Crypto's Frontier - Zero-Knowledge Proofs & The Quantum-Resistant World of Lattice-Based Cryptography

June 14, 2023 — #Cyber Security #CS Basics

Welcome to the cutting edge, crypto-pioneers! Having journeyed through the established lands of symmetric and asymmetric cryptography, digital signatures, and formal attack models, we now set our sights on more advanced and rapidly evolving frontiers. In this post, we'll take a more detailed look at the fascinating world of Zero-Knowledge Proofs (ZKPs) and then venture into Lattice-Based Cryptography, a leading contender in the quest for security in the age of quantum computers.

These topics represent active areas of research and are shaping the future of privacy and security.

Zero-Knowledge Proofs (ZKPs) and Protocols: Proving Without Revealing, Advanced Edition

We previously introduced the magical concept of Zero-Knowledge Proofs: where a Prover can convince a Verifier that they know a secret or that a statement is true, without revealing the secret itself or any information beyond the statement's validity. The core properties remain:

Completeness: If the statement is true, an honest Prover can convince an honest Verifier.
Soundness: If the statement is false, a cheating Prover cannot (except with negligible probability) convince an honest Verifier.
Zero-Knowledge: The Verifier learns nothing other than the fact that the statement is true.

The Ali Baba cave analogy gave us an intuition. Now let's explore some of the different flavors and characteristics of ZKPs.

Types of Zero-Knowledge Proofs:

Interactive vs. Non-Interactive ZKPs (NIZKs):
- Interactive ZKPs: These require multiple rounds of communication (a "conversation") between the Prover and the Verifier. The Verifier often sends random challenges to the Prover, and the Prover's responses demonstrate their knowledge. The Ali Baba cave is an example of an interactive proof. While secure, the need for interaction can be a limitation in some applications.
- Non-Interactive ZKPs (NIZKs): The Prover can generate a single proof string that can be verified by anyone at any time without further interaction with the Prover. This is highly desirable for many real-world systems. Achieving NIZKs often involves:
  - The Random Oracle Model: Assuming access to a truly random function (often approximated by a cryptographic hash function). The Fiat-Shamir heuristic is a common technique to convert certain interactive proofs into non-interactive ones using a random oracle.
  - Common Reference String (CRS) Model: Prover and Verifier have access to a shared, trusted string of random data generated during a setup phase.
Arguments of Knowledge vs. Proofs of Knowledge:
- "Proof" usually implies computational soundness (a cheating prover with unlimited power can't fool the verifier).
- "Argument" implies computational soundness only against computationally bounded provers (which is usually what we care about in practice). Many practical ZKPs are technically "arguments."
ZK-SNARKs (Zero-Knowledge Succinct Non-Interactive Argument of Knowledge):
- Succinct: The generated proofs are very small in size, and verification is very fast, regardless of the complexity of the statement being proven. This is a huge advantage.
- Non-Interactive: As above.
- Argument of Knowledge: The Prover demonstrates knowledge of a secret "witness" that makes the statement true.
- Trusted Setup: Many (though not all) SNARK constructions require an initial trusted setup phase to generate public parameters (like a CRS). If the randomness used in this setup phase is compromised or not properly destroyed, it could allow for the creation of false proofs. This is a significant consideration.
- Applications: Highly popular in privacy-preserving cryptocurrencies (e.g., Zcash) and for scaling blockchain solutions by verifying computations off-chain.
ZK-STARKs (Zero-Knowledge Scalable Transparent Argument of Knowledge):
- Scalable: Proof generation and verification times scale quasi-linearly (or even poly-logarithmically for verification) with the complexity of the computation being proven, making them suitable for very large computations.
- Transparent: They do not require a trusted setup phase. The public parameters are generated using publicly verifiable randomness. This is a major advantage over many SNARKs.
- Security: Often rely on simpler cryptographic assumptions like collision-resistant hash functions, potentially offering stronger post-quantum security properties.
- Proof Size: Proofs can be larger than SNARKs but are improving.

Building Blocks and Underlying Concepts (Very High Level):

The actual construction of ZKP protocols can be quite complex, often drawing from:

Hard Mathematical Problems: Such as the Discrete Logarithm Problem, factoring, or problems on elliptic curves (especially for SNARKs using pairings).
Commitment Schemes: Allowing a party to commit to a value while keeping it hidden, with the ability to reveal it later.
Polynomial Commitments: Committing to a polynomial and then being able to prove evaluations of that polynomial at specific points without revealing the whole polynomial.
Probabilistic Checkable Proofs (PCPs): A more theoretical concept that underpins some advanced ZKP constructions.

More Applications of ZKPs:

Beyond the initial examples of authentication, ZKPs are finding applications in:

Privacy-Preserving Credentials: Proving you have a valid driver's license without revealing your name or address.
Verifiable Computation / Outsourced Computation: Proving that a computation performed by an untrusted party (e.g., a cloud server) was done correctly without needing to re-execute it or reveal the inputs.
Ensuring Fairness in Protocols: Verifying steps in a multi-party computation were performed honestly.
Blockchain and Cryptocurrencies: Enhancing privacy (Zcash, Monero partially) and scalability (rollups).

Lattice-Based Cryptography: Preparing for the Quantum Era

One of the most significant looming challenges in cryptography is the advent of large-scale quantum computers. While not fully realized yet, theoretical quantum algorithms (most notably Shor's algorithm) can efficiently solve the mathematical problems that underpin much of our current public-key infrastructure:

Shor's Algorithm: Can factor large integers quickly, breaking RSA.
Shor's Algorithm (adapted): Can solve the Discrete Logarithm Problem (DLP) and the Elliptic Curve Discrete Logarithm Problem (ECDLP) efficiently, breaking Diffie-Hellman, DSA, ECDH, and ECDSA.

This potential "quantum apocalypse" for current cryptography has spurred intense research into Post-Quantum Cryptography (PQC) – cryptographic algorithms that are believed to be secure against attacks by both classical and quantum computers. Lattice-based cryptography is one ofthe most promising families of PQC candidates.

Introduction to Lattices

Conceptually, a lattice is a regular, infinite grid of points in $n$ -dimensional space. More formally, given a set of linearly independent basis vectors $B = \{\mathbf{b}_1, \mathbf{b}_2, \ldots, \mathbf{b}_n\}$ in $\mathbb{R}^m$ (usually $m \ge n$ ), the lattice $L(B)$ generated by $B$ is the set of all integer linear combinations of these basis vectors: $L(B) = \{ \sum_{i=1}^n x_i \mathbf{b}_i \mid x_i \in \mathbb{Z} \}$

Think of a 2D lattice as the points on an infinite sheet of graph paper, or the arrangement of atoms in a perfect crystal.

Hard Problems in Lattices

The security of lattice-based cryptosystems relies on the presumed computational difficulty of certain problems defined on lattices. Even for quantum computers, these problems are believed to be hard.

Shortest Vector Problem (SVP): Given an arbitrary basis for a lattice $L$ , find the shortest non-zero vector in $L$ .
- Finding an exact solution is NP-hard. Finding an approximate solution within a certain factor is also believed to be hard.
Closest Vector Problem (CVP): Given a lattice $L$ and a target vector $\mathbf{t}$ (not necessarily in $L$ ), find the lattice vector $\mathbf{v} \in L$ that is closest to $\mathbf{t}$ .
- Also NP-hard.
Learning With Errors (LWE): This is a very influential problem for modern lattice-based crypto.
- Informal Description: Suppose you have a secret vector $\mathbf{s}$ . You are given a set of "almost" linear equations. Specifically, you get many samples of the form $(\mathbf{a}_i, b_i)$ , where $\mathbf{a}_i$ is a random vector, and $b_i \approx \langle \mathbf{a}_i, \mathbf{s} \rangle$ (the inner product). The "almost" comes from adding a small, random error term $e_i$ to the inner product: $b_i = \langle \mathbf{a}_i, \mathbf{s} \rangle + e_i \pmod q$ (for some modulus $q$ ).
- The LWE Problem: Given these samples $(\mathbf{a}_i, b_i)$ , find the secret vector $\mathbf{s}$ .
- This problem is believed to be hard, even for quantum computers. Its hardness can be related to worst-case lattice problems like SVP.
Ring-LWE (RLWE): A variant of LWE that operates over polynomial rings, which allows for more efficient constructions (smaller keys, faster operations) while retaining strong security guarantees. Many practical lattice-based schemes are built on RLWE.

Why Lattices for Post-Quantum Cryptography (PQC)?

Quantum Resistance: As mentioned, core lattice problems are believed to be resistant to attacks by Shor's algorithm and other known quantum algorithms.
Versatility: Lattice-based constructions can be used to build a wide array of cryptographic primitives:
- Public-Key Encryption (PKE) schemes
- Key Encapsulation Mechanisms (KEMs)
- Digital Signature schemes
- Fully Homomorphic Encryption (FHE) schemes are almost exclusively built using lattices.
- Pseudorandom functions, identity-based encryption, and more.
Security Proofs: Many lattice-based schemes have strong security proofs based on reductions from worst-case lattice problems (meaning breaking the scheme is at least as hard as solving a known hard lattice problem in the worst case).
Efficiency: While initial schemes were less practical, modern lattice-based cryptography (especially those based on Ring-LWE and Module-LWE) can be very efficient in terms of key sizes, computation speed, and bandwidth, making them competitive with traditional schemes.

Examples of Lattice-Based Schemes

The NIST (National Institute of Standards and Technology) Post-Quantum Cryptography Standardization project has been a major driver in this area. As of late 2023/early 2024, several lattice-based schemes were selected for standardization or as strong candidates:

For Public-Key Encryption/KEMs:
- CRYSTALS-Kyber: A Key Encapsulation Mechanism (KEM) based on Module-LWE. Selected by NIST for standardization.
For Digital Signatures:
- CRYSTALS-Dilithium: A digital signature scheme based on Module-LWE. Selected by NIST for standardization.
- Falcon: Another digital signature scheme based on NTRU lattices (related to Short Integer Solution - SIS problem). Selected by NIST for standardization.
NTRU Encrypt: One of the oldest and most studied lattice-based encryption schemes.

Current Status and Future

Lattice-based cryptography is a vibrant and rapidly developing field. With the NIST PQC standardization process moving forward, we are beginning to see the first generation of standardized quantum-resistant algorithms, many of which are lattice-based. The transition to these new cryptographic standards will be a significant undertaking over the coming years to ensure long-term data security against future quantum threats.

Key Takeaways

Our journey into advanced cryptography reveals tools and concepts that are pushing the boundaries of what's possible in terms of security and privacy:

Zero-Knowledge Proofs offer powerful ways to prove assertions without revealing underlying secrets, with variants like ZK-SNARKs and ZK-STARKs enabling compact and efficient proofs for complex statements, crucial for privacy and scalability in various applications.
Lattice-Based Cryptography stands as a leading candidate for post-quantum cryptography, offering schemes built on mathematical problems believed to be hard even for quantum computers. Its versatility and efficiency make it a cornerstone for future security standards.

These advanced topics highlight the continuous evolution of cryptography, driven by new mathematical insights, emerging computational threats (like quantum computers), and the ever-increasing demand for stronger security and privacy in our digital lives.