Efficient ZeroKnowledge Arguments from TwoTiered Homomorphic Commitments Jens

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Efficient Zero-Knowledge Arguments from Two-Tiered Homomorphic Commitments Jens Groth University College London

Efficient Zero-Knowledge Arguments from Two-Tiered Homomorphic Commitments Jens Groth University College London

Zero-knowledge argument Zero-knowledge: Nothing but truth revealed; witness remains secret Prover Soundness: Statement is

Zero-knowledge argument Zero-knowledge: Nothing but truth revealed; witness remains secret Prover Soundness: Statement is true Verifier

Statements •

Statements •

Results Circuit with N gates Rounds Cramer et al. 1994 3 Groth 2009 5

Results Circuit with N gates Rounds Cramer et al. 1994 3 Groth 2009 5 This paper 7 Range proof [0; 2 N) Rounds Cramer et al. 1994 3 Camenisch et al. 2008 3 This paper 7 Prover Verifier Comm.

Two-tiered commitments •

Two-tiered commitments •

Pedersen commitment •

Pedersen commitment •

Commitments to vectors •

Commitments to vectors •

Bilinear groups •

Bilinear groups •

Commitments to vectors of group elements •

Commitments to vectors of group elements •

Homomorphic property •

Homomorphic property •

Commitments to commitments •

Commitments to commitments •

Two-tiered homomorphic commitments •

Two-tiered homomorphic commitments •

Range proof •

Range proof •

Range proof for [0; 2 N) •

Range proof for [0; 2 N) •

Commitments to the bits •

Commitments to the bits •

Evaluating committed polynomials •

Evaluating committed polynomials •

Proving bits are 0 or 1 •

Proving bits are 0 or 1 •

Range proof conclusion •

Range proof conclusion •

Efficiency •

Efficiency •

Circuit satisfiability •

Circuit satisfiability •

Summary • Two-tiered homomorphic commitments give us efficient zero-knowledge arguments Circuit with N gates

Summary • Two-tiered homomorphic commitments give us efficient zero-knowledge arguments Circuit with N gates Rounds Cramer et al. 1994 3 Groth 2009 5 This paper 7 Range proof [0; 2 N) Rounds Cramer et al. 1994 3 Camenisch et al. 2008 3 This paper 7 Prover Verifier Comm.

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