Briefing
The fundamental challenge in post-quantum cryptography is designing zero-knowledge proof systems that are both quantum-safe and practically efficient, specifically requiring proofs of knowledge for short vectors in lattice-based schemes without incurring massive overhead. The Lantern protocol solves this by introducing a novel, direct method for proving the small Euclidean norm of a vector, leveraging polynomial product proofs to embed the inner product as a hidden constant coefficient. This mechanism eliminates the need for complex, coefficient-by-coefficient range proofs, resulting in a system that produces proofs up to three times smaller than previous lattice-based arguments. The single most important implication is the immediate practical deployability of post-quantum privacy primitives, enabling a new generation of succinct, quantum-resistant decentralized applications.
Context
The academic community has long faced a critical trade-off in building quantum-safe cryptography ∞ while lattice-based assumptions (like Module-LWE and Module-SIS) offer resilience against quantum computers, their associated zero-knowledge proof systems have historically been inefficient. The core theoretical limitation centered on proving that a committed vector possesses a “short” Euclidean norm ∞ a non-linear statement crucial for security ∞ without revealing the vector itself. Prior approaches required cumbersome techniques, such as proving the smallness of every vector coefficient or converting to the Chinese Remainder Theorem (CRT) representation, which resulted in proof sizes that were too large for scalable blockchain or privacy-preserving applications.
Analysis
The Lantern protocol’s core mechanism is a paradigm shift in how lattice vector norms are proved. Instead of proving the smallness of individual coefficients, the protocol proves knowledge of the vector’s inner product with itself, which directly relates to its Euclidean norm. This is achieved by observing that the inner product of two vectors can be constructed to appear as the constant term of a polynomial product involving the vectors’ polynomial representations. The protocol then utilizes a polynomial product proof system, which allows the prover to commit to the polynomials and demonstrate the constant coefficient’s value while keeping the other coefficients (and thus the underlying vector) hidden.
This is coupled with an economical “approximate range proof” to ensure the vector’s norm is small over the integers, not just modulo q. This direct approach bypasses the complex, multi-step proofs of prior art, establishing a more succinct and computationally lighter cryptographic primitive.
Parameters
- Proof Size Reduction ∞ 2x to 3x smaller. A very brief, simple explanation of what it is.
- Basic Proof Size ∞ 13KB. The concrete size of a basic zero-knowledge proof generated by the Lantern protocol.
- Hardness Assumption ∞ Module-SIS and Module-LWE. The foundational lattice problems that secure the protocol against quantum attacks.
Outlook
This foundational research opens new avenues for deploying post-quantum security across decentralized systems. The Lantern framework is designed for black-box integration, meaning it can be immediately plugged into advanced cryptographic constructions such as verifiable encryption, ring signatures, and privacy-preserving identity systems. In the next three to five years, this efficiency gain will be critical for enabling truly private and quantum-resistant decentralized finance (DeFi) and verifiable computation at scale. Future research will focus on optimizing the polynomial commitment scheme further and exploring its application in lattice-based verifiable delay functions (VDFs) to secure decentralized randomness generation.
