A directed acyclic graph (DAG) of addition and multiplication gates over a field, used to represent polynomial computations in a form amenable to cryptographic proof systems. Arithmetic circuits are the canonical intermediate representation for zero-knowledge proof schemes such as zk-SNARKs and STARKs: a computation is first expressed as an arithmetic circuit, then compiled into a system of polynomial constraints (R1CS or Plonkish), and finally proved using a cryptographic proving system. The complexity of a circuit is characterised by its depth (for parallelism) and size (gate count).
Overview
- Arithmetic circuits occupy a central position in modern cryptographic engineering: nearly every practical zero-knowledge proof system compiles a target computation to an arithmetic circuit before generating proof constraints.
- The circuit is defined over a large prime field; inputs and outputs are field elements; intermediate values at each gate are constrained to satisfy the circuit’s wiring.
- Circuit complexity determines prover time and proof size — minimising gate count and depth is an active area of compiler research for zkEVMs and similar systems.
- Domain-specific languages (DSLs) such as Circom, Noir, and Cairo compile human-readable constraint descriptions to arithmetic circuits automatically.
Key Aspects
- Field selection — the prime modulus of the underlying field must be chosen to match the elliptic curve or hash function used in the proving system.
- R1CS encoding — rank-1 constraint systems encode each multiplication gate as a linear-algebraic constraint triplet.
- Plonkish variants — custom gate types (plookup, range checks) extend vanilla arithmetic circuits for efficiency.
- Depth vs width — shallow wide circuits parallelise better; deep narrow circuits minimise gates but impede parallelism.
Applications
- zk-rollup transaction validity proofs in Blockchain layer-2 scaling.
- Privacy-preserving credential verification without revealing underlying attributes.
- Verifiable machine learning inference proofs over neural network weights.
- Homomorphic Encryption circuit compilation for outsourced computation.