A Deterministic Cryptographic Prime Generation Chain over Monogenic Cubic Number Fields and their Generalizations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910200912412672 |
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| author | Jakhar, Anuj Kalwaniya, Ravi |
| author_facet | Jakhar, Anuj Kalwaniya, Ravi |
| contents | Generating primes is a fundamental problem in modern cryptography. Deterministic primality tests work well for special integers such as Mersenne or Proth primes, but these forms are quite restrictive. In this paper, we give a direct method to construct new primes from known ones. Starting with a seed prime $q \equiv 1 \pmod{3}$, we construct an integer $N \equiv 1 \pmod{3}$ satisfying $(2N + 1)^2 \equiv -3 \pmod{q}$. We then prove that $N$ is prime using the structure of monogenic pure cubic fields $K = \mathbb{Q}(\sqrt[3]{d})$. The resulting test requires only a single modular exponentiation and runs in $\tilde{\mathcal{O}}(\log^2 N)$ time. Finally, we show how this construction extends to pure number fields of arbitrary prime degree. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_07581 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Deterministic Cryptographic Prime Generation Chain over Monogenic Cubic Number Fields and their Generalizations Jakhar, Anuj Kalwaniya, Ravi Number Theory 11A51, 11Y11, 11R21, 94A60 Generating primes is a fundamental problem in modern cryptography. Deterministic primality tests work well for special integers such as Mersenne or Proth primes, but these forms are quite restrictive. In this paper, we give a direct method to construct new primes from known ones. Starting with a seed prime $q \equiv 1 \pmod{3}$, we construct an integer $N \equiv 1 \pmod{3}$ satisfying $(2N + 1)^2 \equiv -3 \pmod{q}$. We then prove that $N$ is prime using the structure of monogenic pure cubic fields $K = \mathbb{Q}(\sqrt[3]{d})$. The resulting test requires only a single modular exponentiation and runs in $\tilde{\mathcal{O}}(\log^2 N)$ time. Finally, we show how this construction extends to pure number fields of arbitrary prime degree. |
| title | A Deterministic Cryptographic Prime Generation Chain over Monogenic Cubic Number Fields and their Generalizations |
| topic | Number Theory 11A51, 11Y11, 11R21, 94A60 |
| url | https://arxiv.org/abs/2605.07581 |