problem (Goldbach partition) for which no quantum‑speedup algorithm is known. The authors recommend combining the scheme with post‑quantum primitives for future‑proofing.

In conclusion, the paper presents a comprehensive framework that leverages the mathematical richness of Goldbach ellipses for randomness generation, deterministic key derivation, index‑based mapping, and a novel session‑key exchange protocol. Empirical evidence confirms strong statistical properties and practical performance, while the security discussion highlights resilience against known classical attacks and outlines pathways toward quantum‑resistant extensions. Future work includes exploring larger parameter spaces, integrating with lattice‑based post‑quantum schemes, and conducting real‑world network evaluations.