PRIMES 2.0: CM-based Primality Proving

• PRIMES 2.0 is a deterministic primality-proving framework based on elliptic curves with complex multiplication, certifying primes in structured sequences.
• It leverages Hilbert class polynomials and CM theory to construct curves with predictable orders, reducing proof to verifying a point-multiplication relation.
• The method achieves quasi-quadratic time efficiency and has been used to certify record-sized primes, including a 1,392,250-bit prime from a -15 CM sequence.

Searching arXiv for the specified paper and closely related work on deterministic primality proving with elliptic curves and CM. PRIMES 2.0 denotes a deterministic primality-proving framework based on elliptic curves with complex multiplication (CM), introduced by Abatzoglou, Silverberg, Sutherland, and Wong in “A framework for deterministic primality proving using elliptic curves with complex multiplication” (Abatzoglou et al., 2014). The framework applies to integers lying in special sequences and determines primality or compositeness in deterministic quasi-quadratic time. Its central mechanism is to construct CM elliptic curves modulo the candidate integer, predict the group order via CM theory, and certify primality through a point-multiplication relation of Pomerance type. In the formulation associated with (Abatzoglou et al., 2014), the method was used to find large primes, including a $1\,392\,250$-bit prime arising from a $-15$-CM sequence.

1. Conceptual basis in CM theory

The framework is built on the arithmetic of imaginary quadratic orders and the endomorphism theory of elliptic curves. Let $K=\mathbf{Q}(\sqrt{D})$ be an imaginary quadratic field of discriminant $D<0$, and let $O=O_K$ be its full ring of integers, or an order of conductor $f\mid D$. An elliptic curve $E$ over a number field $M$ is said to have CM by $O$ if there is an isomorphism of rings

$$\operatorname{End}_M(E)\simeq O\subset \operatorname{End}_{\overline M}(E).$$

In the formulation given for PRIMES 2.0, this is also summarized as

$-15$0

A second foundational object is the Hilbert class polynomial. For the full ring of integers $-15$1, the polynomial is

$-15$2

Its degree is the class number $-15$3. Once $-15$4 is known, by complex-analytic or CRT-based methods, it can be reduced modulo an arbitrary $-15$5 to obtain candidate $-15$6-invariants in $-15$7 or, more generally, in $-15$8 (Abatzoglou et al., 2014).

These ingredients place CM theory at the center of the primality test. The class polynomial supplies algebraic moduli data, while the CM endomorphism structure supplies a priori information about Frobenius and hence the anticipated group order of the elliptic curve. This suggests that the method is not merely an elliptic-curve analogue of a probable-prime test, but a certificate-producing framework in which the arithmetic of the chosen CM field controls both construction and verification.

2. Algorithmic structure

The input is an odd integer $-15$9 whose primality is to be decided. The procedure described in (Abatzoglou et al., 2014) is organized into a fixed sequence of steps.

First, one picks a suitable discriminant $K=\mathbf{Q}(\sqrt{D})$0 of small class number $K=\mathbf{Q}(\sqrt{D})$1, and computes or retrieves the Hilbert class polynomial

$K=\mathbf{Q}(\sqrt{D})$2

Second, one reduces $K=\mathbf{Q}(\sqrt{D})$3 modulo $K=\mathbf{Q}(\sqrt{D})$4 and attempts to find a root

$K=\mathbf{Q}(\sqrt{D})$5

If no such root exists, then $K=\mathbf{Q}(\sqrt{D})$6 must be composite, because a prime $K=\mathbf{Q}(\sqrt{D})$7 always supports exactly $K=\mathbf{Q}(\sqrt{D})$8 roots. If a root does exist, one proceeds with one of the resulting $K=\mathbf{Q}(\sqrt{D})$9-invariants.

Third, from $D<0$0 one constructs an elliptic curve

$D<0$1

using the standard bijection between $D<0$2-invariants and curve coefficients, for example Tate normal form or the classical formulas relating $D<0$3 and $D<0$4. By construction,

$D<0$5

and, in the formulation stated for the framework, in fact $D<0$6 (Abatzoglou et al., 2014).

Fourth, if $D<0$7 is prime, the CM method identifies the absolute Frobenius endomorphism $D<0$8 through

$D<0$9

for some integer $O=O_K$0. Consequently,

$O=O_K$1

can be computed entirely in $O=O_K$2 time by solving the norm equation in $O=O_K$3, or by explicit evaluation of modular polynomials of small degree.

Fifth, one chooses a rational prime factor $O=O_K$4 of $O=O_K$5 that exceeds $O=O_K$6; the framework states that one is guaranteed to exist when $O=O_K$7 is prime. Let $O=O_K$8 be a random or small base point in $O=O_K$9. Under the assumption that $f\mid D$0 is prime one has

$f\mid D$1

whereas for any prime divisor $f\mid D$2, if $f\mid D$3 were composite, the reduction modulo $f\mid D$4 would satisfy $f\mid D$5, so that

$f\mid D$6

Thus the single relation $f\mid D$7 in $f\mid D$8 certifies primality. If the test fails, the method can often extract a nontrivial factor of $f\mid D$9 from the false relation (Abatzoglou et al., 2014).

The output is either “$E$0 is composite” or “$E$1 is prime” together with the certificate consisting of the curve $E$2, the point $E$3, and the integer $E$4.

3. Complexity and computational profile

The framework states that all of its steps can be carried out in quasi-quadratic time,

$E$5

where $E$6 suppresses factors polynomial in $E$7 (Abatzoglou et al., 2014). In the notation used in the exposition of the method, this is also written as

$E$8

The complexity decomposition given for the principal subroutines is as follows.

Substep	Complexity
Precompute $E$9	$M$0
Find $M$1	$M$2
Curve construction and “CM-jump”	$M$3
Point multiplication	$M$4

Here the dependence on the class number $M$5 is explicit in the class-polynomial and root-finding stages. The requirement that $M$6 have small class number is therefore not incidental; it is a structural condition on the practical efficiency of the framework. This suggests that the “special sequences” to which the method applies are precisely those for which an appropriate CM discriminant with tractable class polynomial data can be integrated into the primality proof strategy.

Verification is correspondingly efficient. In the large-prime example described in the source, verifying the relation $M$7 requires only $M$8 bit-operations (Abatzoglou et al., 2014). Within the intended setting, the certificate is therefore compact in the computational sense that its validation remains quasi-quadratic in the bit length.

4. The $M$9-CM example and explicit large-prime certification

A concrete illustration appears in §6 of (Abatzoglou et al., 2014). The authors consider the sequence

$O$0

They show how to apply the framework with class number $O$1, so that the Hilbert class polynomial has degree $O$2.

After sieving and then applying the CM test, they obtain that for

$O$3

the integer $O$4 is a $O$5-bit prime, approximately $O$6 decimal digits. The corresponding certificate consists of an elliptic curve $O$7 with CM by $O$8, a base point $O$9, and an integer $$\operatorname{End}_M(E)\simeq O\subset \operatorname{End}_{\overline M}(E).$$0 with

$$\operatorname{End}_M(E)\simeq O\subset \operatorname{End}_{\overline M}(E).$$1

such that $$\operatorname{End}_M(E)\simeq O\subset \operatorname{End}_{\overline M}(E).$$2 (Abatzoglou et al., 2014).

The source characterizes this prime as the largest prime currently known whose primality cannot feasibly be proved using classical methods. It also states that the $$\operatorname{End}_M(E)\simeq O\subset \operatorname{End}_{\overline M}(E).$$3-bit $$\operatorname{End}_M(E)\simeq O\subset \operatorname{End}_{\overline M}(E).$$4-CM prime is currently the largest prime whose primality cannot be proved by any non-elliptic-curve method. In the context of the framework, this example functions as more than an existence proof. It demonstrates that the method is capable of producing rigorous primality certificates for extremely large integers outside the classical families ordinarily associated with specialized tests.

5. Relation to earlier primality-proving methods

The framework is explicitly contrasted with both classical special-form tests and general deterministic or heuristic algorithms (Abatzoglou et al., 2014). Classical Lucas–Lehmer or Pepin-type tests run in $$\operatorname{End}_M(E)\simeq O\subset \operatorname{End}_{\overline M}(E).$$5 time, where $$\operatorname{End}_M(E)\simeq O\subset \operatorname{End}_{\overline M}(E).$$6 denotes the bit length, but they apply only to very special forms such as Fermat, Mersenne, and Proth numbers. By contrast, the CM framework applies to any $$\operatorname{End}_M(E)\simeq O\subset \operatorname{End}_{\overline M}(E).$$7 in a “CM-sequence,” namely a setting in which $$\operatorname{End}_M(E)\simeq O\subset \operatorname{End}_{\overline M}(E).$$8 is integral in $$\operatorname{End}_M(E)\simeq O\subset \operatorname{End}_{\overline M}(E).$$9, and yields a full primality certificate.

The paper also positions the framework relative to AKS and Atkin–Morain ECPP. It states that the first general deterministic polynomial-time algorithm, AKS, runs in $-15$00, while the fastest known proof-found algorithms based on Atkin–Morain with random CM curves have heuristic cost approximately $-15$01. Against this background, PRIMES 2.0 achieves provable $-15$02 on its special sequences (Abatzoglou et al., 2014).

These comparisons identify the method’s domain of strength with precision. It is not presented as a universal replacement for general-purpose primality proving, nor as a mere special-case analogue of Lucas–Lehmer. Rather, it occupies an intermediate position: more structured and sequence-dependent than general algorithms, but vastly more flexible than tests tied to a handful of classical number forms. A plausible implication is that the framework’s significance lies in converting algebraic structure from CM theory into deterministic complexity guarantees that remain unavailable for generic inputs.

6. Scope, certification logic, and interpretive boundaries

The certification logic is Pomerance-style in the sense stated in the source: primality is reduced to a single annihilation relation $-15$03 on an elliptic curve whose order has been predicted from CM data (Abatzoglou et al., 2014). The certificate is explicit and consists of the curve, the point, and the integer $-15$04. This distinguishes the method from empirical prime-generating constructions and from tests that certify only probable primality.

A common misconception would be to treat the framework as a general deterministic quasi-quadratic primality test for arbitrary integers. The source does not make that claim. It states instead that the framework determines primality or compositeness of integers that lie in special sequences, and that it applies to $-15$05 in a “CM-sequence.” Its deterministic quasi-quadratic complexity is therefore sequence-conditional rather than universal (Abatzoglou et al., 2014).

Another potential misconception is that the method depends only on finding a suitable elliptic curve modulo $-15$06. In fact, the decisive ingredient is the CM structure: the discriminant $-15$07, the Hilbert class polynomial $-15$08, the existence of a compatible $-15$09-invariant modulo $-15$10, and the Frobenius relation

$-15$11

Without that algebraic framework, the predicted order $-15$12 and the certified choice of $-15$13 would not be available in the form required by the method.

In summary, PRIMES 2.0 is a deterministic CM-based framework for proving the primality of integers in specially structured sequences, with quasi-quadratic complexity, explicit certificates, and a concrete record-setting application to a $-15$14-bit $-15$15-CM prime (Abatzoglou et al., 2014). Its defining contribution is the synthesis of Hilbert class polynomial methods, CM Frobenius determination, and elliptic-curve point-multiplication certificates into a provable primality-proving pipeline.