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 number‑theoretic construct called the Goldbach ellipse and investigates its suitability for cryptographic purposes. Starting from the Goldbach conjecture, the authors define an ellipse around an even integer 2n by selecting two primes at distances j and k from 2n (with j fixed to 1 and k an odd integer greater than 1). The pair of primes is expressed as (2n‑m, 2n+km), and the integer m obtained for each admissible 2n forms the so‑called m‑sequence. For a concrete example with k = 5, Table 1 lists several even numbers and the corresponding m values; notably, even numbers that are multiples of k do not generate an ellipse, creating gaps in the sequence.

The m‑sequence is then mapped to a binary sequence (b‑sequence) by the rule: if m mod 4 = 1, output +1; if m mod 4 = 3, output ‑1. This conversion yields a sequence of 1’s and –1’s that exhibits strong statistical randomness. The authors compute the autocorrelation function C(i) = (1/N) ∑ b_j · b_{j+i} and show (Figure 2) that C(0)≈1 while C(i) for i≠0 is close to zero, indicating near‑ideal two‑valued autocorrelation. Consequently, long substrings of the b‑sequence are difficult to locate without exhaustive search, a property desirable for cryptographic key material.

To further assess structural complexity, the paper treats the original m‑sequence as a binary indicator of the parity of the number of Goldbach partitions (even → 0, odd → 1) and performs a sliding‑window analysis for n < 2000. Tables 2 and 3 present counts of all substrings of lengths 2 through 20, revealing that as substring length grows the distribution of counts becomes increasingly uniform. A comparative study between k = 1 (the Goldbach circle, a special case of the ellipse) and k = 5 shows that the circle’s substring counts are flatter, suggesting that larger k values introduce more irregularity and thus higher entropy.

Beyond statistical analysis, the authors propose a secure session‑key establishment protocol that leverages Goldbach partitions. A Certification Authority (CA) stores secret primes a (for Alice) and b (for Bob). When a session is requested, the CA computes n = a + b and selects an alternative partition p + q = n, where p will serve as the session key and q as an audit key. The CA encrypts p by XOR‑ing it with the hash of each user’s secret (p ⊕ h(a) to Alice, p ⊕ h(b) to Bob). Because only Alice knows a and only Bob knows b, each can recover p by reversing the XOR with their own hash value. The audit key q, retained by the CA, allows verification that p was indeed derived from the correct partition of n. Once recovered, p can seed a pseudo‑random number generator or be used to generate a d‑sequence for subsequent encrypted communication. The protocol includes suggestions for strengthening against replay attacks by incorporating additional random nonces.

In conclusion, the paper demonstrates that Goldbach ellipse‑derived m‑sequences possess high randomness, as evidenced by autocorrelation and substring‑distribution analyses, making them viable sources for cryptographic keys and indexing structures. Moreover, the inherent multiplicity of Goldbach partitions provides a fresh mathematical foundation for designing authentication and key‑exchange protocols, expanding the toolbox of number‑theoretic cryptography.