以下是检查 STARK 深入流程的一些步骤。 Step1 - 构建跟踪（fib2-example） 标有“红色”的数字是公开信息。 Some assertion: Form: [register, step, value] examples: [0,0,1], [1,0,1], [1,7,987] 第 2 步 - 跟踪证明者 协议参数的选择： pub struct ProofOptions { num_queries: u8, // number of queries to increase reliability blowup_factor: u8, // domain extension factor grinding_factor: u8, // ？nonce security parameter, for the security selection of query position hash_fn: HashFunction, // Hash Function blake3-259/192, sha3-256 field_extension: FieldExtension, // Whether the field extension is conducted fri_folding_factor: u8, // The folding factor of Fri protocol 4-8-16 fri_max_remainder_size: u8, // The max remainder of Fri protocol } 1. AIR 实例化 FibAir { context: AirContext::new(trace_info, degrees, options), result: pub_inputs, } AirContext { options, // protocol parameters trace_info, // trace parameters: row number and col number transition_constraint_degrees, // transition the constraint degree, and if the multiplication of two registers is involved, then degree = 2; similarly, if the multiplication of three registers is involved, then degree = 3; if periodic column exists, then degree = base + period.length – 1 ce_blowup_factor, // constraint computational domain extension factor trace_domain < ce_domain < lde_domain trace_domain_generator: B::get_root_of_unity(log2(trace_length)), lde_domain_generator: B::get_root_of_unity(log2(lde_domain_size)), } 2.验证AIR和Trace（Debug模块）的一致性 2.1 验证基本参数 Trace.width = air.trace_width 2.2 验证断言的有效性（边界cs） Trace[assertion_i.register][assertion_i.step] = assertion_i.value 2.2 验证Trace是否保持Transition cs（调试模块） Loop: For every two adjacent rows: pub struct EvaluationFrame<E: FieldElement> { current: Vec<E>, next: Vec<E>, } Holds: next[0] == cur[0] + cur[1] next[1] == next[0] + cur[1] The number of times of Loop: Trace.length - 1 成绩单 write.into(air, public info) 3. 提交跟踪 域参数选择： StarkDomain { trace_twiddles, /ntt twiddle factor ce_domain_size: air.ce_domain_size(), //constraint computational domain ce_to_lde_blowup: air.lde_domain_size() / air.ce_domain_size(), // LDE domain domain_offset: air.domain_offset(), //domain offset } 3.1 插值 -> LDE -> 评估 LDE 域 3.2 承诺 成绩单 write.into(cm_to_trace) 4.评估CS 4.1 获得线性组合系数 Ok(ConstraintCompositionCoefficients { transition: t_coefficients, boundary: b_coefficients, }) 系数个数与约束数一致 在这个示例（fib2-example）中，有 2 个过渡 cs 和 3 个边界 cs。 4.2 为t-cs和b-cs建立Evaluator 4.2.1 t-cs 1. Obtain the degree of t-cs (In this example, the degree is [1,1], refer to Chapter 2.3) 2. Establish the t-cs group (record the parameters needed for reaching the object of degree -> compose degree（ce_domain_size - 1） Loop: Calculate the evaluate-degree = base * （trace.length - 1） + tracce.length / period.cycle *(period.length - 1) of every t-cs Establish the t-cs group according to the degree: TransitionConstraintGroup { degree, // the degree of t-cs degree_adjustment, // The space to reach the object degree, target_degree = composition_degree + trace_poly_degree(vanish poly-degree), degree_adjustment = (target_degree - evaluation_degree) indexes: vec![], coefficients: vec![], } Record index and distribution coefficient “coef” Cycle each t-cs (In fib2-example, cycle twice) 3. Establish the period value table 4. Set the divisor for t-cs, with all t-cs having the same form z(x) = x^n-1 / x - g^{n-1} // The last value does not satisfy the condition 4.2.2 b-cs 1. The divisor of each b-cs is not necessarily the same 2. Judge that if the number of assertions is the same as the number of coefficients to be distributed assertion.len() == coefficients.len() 3. Establish the b-cs group Loop: Obtain the type and initial step of the assertion Establish the b-cs group BoundaryConstraintGroup { constraints: Vec::new(), divisor, // The divisor corresponding to the boundary degree_adjustment, target//The space to reach the object degree, target_degree = composition_degree + divisor.degree()，degree_adjustment = (target_degree - trace_poly_degree) } Add b-cs forms air::BoundaryConstraint { register: assertion.register, // Corresponding register index poly, // assertion value polynomial, interpolation is required for multiple values; otherwise, it is a single value poly_offset, //offset information. If the assert is in the first step, it is [0,1], otherwise it is [1, T] cc, // Coefficient distributed to this b-cs } prover::BoundaryConstraintGroup { degree_adjustment: group.degree_adjustment(), single_value_constraints: Vec::new(),// corresponding Sigle value assertion small_poly_constraints: Vec::new(),//assertion whose assertion value num is smaller than 63 large_poly_constraints: Vec::new(),//assertion whose assertion value num is larger than 63, the values in ce_domain are needed to conduct Pre_compute }; 4.3 评估 t/s-cs Over ce_domain 4.3.1 定义评估者表 ConstraintEvaluationTable<B: StarkField, E: FieldElement<BaseField = B>> { evaluations: Vec<Vec<E>>, //[ce_domain_size][t_cs_merge_value (0): b_cs_values(1...)] divisors: Vec<ConstraintDivisor<B>>, // [t-cs divisor, b-cs divisor] domain_offset: B, trace_length: usize, #[cfg(debug_assertions)] t_evaluations: Vec<Vec<B>>, #[cfg(debug_assertions)] t_expected_degrees: Vec<usize>, } evaluations[step_i][0] += sum_{j}(evaluation[j] * (coefficients.0 + coefficients.1 * xp) // The divisor is identical. evaluations[step_i][j] += (evframe.cur().state[register] - value) * (coefficients.0 + coefficients.1 * xp) // for single assertion evaluations[step_i][j] += (evframe.cur().state[register] - polynom::eval(&self.poly, x * self.x_offset)) * (coefficients.0 + coefficients.1 * xp) // for multiassertion // for multi-assertion evaluations[step_i][j] += (evframe.cur().state[register] - self.values[values_index]) * (coefficients.0 + coefficients.1 * xp) // for multiassertion // for multi-assertion large poly 5. 评估 CS 的承诺 5.1 建立约束组合多项式 5.2 对组合 Poly 的承诺 例子： Compose_poly = a * x^3 + b * x^2 + c * x + d = (a * x^2 + c) * x^ + (b * x^2 + d) (a * x^2 + c ) 和 (b *x^2 +d) 分别对应两列。 6.建立DEEP组合多项式 一般形式：f(x) = q(x)* t(x) 需要随机检查 z f(z) = q(z) * t(z) f(x),q(x),t(x) 确实分别等于 f(z), q(z), t(z) 计算 Deep_composition = (q(x) - q(z)) / (x - z) 检查 q_q(x) 的 LDT 6.1 选择z which out of Domain(ood) 画一个域外点z。根据 E 的类型，该点要么从基本字段中绘制，要么从 E 定义的扩展字段中绘制。 此处从扩展字段（而不是基本字段）进行采样的目的是为了提高安全性 6.2 在 OOD 点 z 处评估迹和约束多项式 6.2.1 z & z * g 处的 trace_poly ood_frame = {cur: [trace_poly_0(z), trace_poly_1(z)], next: [trace_poly_0(z * g), trace_poly_1(z * g)]} 6.2.2 z 处的合成 Poly iter!(self.columns).map(|p| polynom::eval(p, z^m)).collect() // m is the column num of the composition poly 6.3 建立深度合成多项式 6.3.1 生成随机数 pub struct DeepCompositionCoefficients<E: FieldElement> { /// Trace polynomial composition coefficients $\alpha_i$, $\beta_i$, and $\gamma_i$. pub trace: Vec<(E, E, E)>, /// Constraint column polynomial composition coefficients $\delta_j$. pub constraints: Vec<E>, /// Degree adjustment composition coefficients $\lambda$ and $\mu$. pub degree: (E, E), } 6.3.2 Cal 商数聚 => for trace polynomial T`(x) = alpha * (T(x) - T(z)) // degree = trace.length - 1 T``(x) = beta * (T(x) - T(z * g)) T```(x) = gamma * (T(x) - T(z_conjugate)) merge_trace_composition_poly = T(x) / (x - z) + T``(x) / (x - z * g) + T```(x) / (x - z_conjugate) Degree = trace.lengh - 2 => for composition polynomial compute H'_i(x) = (H_i(x) - H_i(z^m)) / (x - z^m) sum(H`_i(x) * cc_i) Deep_composition_polynomial = merge_trace_composition_poly + sum(H`_i(x) * cc_i) Degree = trace.length - 2 => Deep_composition_polynomial degree adjustment Deep_composition_polynomial = Deep_composition_polynomial * (cc_0 + x * cc_1) Degree = trace.length – 1 6.4 评估深度超过 LDE deep_evaluations<lde_domain_size> = deep_composition_poly.evaluate(&domain); 7.计算Deep的FRI Layer num trace.length() = fold_factor ^ Layer_num + remainder_size; fold_factor = {4, 8, 16} remainder_size < remainder_max_size == 256 8. 确定查询的位置 从 lde_domain 中选择查询的多个位置。 fold_position = source_position % (fold_domain) fold_domain = source_domain / fold_factor 9. 建立证明对象 9.1 生成 FRI 证明 pub struct FriProof { layers: Vec<FriProofLayer>, remainder: Vec<u8>, // last poly <all coefficients> num_partitions: u8, // stored as power of 2 } pub struct FriProofLayer { values: Vec<u8>, paths: Vec<u8>, } 9.2 查询以上位置的迹多边形 与上述类似 Queries::new(trace_proof, trace_states) 9.3 查询以上位置的约束多边形 与上述类似 Queries::new(merkle_proof, evaluations) 9.4 建立 STARK 证明 StarkProof { context: self.context, commitments: self.commitments, ood_frame: self.ood_frame, trace_queries, constraint_queries, fri_proof, pow_nonce: self.pow_nonce, } 第 3 步 - 验证证明 从成绩单中读取 pub-info 以获取相关数据，然后执行验证过程。 1. 一致性检查 验证第 5.2 章中描述的数学关系的一致性 => for Boundary b(z) = q_b(x) * b_divisor(z) => for composition poly t(z) = q_z(x) * t_divisor(z) 2. 实例化 FRI-Verifier 对象 pub struct FriVerifier<B, E, C, H> where B: StarkField, E: FieldElement<BaseField = B>, C: VerifierChannel<E, Hasher = H>, H: ElementHasher<BaseField = B>, { max_poly_degree: usize, domain_size: usize, domain_generator: B, layer_commitments: Vec<H::Digest>, //flod_poly commitment layer_alphas: Vec<E>, // random numbers options: FriOptions, num_partitions: usize, _channel: PhantomData<C>, } 3. 计算查询位置的深度多边形 计算方法同第 6.4 章 4. 执行 FRI VERIFY 流程

Target

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