This work systematically explains how various common constraints can be reduced to the Multivariate Sumcheck Protocol. The core idea is to use random linear combinations and logarithmic-derivative techniques to transform “pointwise verification” of universal assertions into a “global summation” form, which can then be efficiently verified by sumcheck. The main methods include: ZeroCheck: Using equality kernels to reduce the assertion “vanishes everywhere on the Boolean hypercube” into a single summation check. Rational Sumcheck: Introducing auxiliary functions and zero checks to convert rational summations into polynomial summations. Multiset Equality Check: Using logarithmic-derivative identities to reduce multiset equality to rational summations. Permutation Check: By attaching indices to elements, reduce the problem to multiset equality, and then to Rational Sumcheck. LogUp Lookup Argument: Using rational-function representations to verify lookup constraints, avoiding expensive product expressions.
These methods are widely applied in modern zero-knowledge proof systems such as Plonk, HyperPlonk, and Halo2, for range checks, wiring checks, permutation arguments, and lookup validations.
We want to prove that a polynomial/function /function \( f(\overrightarrow{x}) \) vanishes on the Boolean hypercube \( \{0, 1\}^n \), i.e.
\[ \forall \overrightarrow{x} \in \{0, 1\}^n, \quad f(\overrightarrow{x}) = 0. \]
The idea is to reduce this "pointwise vanishing" universal assertion to a single "global summation equals zero" check, so that Multivariate Sumcheck can be used in an interactive/non-interactive protocol:
Intuitively, the point-values of \( f \) on the cube act as "weights," and \( \mathit{eq}(\overrightarrow{x}, \overrightarrow{r}) \) serves as an "interpolation kernel" for sampling at random \( \overrightarrow{r} \). If \( f \) is nonzero, then its multilinear extension \( \tilde{f} \) is a nonzero polynomial, which with overwhelming probability does not vanish at a random \( \overrightarrow{r} \).
To verify that \( f(\overrightarrow{x}) = 0 \) holds for all \( \overrightarrow{x} \in \{0,1\}^n \):
\[ \sum_{\overrightarrow{x} \in \{0,1\}^n} \hat{f}(\overrightarrow{x}) = 0. \]
Batch proof: verifying verifying \( f(\overrightarrow{x}) = 0 \land g(\overrightarrow{x}) = 0 \)
Complexity: If \( f \) is multilinear, then \( \hat{f} \) has degree at most 2 per variable (or remains low-degree). Sumcheck requires \( O(n) \) rounds, each handling \( O(1) \)-degree polynomials, making the process efficient overall.
In many proof systems, we need to verify global summation relations involving rational functions, e.g.,
\[ \begin{align*} \sum_{\overrightarrow{x} \in \{0,1\}^n} \frac{p(\overrightarrow{x})}{q(\overrightarrow{x})} = v. \end{align*} \]
Such constraints arise in division gates, lookup arguments, and compressed polynomials. Since sumcheck natively works only on polynomials, we need a reduction strategy to handle rational functions. The idea is to reduce “rational summation” into a combination of polynomial summation and zero checks:
Thus, rational summations are safely reduced to "pure polynomial summation + vanishing check."
Target assertion:
\[ \begin{align*} \sum_{\overrightarrow{x} \in \{0,1\}^n} \frac{p(\overrightarrow{x})}{q(\overrightarrow{x})} = v. \end{align*} \]
Reduction Steps:
If true, then \( f(\overrightarrow{x}) \) must indeed be the inverse of \( q(\overrightarrow{x}) \).
We want to verify whether two function sets take equal values over the Boolean hypercube \( \{0, 1\}^n \): \(f_0(\overrightarrow{x}), f_1(\overrightarrow{x}), \ldots, f_k(\overrightarrow{x}) = g_0(\overrightarrow{x}), g_1(\overrightarrow{x}), \ldots, g_k(\overrightarrow{x})\)
Such constraints are common in proof systems, e.g., Plonk wiring check. The classical approach is to use ProdCheck (product check) to verify that the generated polynomials on both sides are the same. However, ProdCheck requires additional helper polynomials and interaction in distributed settings, which is inefficient.
Solution idea. By the logarithmic derivative identity:
\[ \prod_i(a_i + X) = \prod_i(b_i + X) \quad \Leftrightarrow \quad \sum_i \frac{1}{a_i + X} = \sum_i \frac{1}{b_i + X}, \]
product equality can be reduced to a rational function summation equality. This allows one to use the Rational Sumcheck Protocol directly, without introducing extra helper polynomials.
Key techniques
Principle: Verify whether \( f_0(\overrightarrow{x}),\ldots,f_k(\overrightarrow{x}) = g_0(\overrightarrow{x}),\ldots,g_k(\overrightarrow{x}) \) for \( \overrightarrow{x} \in \{0,1\}^n \). Choose random \( \beta, \gamma \), and construct linear combinations:
\[ F(\overrightarrow{x}) = \sum_j \gamma^j \cdot f_j(\overrightarrow{x}), \quad G(\overrightarrow{x}) = \sum_j \gamma^j \cdot g_j(\overrightarrow{x}). \]
Since ProdCheck is not well-suited for distributed scenarios (where multiple parties would need to interact to generate helper polynomials), we transform it into rational sumcheck, based on the following theorem:
Theorem (Logarithmic Derivative): Let two sets \( a_i, b_i \) both have size \( n \). Then the following equivalence holds:
\[ \prod_i(a_i + X) = \prod_i(b_i + X) \iff \sum_i \frac{1}{a_i + X} = \sum_i \frac{1}{b_i + X}. \]
By setting \( X = \beta \) as a random challenge, define:
\[ v_1 = \sum_{\overrightarrow{x} \in \{0,1\}^n} \frac{1}{\beta + F(\overrightarrow{x})}, \quad v_2 = \sum_{\overrightarrow{x} \in \{0,1\}^n} \frac{1}{\beta + G(\overrightarrow{x})}. \]
We require \( v_1 = v_2 \), i.e.,
\[ \sum_{\overrightarrow{x} \in \{0,1\}^n} \left( \frac{1}{\beta + F(\overrightarrow{x})} - \frac{1}{\beta + G(\overrightarrow{x})} \right) = 0. \]
After combining terms, this becomes:
\[ \sum_{\overrightarrow{x} \in \{0,1\}^n} \frac{F(\overrightarrow{x}) - G(\overrightarrow{x})}{(\beta + F(\overrightarrow{x})) \cdot (\beta + G(\overrightarrow{x}))} = 0. \]
Let \( p(\overrightarrow{x}) = F(\overrightarrow{x}) - G(\overrightarrow{x}) \), \( q(\overrightarrow{x}) = (\beta + F(\overrightarrow{x})) \cdot (\beta + G(\overrightarrow{x})) \). Then Rational Sumcheck can be used to verify:
\[ \begin{align*} \sum_{\overrightarrow{x} \in \{0,1\}^n} \frac{p(\overrightarrow{x})}{q(\overrightarrow{x})} = 0. \end{align*} \]
The core goal of Permutation Check via Multiset Equality is to verify that two lists \( f(\overrightarrow{x}) \) and \( g(\overrightarrow{x}) \) are permutations of each other (same elements, possibly different order).
Direct comparison is inefficient, so this is achieved through Multiset Equality. The idea is to attach an index tag to each element so that elements from different positions can be distinguished.
Construct two multisets: one consisting of \( f(\overrightarrow{x}) \) paired with its index under the identity permutation \( s_{id}(\overrightarrow{x}) \); the other consisting of \( g(\overrightarrow{x}) \) paired with its index under some permutation \( \sigma \), denoted \( s_{\sigma}(\overrightarrow{x}) \). If these two multisets are equal, then \( g \) is indeed a permutation of \( f \).
Formally, with input space \( \{0,1\}^n \):
\[ F = \{ (s_{id}(\overrightarrow{x}), f(\overrightarrow{x})) \mid \overrightarrow{x} \in \{0,1\}^n \}, \]
\[ G = \{ (s_{\sigma}(\overrightarrow{x}), g(\overrightarrow{x})) \mid \overrightarrow{x} \in \{0,1\}^{n} \}. \]
Here, \( s_{id}(\overrightarrow{x}) \) is the index under the identity permutation, and \( s_{\sigma}(\overrightarrow{x}) \) is the index under permutation \(\sigma\).
Permutation Check condition:
\[ F = G. \]
To efficiently check multiset equality, use the Random Linear Combination (RLC) technique: sample a random \(\beta \in \mathbb{F}\), and define
\[ p(\overrightarrow{x}) = s_{id}(\overrightarrow{x}) + \beta \cdot f(\overrightarrow{x}), \quad q(\overrightarrow{x}) = s_{\sigma}(\overrightarrow{x}) + \beta \cdot g(\overrightarrow{x}). \]
Check whether \( \{p(\overrightarrow{x})\} = \{q(\overrightarrow{x})\} \). Thanks to the randomness of \(\beta\), the probability of equality holding without \( F = G \) is negligible.
Permutation Check via Multiset Equality is widely used in zero-knowledge proofs and polynomial interactive protocols, including:
The goal of the LogUp Lookup Argument is to verify subset inclusion \( S \subseteq T \), i.e., that all queried elements \( S \) indeed come from the lookup table \( T \).
Unlike direct comparison via polynomial commitments, LogUp relies on a <strong>rational function identity
\[ \sum_{i \in [P]} \frac{1}{x + s_i} = \sum_{j \in [N]} \frac{m_j}{x + r_j}, \]
where \( m_j \) is a nonnegative integer indicating the multiplicity of element \( T_j \). This amounts to comparing the logarithmic derivatives of two polynomials, avoiding large product constraints.
amounts to comparing the logarithmic derivatives of two polynomials, avoiding large product constraints.
Formally, for query set \( S = (S_1, \ldots, S_p) \in \mathbb{F}^P \) and table \( T = (T_1, \ldots, T_N) \in \mathbb{F}^N \):
If \( S \subseteq T \), then there exists \( \overrightarrow{m} \in \mathbb{F}^N \) such that:
\[ \sum_{i \in [P]} \frac{1}{x + S_i} = \sum_{j \in [N]} m_j \cdot \frac{1}{x + T_j} \]
To handle multi-dimensional lookups, introduce a random linear combination parameter \( \alpha \in \mathbb{F} \) to merge multiple lookup relations into one:
\[ X + \alpha \cdot Y \subseteq T_x + \alpha \cdot T_y. \]
Define:
\[ A_i := \prod_{j=1}^t \frac{1}{\beta + s_j(i)}, \quad B_j := \prod_{j=1}^t \frac{1}{\beta + T_j(i)}, \]
with \( \beta \in \mathbb{F} \) a random challenge.
The final core constraint to verify is:
\[ \sum_{i \in [P]} A_i = \sum_{j \in [N]} m_j \cdot B_j. \tag{1} \]
Additionally, correctness of the denominators must be ensured:
\[ A(i) \cdot (SC(i) + \beta) = 1, \tag{2} \] \[ B(j) \cdot (CT(j) + \beta) = 1. \tag{3} \]
Here’s the English translation of your text, keeping the Markdown format intact:
The LogUp Lookup Argument has become the mainstream technique for lookup constraints in modern zero-knowledge proof systems. Typical applications include:
Similar to HyperPlonk, HyperPianist reduces the following relations to the Multivariate Sumcheck Protocol. However, to achieve non-interactive distributed proofs, it adopts a different reduction strategy to convert them into the Sumcheck Protocol.
The Multivariate Sumcheck Protocol serves as a unified reduction framework that can accommodate various relation checks, including zero checks, rational constraints, set consistency, permutation, and lookup. Its advantages are:
Overall, the Multivariate Sumcheck Protocol has become the most versatile reduction tool in modern zero-knowledge proofs, providing a solid foundation for designing efficient and secure proof systems.
Subscribe to the ZKM Blog to stay updated with the latest research by ZKM.
This work systematically explains how various common constraints can be reduced to the Multivariate Sumcheck Protocol. The core idea is to use random linear combinations and logarithmic-derivative techniques to transform “pointwise verification” of universal assertions into a “global summation” form, which can then be efficiently verified by sumcheck. The main methods include: ZeroCheck: Using equality kernels to reduce the assertion “vanishes everywhere on the Boolean hypercube” into a single summation check. Rational Sumcheck: Introducing auxiliary functions and zero checks to convert rational summations into polynomial summations. Multiset Equality Check: Using logarithmic-derivative identities to reduce multiset equality to rational summations. Permutation Check: By attaching indices to elements, reduce the problem to multiset equality, and then to Rational Sumcheck. LogUp Lookup Argument: Using rational-function representations to verify lookup constraints, avoiding expensive product expressions.
These methods are widely applied in modern zero-knowledge proof systems such as Plonk, HyperPlonk, and Halo2, for range checks, wiring checks, permutation arguments, and lookup validations.
We want to prove that a polynomial/function /function \( f(\overrightarrow{x}) \) vanishes on the Boolean hypercube \( \{0, 1\}^n \), i.e.
\[ \forall \overrightarrow{x} \in \{0, 1\}^n, \quad f(\overrightarrow{x}) = 0. \]
The idea is to reduce this "pointwise vanishing" universal assertion to a single "global summation equals zero" check, so that Multivariate Sumcheck can be used in an interactive/non-interactive protocol:
Intuitively, the point-values of \( f \) on the cube act as "weights," and \( \mathit{eq}(\overrightarrow{x}, \overrightarrow{r}) \) serves as an "interpolation kernel" for sampling at random \( \overrightarrow{r} \). If \( f \) is nonzero, then its multilinear extension \( \tilde{f} \) is a nonzero polynomial, which with overwhelming probability does not vanish at a random \( \overrightarrow{r} \).
To verify that \( f(\overrightarrow{x}) = 0 \) holds for all \( \overrightarrow{x} \in \{0,1\}^n \):
\[ \sum_{\overrightarrow{x} \in \{0,1\}^n} \hat{f}(\overrightarrow{x}) = 0. \]
Batch proof: verifying verifying \( f(\overrightarrow{x}) = 0 \land g(\overrightarrow{x}) = 0 \)
Complexity: If \( f \) is multilinear, then \( \hat{f} \) has degree at most 2 per variable (or remains low-degree). Sumcheck requires \( O(n) \) rounds, each handling \( O(1) \)-degree polynomials, making the process efficient overall.
In many proof systems, we need to verify global summation relations involving rational functions, e.g.,
\[ \begin{align*} \sum_{\overrightarrow{x} \in \{0,1\}^n} \frac{p(\overrightarrow{x})}{q(\overrightarrow{x})} = v. \end{align*} \]
Such constraints arise in division gates, lookup arguments, and compressed polynomials. Since sumcheck natively works only on polynomials, we need a reduction strategy to handle rational functions. The idea is to reduce “rational summation” into a combination of polynomial summation and zero checks:
Thus, rational summations are safely reduced to "pure polynomial summation + vanishing check."
Target assertion:
\[ \begin{align*} \sum_{\overrightarrow{x} \in \{0,1\}^n} \frac{p(\overrightarrow{x})}{q(\overrightarrow{x})} = v. \end{align*} \]
Reduction Steps:
If true, then \( f(\overrightarrow{x}) \) must indeed be the inverse of \( q(\overrightarrow{x}) \).
We want to verify whether two function sets take equal values over the Boolean hypercube \( \{0, 1\}^n \): \(f_0(\overrightarrow{x}), f_1(\overrightarrow{x}), \ldots, f_k(\overrightarrow{x}) = g_0(\overrightarrow{x}), g_1(\overrightarrow{x}), \ldots, g_k(\overrightarrow{x})\)
Such constraints are common in proof systems, e.g., Plonk wiring check. The classical approach is to use ProdCheck (product check) to verify that the generated polynomials on both sides are the same. However, ProdCheck requires additional helper polynomials and interaction in distributed settings, which is inefficient.
Solution idea. By the logarithmic derivative identity:
\[ \prod_i(a_i + X) = \prod_i(b_i + X) \quad \Leftrightarrow \quad \sum_i \frac{1}{a_i + X} = \sum_i \frac{1}{b_i + X}, \]
product equality can be reduced to a rational function summation equality. This allows one to use the Rational Sumcheck Protocol directly, without introducing extra helper polynomials.
Key techniques
Principle: Verify whether \( f_0(\overrightarrow{x}),\ldots,f_k(\overrightarrow{x}) = g_0(\overrightarrow{x}),\ldots,g_k(\overrightarrow{x}) \) for \( \overrightarrow{x} \in \{0,1\}^n \). Choose random \( \beta, \gamma \), and construct linear combinations:
\[ F(\overrightarrow{x}) = \sum_j \gamma^j \cdot f_j(\overrightarrow{x}), \quad G(\overrightarrow{x}) = \sum_j \gamma^j \cdot g_j(\overrightarrow{x}). \]
Since ProdCheck is not well-suited for distributed scenarios (where multiple parties would need to interact to generate helper polynomials), we transform it into rational sumcheck, based on the following theorem:
Theorem (Logarithmic Derivative): Let two sets \( a_i, b_i \) both have size \( n \). Then the following equivalence holds:
\[ \prod_i(a_i + X) = \prod_i(b_i + X) \iff \sum_i \frac{1}{a_i + X} = \sum_i \frac{1}{b_i + X}. \]
By setting \( X = \beta \) as a random challenge, define:
\[ v_1 = \sum_{\overrightarrow{x} \in \{0,1\}^n} \frac{1}{\beta + F(\overrightarrow{x})}, \quad v_2 = \sum_{\overrightarrow{x} \in \{0,1\}^n} \frac{1}{\beta + G(\overrightarrow{x})}. \]
We require \( v_1 = v_2 \), i.e.,
\[ \sum_{\overrightarrow{x} \in \{0,1\}^n} \left( \frac{1}{\beta + F(\overrightarrow{x})} - \frac{1}{\beta + G(\overrightarrow{x})} \right) = 0. \]
After combining terms, this becomes:
\[ \sum_{\overrightarrow{x} \in \{0,1\}^n} \frac{F(\overrightarrow{x}) - G(\overrightarrow{x})}{(\beta + F(\overrightarrow{x})) \cdot (\beta + G(\overrightarrow{x}))} = 0. \]
Let \( p(\overrightarrow{x}) = F(\overrightarrow{x}) - G(\overrightarrow{x}) \), \( q(\overrightarrow{x}) = (\beta + F(\overrightarrow{x})) \cdot (\beta + G(\overrightarrow{x})) \). Then Rational Sumcheck can be used to verify:
\[ \begin{align*} \sum_{\overrightarrow{x} \in \{0,1\}^n} \frac{p(\overrightarrow{x})}{q(\overrightarrow{x})} = 0. \end{align*} \]
The core goal of Permutation Check via Multiset Equality is to verify that two lists \( f(\overrightarrow{x}) \) and \( g(\overrightarrow{x}) \) are permutations of each other (same elements, possibly different order).
Direct comparison is inefficient, so this is achieved through Multiset Equality. The idea is to attach an index tag to each element so that elements from different positions can be distinguished.
Construct two multisets: one consisting of \( f(\overrightarrow{x}) \) paired with its index under the identity permutation \( s_{id}(\overrightarrow{x}) \); the other consisting of \( g(\overrightarrow{x}) \) paired with its index under some permutation \( \sigma \), denoted \( s_{\sigma}(\overrightarrow{x}) \). If these two multisets are equal, then \( g \) is indeed a permutation of \( f \).
Formally, with input space \( \{0,1\}^n \):
\[ F = \{ (s_{id}(\overrightarrow{x}), f(\overrightarrow{x})) \mid \overrightarrow{x} \in \{0,1\}^n \}, \]
\[ G = \{ (s_{\sigma}(\overrightarrow{x}), g(\overrightarrow{x})) \mid \overrightarrow{x} \in \{0,1\}^{n} \}. \]
Here, \( s_{id}(\overrightarrow{x}) \) is the index under the identity permutation, and \( s_{\sigma}(\overrightarrow{x}) \) is the index under permutation \(\sigma\).
Permutation Check condition:
\[ F = G. \]
To efficiently check multiset equality, use the Random Linear Combination (RLC) technique: sample a random \(\beta \in \mathbb{F}\), and define
\[ p(\overrightarrow{x}) = s_{id}(\overrightarrow{x}) + \beta \cdot f(\overrightarrow{x}), \quad q(\overrightarrow{x}) = s_{\sigma}(\overrightarrow{x}) + \beta \cdot g(\overrightarrow{x}). \]
Check whether \( \{p(\overrightarrow{x})\} = \{q(\overrightarrow{x})\} \). Thanks to the randomness of \(\beta\), the probability of equality holding without \( F = G \) is negligible.
Permutation Check via Multiset Equality is widely used in zero-knowledge proofs and polynomial interactive protocols, including:
The goal of the LogUp Lookup Argument is to verify subset inclusion \( S \subseteq T \), i.e., that all queried elements \( S \) indeed come from the lookup table \( T \).
Unlike direct comparison via polynomial commitments, LogUp relies on a <strong>rational function identity
\[ \sum_{i \in [P]} \frac{1}{x + s_i} = \sum_{j \in [N]} \frac{m_j}{x + r_j}, \]
where \( m_j \) is a nonnegative integer indicating the multiplicity of element \( T_j \). This amounts to comparing the logarithmic derivatives of two polynomials, avoiding large product constraints.
amounts to comparing the logarithmic derivatives of two polynomials, avoiding large product constraints.
Formally, for query set \( S = (S_1, \ldots, S_p) \in \mathbb{F}^P \) and table \( T = (T_1, \ldots, T_N) \in \mathbb{F}^N \):
If \( S \subseteq T \), then there exists \( \overrightarrow{m} \in \mathbb{F}^N \) such that:
\[ \sum_{i \in [P]} \frac{1}{x + S_i} = \sum_{j \in [N]} m_j \cdot \frac{1}{x + T_j} \]
To handle multi-dimensional lookups, introduce a random linear combination parameter \( \alpha \in \mathbb{F} \) to merge multiple lookup relations into one:
\[ X + \alpha \cdot Y \subseteq T_x + \alpha \cdot T_y. \]
Define:
\[ A_i := \prod_{j=1}^t \frac{1}{\beta + s_j(i)}, \quad B_j := \prod_{j=1}^t \frac{1}{\beta + T_j(i)}, \]
with \( \beta \in \mathbb{F} \) a random challenge.
The final core constraint to verify is:
\[ \sum_{i \in [P]} A_i = \sum_{j \in [N]} m_j \cdot B_j. \tag{1} \]
Additionally, correctness of the denominators must be ensured:
\[ A(i) \cdot (SC(i) + \beta) = 1, \tag{2} \] \[ B(j) \cdot (CT(j) + \beta) = 1. \tag{3} \]
Here’s the English translation of your text, keeping the Markdown format intact:
The LogUp Lookup Argument has become the mainstream technique for lookup constraints in modern zero-knowledge proof systems. Typical applications include:
Similar to HyperPlonk, HyperPianist reduces the following relations to the Multivariate Sumcheck Protocol. However, to achieve non-interactive distributed proofs, it adopts a different reduction strategy to convert them into the Sumcheck Protocol.
The Multivariate Sumcheck Protocol serves as a unified reduction framework that can accommodate various relation checks, including zero checks, rational constraints, set consistency, permutation, and lookup. Its advantages are:
Overall, the Multivariate Sumcheck Protocol has become the most versatile reduction tool in modern zero-knowledge proofs, providing a solid foundation for designing efficient and secure proof systems.
Subscribe to the ZKM Blog to stay updated with the latest research by ZKM.
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