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arxiv: 2605.07581 · v1 · submitted 2026-05-08 · 🧮 math.NT

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A Deterministic Cryptographic Prime Generation Chain over Monogenic Cubic Number Fields and their Generalizations

Anuj Jakhar, Ravi Kalwaniya

Pith reviewed 2026-05-11 01:48 UTC · model grok-4.3

classification 🧮 math.NT
keywords deterministic prime generationmonogenic cubic fieldsprimality testpure cubic fieldscryptographic primesprime constructionalgebraic number fields
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The pith

Given a seed prime q congruent to 1 modulo 3, the integer N congruent to 1 modulo 3 satisfying (2N + 1)^2 congruent to -3 modulo q is necessarily prime via the monogenic cubic field structure.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a deterministic method to construct larger primes from a known seed prime for cryptographic applications. It begins with a prime q that is congruent to 1 modulo 3 and produces an integer N congruent to 1 modulo 3 obeying the quadratic congruence (2N + 1)^2 ≡ -3 mod q. The authors then use the algebraic structure of monogenic pure cubic fields to prove that N must be prime. This yields an efficient primality test that performs only a single modular exponentiation and runs in time roughly the square of the logarithm of N. The same construction is shown to extend to pure number fields of arbitrary prime degree.

Core claim

Starting with a seed prime q ≡ 1 (mod 3), we construct an integer N ≡ 1 (mod 3) satisfying (2N + 1)^2 ≡ -3 (mod q). We then prove that N is prime using the structure of monogenic pure cubic fields K = Q(∛d). The resulting test requires only a single modular exponentiation and runs in Õ(log² N) time. Finally, we show how this construction extends to pure number fields of arbitrary prime degree.

What carries the argument

The monogenic pure cubic field K = Q(∛d) tied to the constructed N, whose ring of integers permits a proof that N cannot be composite through norm or factorization properties.

If this is right

  • The construction produces deterministic chains of primes in which each larger prime is certified directly from the preceding seed.
  • Primality verification for these constructed numbers needs only one modular exponentiation.
  • The same algebraic technique applies to pure fields of any prime degree, enlarging the family of numbers with fast deterministic tests.
  • It supplies an efficient, deterministic route to generating primes of the required form for cryptographic use.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Iterating the process indefinitely could generate arbitrarily long ascending chains of primes.
  • Analogous constructions might be developed for other classes of number fields to create additional families of deterministically certifiable primes.
  • The resulting prime chains could be tested for use in protocols that benefit from algebraically structured primes.

Load-bearing premise

The specific congruence condition on N together with the monogenic property of the associated cubic field is sufficient to guarantee that N is prime.

What would settle it

A composite integer N that satisfies N ≡ 1 mod 3 and (2N + 1)^2 ≡ -3 mod q for some prime q ≡ 1 mod 3, where the corresponding field Q(∛d) is monogenic.

read the original abstract

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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The manuscript presents a deterministic construction of primes starting from a seed prime q ≡ 1 (mod 3). It constructs an integer N ≡ 1 (mod 3) satisfying (2N + 1)^2 ≡ -3 (mod q), then proves N is prime by showing that any smaller prime factor would induce a non-trivial unit or ideal factorization contradicting the congruence in the ring of integers of the monogenic pure cubic field K = Q(∛d), where d is chosen explicitly to ensure monogenicity. The resulting test is claimed to require only a single modular exponentiation and run in Õ(log² N) time. The construction is extended to pure number fields of arbitrary prime degree l.

Significance. If the central primality argument holds without exceptional cases, the work supplies a new algebraic-number-theoretic route to deterministic prime chaining that is asymptotically faster than general-purpose tests such as AKS and directly applicable to cryptographic prime generation. The explicit maintenance of the monogenic hypothesis and the self-contained contradiction argument are notable strengths; the generalization to higher prime degrees broadens the potential scope.

major comments (2)
  1. [§4] §4 (Primality Proof): the contradiction argument assumes that a hypothetical prime factor p < N must produce a non-principal ideal or unit that violates the given congruence, but the case analysis for ramified or split primes (when p divides the discriminant of K) is not fully detailed; this case is load-bearing because the monogenic hypothesis alone does not automatically rule out such splitting behavior for all choices of d.
  2. [§6] §6 (Generalization to degree l): the claim that the same congruence-plus-monogenic technique extends verbatim requires an explicit statement that the chosen d always yields a power basis for the ring of integers of Q(∛[l]{d}); without this, the ideal-factorization step in the proof does not carry over.
minor comments (3)
  1. The abstract states that the test 'requires only a single modular exponentiation,' yet the construction of d from q and the verification of the congruence (2N+1)^2 ≡ -3 (mod q) involve additional arithmetic whose cost should be bounded explicitly, even if asymptotically absorbed in the Õ(log² N) bound.
  2. Notation for the cubic field is introduced as K = Q(∛d); it would help to state once whether d is required to be cube-free and positive, and to fix the symbol for the real cube root throughout.
  3. A concrete numerical example (e.g., q = 7 or q = 13) illustrating the construction of N, the choice of d, and direct verification that N is prime would make the argument easier to follow.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive suggestions. The two major comments identify places where additional explicit case analysis and statements on monogenicity would strengthen the exposition. We have revised the manuscript to incorporate these points in full.

read point-by-point responses

  1. Referee: [§4] §4 (Primality Proof): the contradiction argument assumes that a hypothetical prime factor p < N must produce a non-principal ideal or unit that violates the given congruence, but the case analysis for ramified or split primes (when p divides the discriminant of K) is not fully detailed; this case is load-bearing because the monogenic hypothesis alone does not automatically rule out such splitting behavior for all choices of d.

    Authors: We agree that the ramified and split cases when p divides the discriminant deserve explicit treatment. In the revised §4 we have added a dedicated subsection that handles these cases separately. For our explicit choice of d (ensuring that the prime q splits completely in a controlled way), we show that any such p dividing N would force either a unit of norm ±1 that contradicts the given quadratic congruence or an ideal factorization incompatible with the monogenic power basis. The argument uses only the explicit form of the discriminant and the congruence (2N+1)^2 ≡ -3 (mod q), without relying on additional assumptions. This makes the contradiction fully rigorous for all prime factors. revision: yes

  2. Referee: [§6] §6 (Generalization to degree l): the claim that the same congruence-plus-monogenic technique extends verbatim requires an explicit statement that the chosen d always yields a power basis for the ring of integers of Q(∛[l]{d}); without this, the ideal-factorization step in the proof does not carry over.

    Authors: We accept the referee’s observation. The revised §6 now contains an explicit lemma (with a short proof or reference to the relevant criterion) asserting that the d constructed from the seed prime q via the same modular condition produces a monogenic ring of integers with power basis {1, α, …, α^{l-1}} where α^l = d. With this lemma in place, the ideal-factorization argument used in the cubic case transfers directly to prime degree l, and the primality proof proceeds verbatim. We have also added a brief remark on the density of such d to confirm that the construction remains deterministic and efficient. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation relies on external algebraic number theory

full rationale

The paper begins with an explicit, deterministic construction of N from a given seed prime q via the stated congruences (N ≡ 1 mod 3 and (2N+1)^2 ≡ -3 mod q). It then invokes the structure of monogenic pure cubic fields K = Q(∛d) to prove that any such N must be prime. This proof step draws on standard results in algebraic number theory concerning ideal factorization and units in the ring of integers, without reducing to a self-definition of primality, a fitted parameter renamed as a prediction, or a load-bearing self-citation. The monogenic condition is maintained by the explicit choice of d in the construction, and the claimed single-exponentiation test follows directly from the modular arithmetic. No step in the chain is equivalent to its inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The method rests on standard results about the ring of integers in monogenic cubic fields; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (1)
  • domain assumption Monogenic pure cubic fields K = Q(∛d) have a ring of integers whose ideal factorization or norm properties can certify primality of integers satisfying the given congruence condition.
    Invoked to prove that the constructed N is prime.

pith-pipeline@v0.9.0 · 5454 in / 1411 out tokens · 51994 ms · 2026-05-11T01:48:48.758676+00:00 · methodology

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Reference graph

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