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Proof Systems¶

Cryptographic proof systems from foundational theory through modern production SNARKs. Notes cover construction, security assumptions, setup requirements, and proof/verifier complexity.

Foundational Theory¶

Note	Description
GMR85 — Knowledge Complexity	Goldwasser-Micali-Rackoff (1985): the paper that defined zero-knowledge
NIZK (Blum-Feldman-Micali)	First non-interactive ZK construction in the common reference string model
Fiat-Shamir Transform	Heuristic to compile interactive proofs into non-interactive ones via a hash function
DI06 — Scalable MPC	Damgard-Ishai MPC-in-the-head approach (basis of Ligero and descendants)

Early SNARKs¶

Note	Description
QSP / GGPR13	Quadratic Span Programs — theoretical basis for practical SNARKs
Pinocchio	First practically efficient zk-SNARK; introduced the R1CS/QAP pipeline (2013)
Groth16	Highly efficient pairing-based SNARK with a circuit-specific trusted setup; still widely deployed

PLONK Family¶

Note	Description
PLONK	Universal and updatable trusted-setup SNARK using PLONKish arithmetisation (2019)
TurboPLONK	PLONK extension with custom gates
plookup	Lookup argument for efficiently proving table membership inside PLONK
Sonic	Universal SNARK with a universal SRS predating PLONK
HyperPlonk	PLONK variant over multilinear polynomials for improved prover time
HONK	Aztec's PLONK successor using Gemini/ShPLONK; production-ready
UltraHonk	Ultra-optimised HONK variant used in Noir's default backend

Transparent / STARK-based¶

Note	Description
FRI	Fast Reed-Solomon IOP — the polynomial commitment scheme powering zk-STARKs
zk-STARKs	Transparent, post-quantum proofs based on FRI and AIR arithmetisation

IOP / Multilinear / Vector Commitments¶

Note	Description
Bulletproofs	Transparent range proofs and arithmetic circuits; no trusted setup, logarithmic proof size
Bulletproofs+	Improved Bulletproofs with shorter proofs and faster verification
Ligero	MPC-in-the-head proof system based on linear codes
Ligero++	Improved Ligero with better concrete efficiency
Spartan	Transparent SNARK using sum-check and multilinear polynomials; no FFTs
Hyrax	Doubly-efficient zkSNARK using Pedersen commitments and the sum-check protocol
Libra	Linear-time prover SNARK using the sum-check protocol over layered circuits

Dynamic & Designated-Verifier¶

Note	Description
DV-Dynamic-zkSNARKs	Designated-verifier dynamic zk-SNARKs with sublinear proof updates; applied to dynamic proofs of index (Wang et al. 2026)