Analysis of the Principles of Binius STARKs and Thoughts on Its Optimization

1 Introduction

One of the main reasons for the inefficiency of STARKs is that most of the values in actual programs are relatively small, such as indexes in for loops, boolean values, counters, etc. However, to ensure the security of proofs based on Merkle trees, many additional redundant values occupy the entire field when using Reed-Solomon coding to expand the data, even though the original values themselves are very small. To address this issue, reducing the size of the field has become a key strategy.

As shown in Table 1, the encoding bit width of the first generation STARKs is 252 bits, the second generation STARKs has an encoding bit width of 64 bits, and the third generation STARKs has an encoding bit width of 32 bits, but the 32-bit encoding still has a lot of wasted space. In contrast, the binary field allows direct bit manipulation, making the encoding compact and efficient without any wasted space, namely the fourth generation STARKs.

Table 1: STARKs Derivation Path

| Algebra | Bit Width | Field | |------|------|------| | 1st Generation | 252bit | Prime Field | | 2nd Generation | 64bit | Goldilocks | | 3rd Generation | 32bit | Prime Field | | 4th Generation | 1bit | Binary Domain |

Compared to the finite fields discovered in recent years such as Goldilocks, BabyBear, and Mersenne31, the study of binary fields can be traced back to the 1980s. Currently, binary fields are widely used in cryptography, with typical examples including:

• Advanced Encryption Standard ( AES ), based on F28 field;

• Galois Message Authentication Code ( GMAC ), based on F2128 field;

• QR code, uses Reed-Solomon encoding based on F28;

• The original FRI and zk-STARK protocols, as well as the Grøstl hash function that entered the SHA-3 finals, which is based on the F28 field, is a hash algorithm that is very suitable for recursion.

When smaller fields are used, the extension field operation becomes increasingly important for ensuring security. The binary field used by Binius relies entirely on extension fields to guarantee its security and practical usability. Most polynomials involved in Prover calculations do not need to enter the extension field and can operate solely in the base field, achieving high efficiency in small fields. However, random point checks and FRI calculations still need to delve into larger extension fields to ensure the necessary security.

When constructing proof systems based on binary fields, there are two practical issues: the size of the field used to represent the trace in STARKs should be greater than the degree of the polynomial; and when committing to a Merkle tree in STARKs, Reed-Solomon encoding must be performed, and the size of the field used should be greater than the size after encoding expansion.

Binius has proposed an innovative solution that addresses these two issues separately and achieves the representation of the same data in two different ways: first, by using multivariate ( specifically multivariate ) polynomials instead of univariate polynomials, representing the entire computation trajectory through its values on "hypercubes" (; secondly, since the length of each dimension of the hypercube is 2, standard Reed-Solomon extension like STARKs cannot be performed, but the hypercube can be viewed as square ), and Reed-Solomon extension can be based on this square. This approach greatly enhances coding efficiency and computational performance while ensuring security.

2 Principle Analysis

The construction of most SNARKs systems currently typically includes the following two parts:

• Information-Theoretic Polynomial Interactive Oracle Proof, PIOP(: PIOP, as the core of the proof system, transforms the input computational relations into verifiable polynomial equalities. Different PIOP protocols allow the prover to send polynomials incrementally through interaction with the verifier, enabling the verifier to validate the correctness of the computation by querying a limited number of polynomial evaluation results. Existing PIOP protocols include: PLONK PIOP, Spartan PIOP, and HyperPlonk PIOP, each of which has a different approach to handling polynomial expressions, thus affecting the overall performance and efficiency of the SNARK system.

• Polynomial Commitment Scheme ) Polynomial Commitment Scheme, PCS (: The polynomial commitment scheme is used to prove whether the polynomial equations generated by PIOP hold true. PCS is a cryptographic tool through which the prover can commit to a certain polynomial and later verify the evaluation results of that polynomial while hiding other information about the polynomial. Common polynomial commitment schemes include KZG, Bulletproofs, FRI ) Fast Reed-Solomon IOPP (, and Brakedown, among others. Different PCS have different performance, security, and applicability.

According to specific requirements, choose different PIOP and PCS, and combine them with suitable finite fields or elliptic curves to construct proof systems with different attributes. For example:

• Halo2: a combination of PLONK PIOP and Bulletproofs PCS, based on the Pasta curve. Halo2 is designed with a focus on scalability and the removal of trusted setup from the ZCash protocol.

• Plonky2: combines PLONK PIOP and FRI PCS, based on the Goldilocks field. Plonky2 is designed for efficient recursion. When designing these systems, the chosen PIOP and PCS must match the finite field or elliptic curve being used to ensure the correctness, performance, and security of the system. The choices of these combinations not only affect the proof size and verification efficiency of the SNARK but also determine whether the system can achieve transparency without a trusted setup, and whether it can support extended features such as recursive proofs or aggregate proofs.

Binius: HyperPlonk PIOP + Brakedown PCS + binary fields. Specifically, Binius includes five key technologies to achieve its efficiency and security. First, the arithmetic based on tower of binary fields ) towers of binary fields ( forms the foundation of its computation, enabling simplified operations within the binary fields. Second, in its interactive Oracle proof protocol ) PIOP (, Binius adapted HyperPlonk product and permutation checks, ensuring secure and efficient consistency checks between variables and their permutations. Third, the protocol introduces a new multilinear shift proof, optimizing the efficiency of verifying multilinear relationships over small fields. Fourth, Binius employs an improved Lasso lookup proof, providing flexibility and strong security for the lookup mechanism. Finally, the protocol uses a small field polynomial commitment scheme ) Small-Field PCS (, allowing it to implement an efficient proof system over binary fields while reducing the overhead typically associated with large fields.

) 2.1 Finite Fields: Arithmetic based on towers of binary fields

Towering binary fields are key to achieving fast verifiable computation, mainly due to two aspects: efficient computation and efficient arithmetic. Binary fields inherently support highly efficient arithmetic operations, making them ideal for cryptographic applications that are sensitive to performance requirements. Moreover, the structure of binary fields supports a simplified arithmetic process, meaning that operations performed over binary fields can be expressed in a compact and easily verifiable algebraic form. These features, along with the ability to fully leverage their hierarchical characteristics through tower structures, make binary fields particularly suitable for scalable proof systems like Binius.

In this context, "canonical" refers to the unique and direct representation of elements in a binary field. For example, in the simplest binary field F2, any k-bit string can be directly mapped to a k-bit binary field element. This is different from prime fields, which cannot provide such a canonical representation within a given bit length. Although a 32-bit prime field can fit within 32 bits, not every 32-bit string can uniquely correspond to a field element, while binary fields possess the convenience of this one-to-one mapping. In the prime field Fp, common reduction methods include Barrett reduction, Montgomery reduction, and special reduction methods for specific finite fields such as Mersenne-31 or Goldilocks-64. In the binary field F2k, commonly used reduction methods include special reduction ( as used in AES ), Montgomery reduction ### as used in POLYVAL (, and recursive reduction ) such as Tower (. The paper "Exploring the Design Space of Prime Field vs. Binary Field ECC-Hardware Implementations" points out that binary fields do not require carry in both addition and multiplication operations, and the squaring operation in binary fields is very efficient because it follows the simplification rule of )X + Y (2 = X2 + Y 2.

As shown in Figure 1, a 128-bit string: this string can be interpreted in various ways within the context of binary fields. It can be viewed as a unique element in the 128-bit binary field, or parsed as two 64-bit tower field elements, four 32-bit tower field elements, sixteen 8-bit tower field elements, or 128 F2 field elements. This flexibility in representation does not require any computational overhead, just a typecast of the bit string )typecast(, which is a very interesting and useful property. At the same time, small field elements can be packed into larger field elements without additional computational overhead. The Binius protocol takes advantage of this feature to improve computational efficiency. Furthermore, the paper "On Efficient Inversion in Tower Fields of Characteristic Two" discusses the computational complexity of multiplication, squaring, and inversion operations in an n-bit tower binary field ) decomposed into m-bit subfields (.

![Tower Binary Domain])https://img-cdn.gateio.im/webp-social/moments-a5d031be4711f29ef910057f4e44a118.webp(

Figure 1: Tower Binary Field

) 2.2 PIOP: Adapted HyperPlonk Product and Permutation Check------Applicable to binary field

The PIOP design in the Binius protocol draws on HyperPlonk and employs a series of core checking mechanisms to verify the correctness of polynomials and multivariable sets. These core checks include:

1. GateCheck: Verify if the confidential witness ω and public input x satisfy the circuit computation relationship C(x,ω)=0, to ensure the correct operation of the circuit.

2. PermutationCheck: Verify whether the evaluation results of two multivariable polynomials f and g on the Boolean hypercube are a permutation relation f###x( = f)π(x)(, to ensure the consistency of the arrangement between the polynomial variables.

3. LookupCheck: Verify whether the evaluation of the polynomial is within the given lookup table, i.e., f(Bµ) ⊆ T)Bµ(, ensuring that certain values are within the specified range.

4. MultisetCheck: Check if two multivariable sets are equal, that is, {)x1,i,x2,(}i∈H = {)y1,i,y2,(}i∈H, ensuring consistency among multiple sets.

5. ProductCheck: Check whether the evaluation of the rational polynomial on the Boolean hypercube is equal to a given declared value ∏x∈Hµ f)x( = s, to ensure the correctness of the polynomial product.

6. ZeroCheck: Verifying whether a multivariable polynomial is zero at any point on the Boolean hypercube ∏x∈Hµ f)x( = 0, ∀x ∈ Bµ, to ensure the distribution of the polynomial's zeros.

7. SumCheck: Verifying whether the sum of a multivariate polynomial equals the declared value ∑x∈Hµ f)x( = s. It reduces the computational complexity for the verifier by transforming the multivariate polynomial evaluation problem into a single-variable polynomial evaluation. Additionally, SumCheck allows batching by introducing random numbers to construct linear combinations for processing multiple sum verification instances.

8. BatchCheck: Based on SumCheck, verifies the correctness of multiple multivariate polynomial evaluations to improve protocol efficiency.

Although Binius and HyperPlonk share many similarities in protocol design, Binius has made improvements in the following three areas:

• ProductCheck Optimization: In HyperPlonk, ProductCheck requires that the denominator U is non-zero everywhere on the hypercube, and the product must equal a specific value; Binius simplifies this check process by specializing this value to 1, thereby reducing computational complexity.

• Handling of the zero division problem: HyperPlonk fails to adequately address the zero division case, making it impossible to assert the non-zero problem of U on the hypercube; Binius correctly handles this issue, allowing Binius's ProductCheck to continue processing even when the denominator is zero, enabling generalization to any product value.

• Cross-column PermutationCheck: HyperPlonk does not have this feature; Binius supports PermutationCheck across multiple columns, allowing Binius to handle more complex polynomial arrangement scenarios.

Therefore, Binius has improved the existing PIOPSumCheck mechanism, enhancing the flexibility and efficiency of the protocol, especially in providing stronger functional support when handling more complex multivariable polynomial verifications. These improvements not only address the limitations in HyperPlonk but also lay the foundation for future proof systems based on binary fields.

) 2.3 PIOP: New multilinear shift argument------Applicable to boolean hypercube

In the Binius protocol, the construction and handling of virtual polynomials is key.