What is MAX

MAX (Mathematical Authentication eXchange) is an ecosystem that brings together:

• Number theory (especially prime numbers),
• Post-quantum cryptography standardized by NIST,
• Deterministic software for real devices and servers.

The goal is to build mathematical digital identities (MAX IDs) that do not depend on emails, usernames, or passwords, but on non-invertible numerical structures generated directly on the user’s device.

MAX Prime Theory: prime numbers, modular signatures and enrichment

At the core of the ecosystem lies a new theory of prime numbers, the MAX Prime Theory, derived from the study of a pair of twin quadratic polynomials. These polynomials:

• generate prime numbers in a deterministic way (same input → same primes),
• show an enrichment factor relative to the classical Bateman–Horn estimate (in many ranges, up to 7–10× more primes than expected),
• allow the construction of prime sets with the same modular signature by applying modular filters to the initial parameter.

These results are documented in the papers published on Zenodo:

• Twin Quadratic Polynomials (link)
• Modular Sets and Modular Signatures (link)
• Infinitude of Twin Quadratic Polynomials (link)

The key idea is that, assuming Bateman–Horn, prime numbers can be “catalogued” by modular signatures, suggesting a possible hidden order inside the set of primes.

Mathematical Applied to eXcryption: Merkle Tree and non-invertibility

Building on these polynomials, the project MAX — Mathematical Applied to eXcryption explores how to turn these structures into deterministic and non-invertible cryptographic keys.

The approach constructs a Merkle Tree from about 300 prime numbers generated by the MAX engine and demonstrates that, given only the Merkle Root, it is practically impossible to recover:

• the internal primes,
• the initial parameters,
• and especially the user’s secret phrases.

This Root becomes the logical foundation for a universal mathematical identity: the MAX ID.

MX²: the cryptographic container that links everything together

To connect the mathematical layer with modern cryptography, the protocol MX² (MAX to eXcryption Container) was created. It uses well-established algorithms:

• Argon2id to derive a secure seed from two 80-character phrases,
• ChaCha20-DRBG to obtain a deterministic stream of bits,
• XChaCha20-Poly1305 for authenticated encryption and decryption,
• HKDF to separate different purposes (MAX ID, Login, Chat, Vault, Signature).

The result is a portable container that stores encrypted phrases and proves that all keys are generated locally. The MX² Rust code is available on GitHub (link) and can be independently verified.

MAX ID, post-quantum keys and the MAX App

From the same MX² seed, the MAX engine generates:

• the prime numbers used in the MAX ID Merkle Tree,
• SPHINCS+ keys for post-quantum signatures (login, registration, file signing),
• FrodoKEM keys for key exchange in MAX Chat,
• MX² symmetric keys for local encryption (MAX Vault) and messages.

Everything happens deterministically on the device:

• phrases are never sent to the server,
• private keys are never stored in the cloud,
• the server only sees encrypted packets or SPHINCS+ signatures to verify.

From this architecture emerges the MAX App, which integrates:

• MAX Login for post-quantum QR-based access,
• MAX Chat for E2E messages with FrodoKEM + MX²,
• MAX Prime for deterministic prime generation (customizable),
• MAX Vault for protecting files and notes locally,
• MAX Signature for SPHINCS+ signing of files and documents.

Everything is connected through a single mathematical identity (MAX ID) and a single cryptographic container (MX²), without ever exposing the user’s secrets.