Analysis of Binius STARKs Principles and Optimization Thoughts

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 indices in for loops, boolean values, counters, and so on. However, to ensure the security of proofs based on Merkle trees, many additional redundant values occupy the entire field when data is expanded using Reed-Solomon encoding, even though the original values themselves are very small. To solve this problem, reducing the size of the field has become a key strategy.

As shown in Table 1, the encoding bit width of the 1st generation STARKs is 252 bits, the 2nd generation STARKs has an encoding bit width of 64 bits, and the 3rd generation STARKs has an encoding bit width of 32 bits. However, the 32-bit encoding width still has a lot of wasted space. In contrast, the binary field allows for direct bit manipulation, making the encoding compact and efficient without any wasted space, which is the 4th generation STARKs.

| Time | STARKs | Encoding Bit Width | |------|--------|----------| | 2018 | 1st Generation | 252bit | | 2021 | 2nd Generation | 64bit | | 2022 | 3rd Generation | 32bit | | 2023 | 4th Generation | 1bit |

Table 1: STARKs Derivation Path

Compared to recent findings in finite fields such as Goldilocks, BabyBear, and Mersenne31, research on 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, using Reed-Solomon encoding based on F28;

• The original FRI and zk-STARK protocols, as well as the Grøstl hash function that made it to the finals of SHA-3, are based on the F28 field and are a hash algorithm very suitable for recursion.

When using smaller fields, 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 computations 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 computations still need to delve into larger extension fields to ensure the required security.

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

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

2 Principle Analysis

The construction of most current SNARKs systems usually includes the following two parts:

• Information-Theoretic Polynomial Interactive Oracle Proof, PIOP(: PIOP, as the core of the proof system, transforms the input computational relationships into verifiable polynomial equalities. Different PIOP protocols allow the prover to gradually send polynomials through interaction with the verifier, enabling the verifier to validate the correctness of the computation by querying the evaluation results of a small number of polynomials. Existing PIOP protocols include: PLONK PIOP, Spartan PIOP, and HyperPlonk PIOP, each of which processes polynomial expressions differently, thereby affecting the performance and efficiency of the entire SNARK system.

• Polynomial Commitment Scheme ): The Polynomial Commitment Scheme is used to prove whether polynomial equations generated by PIOP hold true. PCS is a cryptographic tool that allows the prover to commit to a certain polynomial and later verify the evaluation result 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 varying performance, security, and applicable scenarios.

According to specific requirements, different PIOP and PCS can be selected, and combined with suitable finite fields or elliptic curves, to construct proof systems with different attributes. For example:

• Halo2: Combined from PLONK PIOP and Bulletproofs PCS, and based on the Pasta curve. When designing Halo2, emphasis was placed on scalability and the removal of the trusted setup in the ZCash protocol.

• Plonky2: combines PLONK PIOP with FRI PCS and is 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 used to ensure the correctness, performance, and security of the system. The choice of these combinations not only affects the proof size and verification efficiency of SNARK but also determines 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 towers of binary fields ( towers of binary fields ) forms the foundation of its computation, enabling simplified operations within the binary fields. Second, Binius adapted HyperPlonk product and permutation checks in its interactive Oracle proof protocol ( PIOP ), 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 version of the Lasso lookup proof, providing flexibility and robust security for the lookup mechanism. Finally, the protocol utilizes a small field polynomial commitment scheme ( Small-Field PCS ), allowing for an efficient proof system over binary fields and reducing the overhead typically associated with large fields.

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

Towered binary fields are key to achieving fast verifiable computation, primarily due to two aspects: efficient computation and efficient arithmetic. Binary fields inherently support highly efficient arithmetic operations, making them an ideal choice for cryptographic applications that are sensitive to performance requirements. Additionally, the structure of binary fields supports a simplified arithmetic process, meaning that operations performed over binary fields can be represented in a compact and easily verifiable algebraic form. These characteristics, combined with the ability to fully leverage their hierarchical properties through a tower structure, make binary fields particularly suitable for scalable proof systems such as Binius.

The term "canonical" refers to the unique and direct representation of an element in a binary field. For example, in the most basic binary field F2, any k-bit string can be directly mapped to a k-bit binary field element. This differs from prime fields, which cannot provide such a canonical representation within a given bit length. Although a 32-bit prime field can be contained in 32 bits, not every 32-bit string can uniquely correspond to a field element, while binary fields have 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, common reduction methods include special reduction ) as used in AES (, Montgomery reduction ) as used in POLYVAL ###, and recursive reduction ( as in 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 for addition and multiplication operations, and the square operation in binary fields is very efficient because it follows the simplified rule of (X + Y )2 = X2 + Y2.

As shown in Figure 1, a 128-bit string: this string can be interpreted in multiple ways within the context of binary fields. It can be considered a unique element in a 128-bit binary field, or parsed as two 64-bit tower field elements, four 32-bit tower field elements, 16 8-bit tower field elements, or 128 F2 field elements. This flexibility in representation incurs no computational overhead, merely typecasting 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. Additionally, the paper "On Efficient Inversion in Tower Fields of Characteristic Two" explores the computational complexity of multiplication, squaring, and inversion operations in n-bit tower binary fields that can be 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 fields

The PIOP design in the Binius protocol draws on HyperPlonk, employing 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 arithmetic relationship C(x, ω)=0, to ensure the correct operation of the circuit.

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

3. LookupCheck: Verify whether the evaluation of the polynomial is within the given lookup table, that is, 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 a rational polynomial on the Boolean hypercube equals a declared value ∏x∈Hµ f)x( = s, to ensure the correctness of the polynomial product.

6. ZeroCheck: Verify 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 roots.

7. SumCheck: Detecting whether the sum of a multivariable polynomial is equal to the declared value ∑x∈Hµ f)x( = s. By transforming the evaluation problem of multivariable polynomials into that of univariate polynomial evaluation, it reduces the computational complexity for the verifier. Furthermore, SumCheck allows for batching by introducing random numbers and constructing linear combinations to achieve batching of multiple sum verification instances.

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

Although Binius and HyperPlonk have 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 that the product must equal a specific value; Binius simplifies this check by specializing that value to 1, thus reducing computational complexity.

• Handling of the division by zero problem: HyperPlonk fails to adequately address the division by zero situation, leading to the inability to assert the non-zero problem of U on the hypercube; Binius correctly handles this issue, allowing Binius's ProductCheck to continue processing even in cases where 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 situations.

Therefore, Binius has improved the existing PIOPSumCheck mechanism, enhancing the flexibility and efficiency of the protocol, especially in providing stronger functional support when dealing with more complex multivariable polynomial verification. 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 are crucial.