Proof of knowledge

As explained in Elliptic Curve Cryptography for Developers

A proof of knowledge is a cryptographic proof that demonstrates a prover possesses a witness satisfying a specified statement.

A proof of knowledge demonstrates that a prover possesses a private witness satisfying a public statement, without revealing the witness. Noninteractive zero-knowledge proofs use public data, a common reference string (CRS), and a public proof; verification checks consistency and aims to be short while keeping the proof small. section 1.3.4 section 19.7

A SNARK-style construction organizes the proof around a quadratic arithmetic program (QAP):

• Represent computation as addition and multiplication gates whose inputs and outputs are QAP wires. section 19.7
• Separate public statement values from private witness values and permute the statement wires into the order required by the CRS. section 19.5.4
• Compute line values, cross-term coefficients, and the h(x) coefficients used to encode the QAP relation. section 19.5.4 section 19.7
• Combine CRS points with these coefficients to construct proof points A, B, and C, then save public values and proof points as a record. section 19.5.4

A medical-record proof record

A contrived record uses the medicine number and dose as public statement values while keeping the patient number in the witness. With inputs 2036, 1700000, and 49, the implementation computes QAP wires, derives cross terms, constructs A, B, and C through elliptic-curve operations, and writes the statement and points to a SNARK record file. section 19.5.4

Figure 19.10 Assignment of values for each record in QAP example

Use the wire-assignment diagram to trace how each record value becomes a public statement, private witness, or derived QAP value. section 19.5.4

How Elliptic Curve Cryptography for Developers covers it

How deep it goes

Builds a QAP-based SNARK proof through wire evaluation, coefficient computation, elliptic-curve multiplication, and record serialization, but defers verifier code.

Examples

Uses a medicine number, dose, patient-dependent witness, QAP wires, cross terms, proof points A/B/C, and a serialized record file.

What it assumes

Assumes familiarity with elliptic curves and pairings while introducing public statements, witnesses, wires, QAP gates, and CRS roles.

What you can do after

Build a SNARK-style proof record from QAP parameters, CRS data, public inputs, private witness values, and pairing-friendly curve operations.

How it connects

Connects zero-knowledge proofs, SNARKs, QAPs, witnesses, public statements, common reference strings, and elliptic-curve pairings.

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Related concepts

• zk-SNARKincludes · A zk-SNARK is a zero-knowledge, succinct, non-interactive argument of knowledge that enables efficient verification of a prover’s knowledge of a computation’s secret witness without revealing it.
• zero-knowledge proofincludes · A zero-knowledge proof is a cryptographic protocol in which a prover demonstrates a statement’s truth without revealing information beyond its validity.
• witnessuses · A witness is private information that enables a prover to establish that a statement is true within a proof system.
• discrete logarithmuses · A discrete logarithm is an exponent or scalar that relates a specified group element to a base or generator under the group operation.
• protocolproduced by · A protocol is a defined set of rules and conventions governing communication or interaction among systems, components, or participants.
• common reference stringused by · A common reference string is publicly available setup data shared by participants in a cryptographic proof system.

This explainer was generated by Manning's knowledge graph from the text of Elliptic Curve Cryptography for Developers. The section links point to the parts of the book it draws on.