How to Optimize Arithmetic Operations in Circuits

Arithmetic optimization in zero-knowledge circuits is the process of minimizing the number of constraints or gates required to prove a computation. In ZK systems like Circom, Halo2, or Noir, each arithmetic operation (+, -, *, comparisons) contributes to the final proof size and generation time. The primary goal is to write circuits that are efficient by design, reducing prover overhead and making applications more scalable and cost-effective. This is critical for on-chain verification where gas costs are proportional to proof complexity.

A foundational optimization technique is constant folding, where operations involving only constant values are computed at compile time rather than represented as circuit constraints. For example, in a Circom template, writing signal output <== 5 * 7; creates a multiplication constraint. Pre-computing this to signal output <== 35; eliminates an unnecessary constraint. Similarly, leveraging bitwise operations for certain checks can be far more efficient than arithmetic comparisons, as many ZK backends have native support for bit decomposition.

Another key strategy involves reusing computed signals to avoid redundant calculations. If multiple parts of your circuit need the same intermediate value, calculate it once, store it in a signal, and reference that signal elsewhere. This is analogous to common subexpression elimination in compiler design. Furthermore, understanding the underlying proof system is crucial: some backends handle native field operations (modulo a prime) cheaply but make non-native operations (like integer division or floating point) extremely expensive, guiding your choice of numerical representation.

When dealing with loops and iterations, unrolling can have significant implications. A fixed loop with a known number of iterations allows the compiler to apply optimizations across iterations. In contrast, a dynamic loop whose bounds are a variable input often requires more complex and less efficient circuit logic. Designing algorithms with fixed, minimal iteration counts is a high-impact optimization. Libraries like zk-SNARK-friendly hash functions (Poseidon, Rescue) are also explicitly designed to minimize circuit complexity compared to traditional hashes like SHA-256.

Finally, always profile and benchmark your circuit. Use the tools provided by your framework (e.g., Circom's --r1cs flag to output constraint count) to measure the impact of your optimizations. A 20% reduction in constraints can translate to a similar reduction in prover time and cost. The iterative process of writing, measuring, and refining is essential for building production-ready ZK applications that are both secure and performant.

Zero-knowledge circuits, built with frameworks like Circom or Halo2, represent computations as a system of polynomial constraints over a finite field. Unlike traditional programming, the primary cost metric is the number of constraints generated, which directly impacts proof generation time and size. Before optimizing, you must understand the core arithmetic types: native field operations (addition, multiplication) are cheap, while non-native operations (comparisons, range checks, integer division) are extremely expensive. The finite field modulus, such as the BN254 scalar field's ~254-bit prime, defines the universe of possible values and overflow behavior.

Efficient circuit design requires a paradigm shift from runtime performance to constraint minimization. A common optimization is replacing division with multiplication by a modular inverse, as the latter is a single constraint. For example, computing a / b is expensive, but calculating b_inv (where b * b_inv ≡ 1 mod p) and then a * b_inv is far cheaper. Similarly, boolean constraints are used to enforce that a variable is 0 or 1, which is essential for conditional logic and comparisons. Libraries like circomlib provide optimized templates (IsZero, LessThan) that implement these patterns.

Understanding the cost of non-native operations is critical. Checking if a 32-bit integer a is less than b cannot use CPU comparison operators. It must be proven using a range check and a subtraction circuit, often requiring hundreds of constraints. Precomputation and witness optimization can help. If a value is known at compile-time (a constant), it can be hardcoded, generating zero constraints. Moving computations outside the circuit when possible—performing them in the external application and providing the result as a witness—is a fundamental optimization strategy.

Modular arithmetic also requires careful handling of overflow. In a field modulo p, the result of a + b is (a + b) % p. This is not integer overflow; values wrap around the field modulus. To emulate integer arithmetic, you must explicitly manage carries using multiple field elements. For example, representing a 256-bit integer in a 254-bit field requires splitting it into high and low limbs. Operations then must account for carries between limbs, significantly increasing constraint count. This makes big integer math one of the most expensive operations in ZK circuits.

Finally, leverage existing circuit libraries and audit their code. Don't implement a cryptographic hash like Poseidon or SHA-256 from scratch; use the battle-tested, optimized templates from circomlib or framework-specific libraries. Examine how they use techniques like linear combination to reduce multiplicative constraints. Profiling your circuit with the compiler's output (e.g., Circom's --r1cs flag to count constraints) is non-negotiable. It allows you to identify bottlenecks, such as a single comparison operation consuming 30% of your constraints, and target optimizations effectively.

Zero-knowledge circuits operate on a constraint system, where every arithmetic operation (addition, multiplication, etc.) must be expressed as a polynomial equation that the prover must satisfy. Each of these equations is a constraint. The total number of constraints directly impacts performance: more constraints mean a larger proof, longer proving times, and higher costs for the prover. Different proving systems (like Groth16, PLONK, or Halo2) have different cost models, but minimizing constraints is a universal goal for efficiency.

Not all operations are equal. A naive implementation might use many constraints for a simple check. For example, verifying a signature might involve checking if a point is on a curve, which requires a field multiplication—a high-cost operation. Optimization often involves trading constraint count for other resources, like adding more witness variables or using precomputed values. The key is to profile your circuit to identify bottlenecks, which are typically complex non-linear operations like elliptic curve pairings or hash functions.

A practical optimization is constraint reduction through algebraic manipulation. Instead of using a generic is_zero check (which can be expensive), you might leverage the fact that a * b = 0 implies either a = 0 or b = 0. Another technique is batching similar operations. If you need to check multiple signatures, you can often aggregate them into a single multi-scalar multiplication or a single pairing check, dramatically reducing the constraint count compared to verifying each one individually.

Real-world optimization requires choosing the right primitives. Using a SNARK-friendly hash function like Poseidon or Rescue over SHA-256 can reduce constraints by orders of magnitude. Similarly, selecting efficient elliptic curves (such as BN254 or BLS12-381 for pairings) is critical. Always benchmark with tools specific to your proving backend (e.g., criterion for Rust-based bellman or arkworks) to measure the actual constraint count and proving time of different implementations.

Finally, consider the cost trade-offs between the prover and verifier. Some optimizations that speed up proving might make the verification key slightly larger or verification slightly slower. The optimal balance depends on your application's threat model and user flow. For a decentralized application where many users verify a single proof, verifier efficiency is paramount. For a private client application, prover time may be the primary concern. Profile both sides of the computation.

Zero-knowledge circuits operate within a finite field, where every variable is a field element. The performance and cost of generating a proof are directly tied to the number of arithmetic constraints your circuit creates. An optimization strategy focuses on minimizing these constraints by reducing the total count of addition, multiplication, and comparison operations. This is critical because each multiplication typically creates one constraint in Rank-1 Constraint System (R1CS)-based proving systems like Groth16, while newer systems like Plonk and Halo2 have more complex but analogous costs.

The most impactful optimization is reducing non-linear operations. Multiplication and comparison (greater-than, less-than) are far more expensive than addition. For example, checking if a value a is within a range [0, 2^n) using bit decomposition creates n constraints. A common technique is to use lookup tables where supported (e.g., in Halo2) to prove a value is in a pre-defined set with a single constraint. Another method is to redesign logic to use boolean algebra and additive checks instead of comparative ones where possible.

Circuit-friendly data types are essential. Represent values in their native field element form and avoid unnecessary conversions. For instance, if you only need to prove knowledge of a 32-byte value, keep it as a single field element rather than 256 individual bit variables. Use conditional selection wisely: the pattern result = a * selector + b * (1 - selector) uses one multiplication to choose between a and b, which is cheaper than using an if-else branch that might create separate constraint paths.

Leverage constant folding and compile-time computation. Any operation where all inputs are constants known at circuit compilation time should be pre-computed outside the circuit. Your code should pass in the resulting constant, not perform the calculation inside a gadget. Similarly, batch linear combinations. Instead of computing sum = a + b + c + d as three separate additions, some frameworks allow you to define a single linear combination constraint, which is more efficient.

Here is a practical example in a pseudo-circuit syntax, contrasting a naive and an optimized approach for checking a value is non-zero:

Naive (Expensive): is_nonzero = witness inverse = value.inverse() constrain(value * is_nonzero == 1) // Requires expensive inverse

Optimized (Cheaper): is_nonzero = witness constrain(value * is_nonzero == 1 - (1 - value * is_nonzero) * ...) // Uses a boolean constraint pattern

The optimized version avoids the costly inversion operation, replacing it with a constraint that enforces boolean behavior.

Finally, always profile your circuit. Use your proving framework's tools to count constraints and identify bottlenecks. Optimization is an iterative process of identifying the most expensive operations—often found in cryptographic primitives, hash functions, or range checks—and applying these patterns: replacing multiplications with additions, using lookups, and pre-computing constants. The goal is to write circuits that are not just functionally correct but also constraint-minimal for faster, cheaper proofs.

Effective circuit optimization requires a multi-layered approach. Start by selecting the right proving system and field for your application—BLS12-381 for pairing-based proofs, BN254 for compatibility, or Grumpkin for native 256-bit operations. Within your chosen framework, leverage high-level techniques like constraint reduction through algebraic simplification, strategic use of public inputs, and lookup arguments for complex operations. Remember that every non-linear constraint, especially those from multiplications or comparisons, has a direct and significant impact on proving time and cost.

The next step is to implement low-level optimizations within your constraint system. This includes using windowed exponentiation for repeated multiplications, employing Karatsuba multiplication for large integers, and pre-computing constants outside the circuit. For finite field arithmetic, ensure you are using the most efficient modular reduction algorithms available in your library, such as the Montgomery reduction. Profiling your circuit with tools like snarkjs or your proving system's built-in profiler is non-negotiable for identifying bottlenecks.

To solidify these concepts, review and experiment with optimized code. For example, compare a naive square-and-multiply exponentiation with a windowed method in a framework like Circom or Halo2. Analyze real-world circuits from audited projects like zkSync Era's bridge or Aztec's private payment system to see these patterns in production. The 0xPARC circuit optimization guide and Ethereum Foundation's Privacy and Scaling Explorations team publications are excellent resources for advanced patterns.

Your optimization journey should be iterative. Begin with a correct, simple implementation, then measure its performance. Systematically apply the techniques discussed—constraint reduction, algorithmic improvement, and backend tuning—measuring the impact of each change. Keep detailed benchmarks; a 20% reduction in constraints can translate to a 30% reduction in proving cost on mainnet. Always verify that optimizations do not compromise the security or correctness of your circuit's logic.

Looking forward, stay informed about advancements in proof systems. Plonkish arithmetization and custom gate design in Halo 2, STARKs with their rapid proof generation, and new lookup argument constructions like LogUp are actively pushing the boundaries of what's efficient. Engaging with the research community through forums like the ZKProof standardization effort and implementing prototypes with new libraries will keep your skills at the forefront. The goal is to build applications that are not just possible, but practical for end-users.