This paper presents a novel zero-knowledge proof scheme constructed upon polynomial multi-variable rings. The core claim is to design a scheme that significantly enhances proof efficiency and security while addressing the computational complexity bottlenecks prevalent in existing approaches. The proposed mechanism leverages the unique properties of multi-variable polynomial rings to establish a streamlined proof and verification process, minimizing the risk of information leakage. Unlike traditional zero-knowledge proofs that often rely heavily on large number arithmetic, this scheme utilizes polynomial operations, leading to potentially improved performance. This work contributes to the field of cryptographic primitives by offering a new design paradigm rooted in algebraic structures, potentially unlocking avenues for more efficient and practical zero-knowledge proofs. The key contributions are a novel construction and a theoretical analysis demonstrating the security and efficiency gains. The scheme operates by encoding the statement to be proven as a polynomial equation in a multi-variable ring, and the prover generates a proof that allows the verifier to confirm the equation's validity without learning any information beyond the proof itself. This design aims to provide a more scalable and practical solution for zero-knowledge proof applications.
No quasiperfect number ($σ(n) = 2n + 1$) is known, and its number of distinct prime factors is bounded below; the bound $ω\ge 7$ of Hagis--Cohen has stood since 1982, obstructed by a family of ``deep leaves'' on which pure enumeration cannot terminate (the scan bound for the intermediate prime reaches $8 \times 10^8$, and the exponent dimension is unbounded). This paper clears that obstruction with three lemmas at the level of secondary-school algebra --- a discriminant criterion, a quadratic-residue sieve, and a multilinear resolver --- which eliminate the last prime $q$, the intermediate prime $p$, and the exponent dimension respectively, turning a non-terminating search into a finite decision. On this basis all 381 stems of ``$3 \mid n$ and $ω= 7$'' and their $79{,}751{,}212$ deep leaves are eliminated, with the ledger closing exactly and zero solutions throughout; the complementary case ``$3 \nmid n$ and $ω= 7$'' collapses to a single stem, which is eliminated directly, so that the proof does not rest on any theorem whose published record we could not independently re-verify. Together with the machine elimination of $ω\le 6$ (Theorem B4), this yields the main theorem: \emph{any quasiperfect number, if one exists, satisfies $ω(n) \ge 8$} --- the first advance of this bound since Hagis--Cohen 1982. The full computation has been reproduced by seven separately closed ledgers across three algorithmic architectures (CPU and GPU), all with zero solutions and exact ledger closure, and the lemma layer is formalized in Lean (259 theorems, zero \texttt{sorry}). A 2023 preprint of Zemann reported the same bound by a different computation; our audit of its public code found a coverage gap of 35 feasible exponents, so the elimination given here is, to our knowledge, the first complete proof. Code, ledgers, and Lean sources are available from the authors.
In 2021, Masson, Sanso, and Zhang introduced the Bandersnatch curve associated to the BLS12-381 pairing-friendly curve, an elliptic curve designed for zero-knowledge proofs requiring circuits with a curve arithmetic. This type of curve is useful for privacy-preserving protocols, and more generally for succinct validity proof using pairing-based SNARKs. An embedded curve is defined over a field whose order is the group order of its associated curve. In this way, the pairing-friendly curve is used to express a zero-knowledge proof (such as a SNARK) of a statement taking place on the embedded curve. Contrary to the previous embedded curves (such as CØCØ, JubJub), Bandersnatch was built with the complex multiplication (CM) method, in order to ensure a very small discriminant (-8, whose magnitude is small), and thus efficient scalar multiplication thanks to the GLV technique. The algorithm provided by Masson, Sanso, and Zhang for searching this type of curves requires computation of Hilbert class polynomials, making the search of curve slow. It was not known whether Bandersnatch was an exceptional curve or whether comparable curves exist, of larger discriminants. This paper highlights the technicalities of the CM method already in use in the 90s to generate curve parameters of chosen order. This old technique allows revisiting the curve search of Bandersnatch, providing a dramatic speed-up improvement. This paper presents two algorithms: one to generate embedded elliptic curves of SNARK-friendly elliptic curves, with a variable discriminant; a second to generate families (parameterized by polynomials) with a fixed discriminant. When the (negative) discriminant is -3 modulo 4, it is possible to obtain a prime-order curve, and form a cycle. To illustrate this, we apply the technique first to generate more embedded curves like Bandersnatch with BLS12-381, such as a curve of discriminant -6673027, defining a plain twist-secure cycle. We also comment on the scarcity of Bandersnatch-like CM curves, and recall that with this generic algorithm, it is only a question of core-hours to find them. Second, we show the link between a paper of Ben Smith in 2015 and the work of Dai, Lin, Zhao, and Zhou in 2023, obtaining prime-order parameterized families of embedded curves of fixed discriminant, such as -3 for BLS and KSS18 curves. With KSS16 curves, the discriminant -4 is also possible (the curve has an even order). The technique can work with any KSS, Scott–Guillevic, Gasnier–Guillevic, or other fixed-discriminant parameterized family of pairing-friendly curves. This paper provides a more general point of view on embedded curves such as Bandersnatch, putting into perspective the works of Masson, Sanso, and Zhang, and Sanso and El Housni. The Python/SageMath scripts are available at https://gitlab.inria.fr/zk-curves/cm-embedded-curves/.