Elliptic Curve Pairing

The core property of a pairing is bilinearity. For points P, Q on curve E1 and E2 and scalars a, b, the pairing satisfies e(aP, bQ) = e(P, Q)^(a*b). This allows moving exponents between groups, which is impossible with standard elliptic curve operations.

• Key Use: Enables verification of complex relationships without revealing secrets.
• Example: In BLS signatures, it allows aggregation of many signatures into one verifiable unit.

A useful pairing must be non-degenerate, meaning e(P, Q) ≠ 1 for some inputs, ensuring the output is meaningful. Modern pairing-friendly curves (like BN254 and BLS12-381) are engineered for computational efficiency.

• Optimized Fields: They operate over specific finite fields that balance security and the cost of the pairing operation.
• Practical Impact: Enables zk-SNARKs and identity-based encryption to be practical on blockchains.

Most blockchain applications use Type 3 pairings, which operate on two distinct source groups G1 and G2 where there is no efficiently computable isomorphism between them. This asymmetry provides stronger security assumptions.

• G1 & G2: Typically, G1 has points with coordinates in the base field (shorter), and G2 in an extension field.
• Advantage: More secure and efficient than symmetric (Type 1) pairings for decentralized settings.

Pairings are the cryptographic engine behind zk-SNARK proving systems. They allow a verifier to check a quadratic arithmetic program (QAP) proof by evaluating a single pairing equation.

• Role: The pairing performs the final elliptic curve check on encoded polynomials, verifying the proof is valid without learning anything else.
• Foundation: Used in Zcash's original Sprout protocol and Ethereum's pre-compiles for verifiable computation.

The BLS signature scheme leverages pairings to aggregate multiple signatures into a single, constant-sized signature. This is possible due to bilinearity.

• Process: Individual signatures (points on G1) can be added together. The verifier uses a pairing to check the aggregate against the aggregate public key (on G2).
• Blockchain Use: Critical for scaling consensus in Ethereum 2.0 (attestation aggregation) and reducing on-chain data.

Pairings enable Identity-Based Encryption (IBE) and signatures, where a public key can be any string (like an email address). A trusted authority uses a master secret to generate corresponding private keys.

• Mechanism: The pairing allows derivation and verification based on a public master public key.
• Application Concept: Simplifies key management, though requires a trusted setup, making it more common in enterprise than permissionless blockchain contexts.

Beyond specific applications, pairings enable the construction of powerful cryptographic building blocks used across protocols. These include:

• Identity-Based Encryption (IBE): Allows encryption using an identity (e.g., an email) as a public key.
• Functional Encryption: Grants decryption power for specific functions of the encrypted data.
• Threshold Cryptography: Enables distributed key generation and signing, enhancing security for multi-party computations like distributed validators.

Not all elliptic curves support efficient pairings. Special pairing-friendly curves are required, each with different security and performance trade-offs. The choice of curve is a fundamental design decision for any protocol using this cryptography.

• BN254 (Barreto-Naehrig): Widely used in early ZK projects (e.g., Zcash, early zkSync).
• BLS12-381: The current industry standard for new projects (Ethereum 2.0, Filecoin, Chia) due to better security margins.
• Performance: Pairing operations are computationally intensive, making proof generation a bottleneck that hardware acceleration aims to solve.

Implementing pairing operations is notoriously complex, and even mathematically correct code can leak secret information through side-channel attacks. These include:

• Timing attacks: Execution time variations can reveal secret scalar values.
• Power analysis: Fluctuations in power consumption during computation can be analyzed to extract keys.
• Fault injection: Introducing hardware glitches can cause incorrect computations that reveal secrets.

Mitigation requires constant-time algorithms and hardware-level countermeasures.

Many pairing-based systems, especially zk-SNARKs, require a trusted setup to generate public parameters (a Common Reference String or Structured Reference String). This ceremony produces a toxic waste—secret random numbers that must be destroyed. If compromised, an attacker could generate fraudulent proofs. The security of the entire system depends on the integrity of this one-time event, making multi-party computation (MPC) ceremonies with many participants the standard for reducing trust.

Pairing security rests on hard mathematical problems like the Elliptic Curve Discrete Logarithm Problem (ECDLP) and extensions like the Bilinear Diffie-Hellman (BDH) assumption. The primary long-term risk is cryptographic obsolescence:

• Quantum threat: Pairing-friendly curves are vulnerable to Shor's algorithm, necessitating migration to post-quantum cryptography.
• Algorithmic advances: New mathematical breakthroughs could weaken the underlying problems faster than anticipated.

Systems must be designed with upgradeability in mind.

Not all elliptic curves are suitable for efficient pairings. Special pairing-friendly curves like BN254, BLS12-381, and BLS12-377 are used, each with its own security considerations:

• Subgroup attacks: Ensuring operations occur within the correct cryptographic subgroups of the curve to avoid small-subgroup confinement attacks.
• Curve rigidity: Some curves (e.g., BN254) were generated with unexplained parameters, raising concerns about hidden vulnerabilities or backdoors. Modern curves like BLS12-381 prioritize rigid, transparent generation to avoid this.

Even a perfectly secure pairing implementation can be undermined by errors in the broader cryptographic protocol. Key risks include:

• Signature malleability: In BLS signatures, proofs must be verified to prevent forgery via rogue key attacks.
• Proof system soundness: In zk-SNARKs, the security of the Quadratic Arithmetic Program (QAP) and polynomial commitment scheme is paramount; a flaw here breaks the entire zero-knowledge property.
• Randomness failures: Using non-cryptographic or predictable randomness during proof or key generation can completely compromise security.

The formal mathematical property that defines a pairing. A bilinear map is a function that takes two inputs from separate groups and maps them to an element in a third group, with the property of bilinearity. This means the map is linear in each of its arguments.

• Formal Definition: For a map e: G₁ × G₂ → G_T, and scalars a, b: e(aP, bQ) = e(P, Q)^(a*b).
• Core Utility: This property enables checking multiplicative relationships in the exponent, which is impossible with standard ECC alone.

The fundamental hard problem underlying most asymmetric cryptography. Given a group element h = g^x, finding the integer exponent x is the DLP. In pairing-friendly elliptic curves, this problem must be hard in all three groups (G₁, G₂, G_T) for security.

• Pairing Impact: While pairings rely on the hardness of DLP, they also introduce new problem assumptions like the Bilinear Diffie-Hellman (BDH) and Decisional Bilinear Diffie-Hellman (DBDH) problems, which are used to prove security of pairing-based schemes.

Not all elliptic curves support efficient, secure pairings. Pairing-friendly curves are specially constructed families with the necessary properties.

• Key Properties: Require a small embedding degree and prime order subgroups that facilitate the bilinear map.
• Common Families: BN254 (Barreto-Naehrig), BLS12-381 (Barreto-Lynn-Scott), and BLS12-377 are standard curves used in Ethereum, Zcash, and other blockchain systems. Each offers different trade-offs between security and performance.