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[RareSkills ZK Book ↔ Groth.jl Map](@id rareskills-map)

This page stitches together the RareSkills ZK Book material and the production implementations inside Groth.jl. Use it as a compass when moving from conceptual chapters to code.

Pages = ["rareskills-map.md"]
Depth = 2

flowchart LR
    subgraph Book["RareSkills ZK Book concepts"]
        direction TB
        FF["Finite fields<br/>modular arithmetic"]
        SUB["Group theory<br/>subgroups"]
        POLY["Lagrange interpolation<br/>polynomials"]
        ECC["Elliptic curve arithmetic"]
        PAIR["Pairings<br/>final exponentiation"]
        R1CS["Rank-1 constraint systems"]
        QAP["QAP<br/>R1CS to QAP"]
        G16["Trusted setup<br/>Groth16"]
        EX["Worked circuits<br/>tutorials"]
    end

    subgraph Code["Groth.jl implementation"]
        direction TB
        GA["GrothAlgebra/FiniteFields.jl"]
        GG["GrothAlgebra/Group.jl"]
        PA["GrothAlgebra/Polynomial.jl"]
        GC["GrothCurves/BN254Curve.jl"]
        GP["GrothCurves pairing pipeline"]
        PR1["GrothProofs/R1CS.jl"]
        PQ["GrothProofs/QAP.jl"]
        PG["GrothProofs/Groth16.jl"]
        EXJ["GrothExamples/"]
    end

    FF --> GA
    SUB --> GG
    POLY --> PA
    ECC --> GC
    PAIR --> GP
    R1CS --> PR1
    QAP --> PQ
    G16 --> PG
    EX --> EXJ
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Algebraic Foundations

  • RareSkills ZK Book concepts: finite fields and modular arithmetic, group theory, homomorphisms, multiplicative subgroups, polynomial arithmetic, and inner-product algebra.
  • Groth.jl counterparts:
    • GrothAlgebra/FiniteFields.jl defines FiniteFieldElement plus the current fixed-width Montgomery BN254 field backend.
    • GrothAlgebra/Polynomial.jl implements Horner evaluation, interpolation, derivatives, and FFT scaffolding.
    • GrothAlgebra/Group.jl provides curve-agnostic group operations, w-NAF helpers, and MSM support.
  • What changes in code: we lean on fixed-width limbs, concrete extension-field storage, windowed MSM, and FFT-friendly evaluation domains to feed the prover efficiently.

Curves, Towers, and Pairings

  • RareSkills ZK Book concepts: elliptic curve addition, elliptic curves over finite fields, bilinear pairings, Miller loops, and final exponentiation.
  • Groth.jl counterparts:
    • GrothCurves/BN254Curve.jl keeps G1/G2 in Jacobian form with mixed additions and batch normalisation.
    • Tower files (BN254Fp2.jl, BN254Fp6_3over2.jl, BN254Fp12_2over6.jl) encode the extension fields exactly as derived in the book.
    • BN254MillerLoop.jl, BN254FinalExp.jl, BN254Pairing.jl follow the optimal ate pipeline and reuse Frobenius shortcuts.
  • What changes in code: we use sparse Fp12 placement, precomputed Frobenius constants, and the BN254Engine abstraction so future curves can reuse the same interface.

Constraint Systems and QAPs

  • RareSkills ZK Book concepts: arithmetic circuits, rank-1 constraint systems, Lagrange interpolation, Schwartz-Zippel, quadratic arithmetic programs, and R1CS to QAP.
  • Groth.jl counterparts:
    • GrothProofs/R1CS.jl ships multiplication, sum-of-products, affine-product, and square-offset circuits plus randomised fixtures.
    • GrothProofs/QAP.jl records the active constraint points and builds an arkworks-shaped power-of-two domain: constraint rows first, public-input selector rows next, and zero padding last.
  • What changes in code: prove_full uses the coset-only quotient path, while dense/coset parity survives in tests and debugging helpers. The current performance work is focused more on prover hot paths than on broad domain rewrites.

Groth16 Pipeline

  • RareSkills ZK Book concepts: trusted setup, evaluating a QAP on a trusted setup, and Groth16 proving and verification.
  • Groth.jl counterparts:
    • GrothProofs/Groth16.jl wires setup/prove/verify, including the prepared verifier path and batched pairings via pairing_batch.
    • GrothProofs/test/runtests.jl mirrors the book’s witness discipline across multiple circuit families.
    • GrothExamples/ provides notebook-based walkthroughs (AbstractAlgebra-first and Groth package-native R1CS → QAP flows) for side-by-side comparison.

How to Use the Map

  1. Start with the RareSkills ZK Book section you are studying.
  2. Jump to the matching Groth.jl module listed above.
  3. Compare representation choices—projective vs affine, batched MSM, FFT preparation—to understand how the production prover keeps the algebraic guarantees while optimising for performance.

For a focused explanation of why the optimized code no longer always looks like the textbook derivation, see [From Textbook To Optimized Code](@ref textbook-to-optimized).

Update this page whenever new features land or the textbook mapping shifts.