diff --git a/kimchi/src/circuits/expr.rs b/kimchi/src/circuits/expr.rs
index 4bc6977555..b12b858a67 100644
--- a/kimchi/src/circuits/expr.rs
+++ b/kimchi/src/circuits/expr.rs
@@ -1921,6 +1921,14 @@ impl<F: Field> From<u64> for Expr<ConstantExpr<F>> {
     }
 }
 
+impl<F: Field> Mul<F> for Expr<ConstantExpr<F>> {
+    type Output = Expr<ConstantExpr<F>>;
+
+    fn mul(self, y: F) -> Self::Output {
+        Expr::Constant(ConstantExpr::Literal(y)) * self
+    }
+}
+
 //
 // Display
 //
diff --git a/kimchi/src/circuits/gate.rs b/kimchi/src/circuits/gate.rs
index aadfe84589..cfa2f756a5 100644
--- a/kimchi/src/circuits/gate.rs
+++ b/kimchi/src/circuits/gate.rs
@@ -2,7 +2,7 @@
 
 use crate::circuits::{constraints::ConstraintSystem, domains::EvaluationDomains, wires::*};
 use ark_ff::bytes::ToBytes;
-use ark_ff::{FftField, Field};
+use ark_ff::{FftField, Field, One, Zero};
 use ark_poly::{Evaluations as E, Radix2EvaluationDomain as D};
 use num_traits::cast::ToPrimitive;
 use o1_utils::hasher::CryptoDigest;
@@ -10,6 +10,7 @@ use serde::{Deserialize, Serialize};
 use serde_with::serde_as;
 use std::collections::{hash_map::Entry, HashMap, HashSet};
 use std::io::{Result as IoResult, Write};
+use std::ops::Mul;
 
 type Evaluations<Field> = E<Field, D<Field>>;
 
@@ -68,40 +69,70 @@ pub struct SingleLookup<F> {
 /// analogously using `joint_combiner`.
 ///
 /// This function computes that combined value.
-pub fn combine_table_entry<'a, F: Field, I: DoubleEndedIterator<Item = &'a F>>(
-    joint_combiner: F,
-    v: I,
-) -> F {
-    v.rev().fold(F::zero(), |acc, x| joint_combiner * acc + x)
+pub fn combine_table_entry<'a, F, I>(joint_combiner: F, v: I) -> F
+where
+    F: 'a, // Any references in `F` must have a lifetime longer than `'a`.
+    F: Zero + One + Clone,
+    I: DoubleEndedIterator<Item = &'a F>,
+{
+    v.rev()
+        .fold(F::zero(), |acc, x| joint_combiner.clone() * acc + x.clone())
 }
 
-impl<F: Field> SingleLookup<F> {
+impl<F: Copy> SingleLookup<F> {
     /// Evaluate the linear combination specifying the lookup value to a field element.
-    pub fn evaluate<G: Fn(LocalPosition) -> F>(&self, eval: G) -> F {
+    pub fn evaluate<K, G: Fn(LocalPosition) -> K>(&self, eval: G) -> K
+    where
+        K: Zero,
+        K: Mul<F, Output = K>,
+    {
         self.value
             .iter()
-            .fold(F::zero(), |acc, (c, p)| acc + *c * eval(*p))
+            .fold(K::zero(), |acc, (c, p)| acc + eval(*p) * *c)
     }
 }
 
 /// A spec for checking that the given vector belongs to a vector-valued lookup table.
 #[derive(Clone, Serialize, Deserialize)]
-pub struct JointLookup<F> {
-    pub table_id: usize,
-    pub entry: Vec<SingleLookup<F>>,
+pub struct JointLookup<SingleLookup> {
+    pub table_id: i32,
+    pub entry: Vec<SingleLookup>,
 }
 
-impl<F: Field> JointLookup<F> {
+/// A spec for checking that the given vector belongs to a vector-valued lookup table, where the
+/// components of the vector are computed from a linear combination of locally-accessible cells.
+pub type JointLookupSpec<F> = JointLookup<SingleLookup<F>>;
+
+impl<F: Zero + One + Clone> JointLookup<F> {
     // TODO: Support multiple tables
     /// Evaluate the combined value of a joint-lookup.
-    pub fn evaluate<G: Fn(LocalPosition) -> F>(&self, joint_combiner: F, eval: &G) -> F {
-        let mut res = F::zero();
-        let mut c = F::one();
-        for s in self.entry.iter() {
-            res += c * s.evaluate(eval);
-            c *= joint_combiner;
+    pub fn evaluate(&self, joint_combiner: F) -> F {
+        combine_table_entry(joint_combiner, self.entry.iter())
+    }
+}
+
+impl<F: Copy> JointLookup<SingleLookup<F>> {
+    /// Reduce linear combinations in the lookup entries to a single value, resolving local
+    /// positions using the given function.
+    pub fn reduce<K, G: Fn(LocalPosition) -> K>(&self, eval: &G) -> JointLookup<K>
+    where
+        K: Zero,
+        K: Mul<F, Output = K>,
+    {
+        JointLookup {
+            table_id: self.table_id,
+            entry: self.entry.iter().map(|s| s.evaluate(eval)).collect(),
         }
-        res
+    }
+
+    /// Evaluate the combined value of a joint-lookup, resolving local positions using the given
+    /// function.
+    pub fn evaluate<K, G: Fn(LocalPosition) -> K>(&self, joint_combiner: K, eval: &G) -> K
+    where
+        K: Zero + One + Clone,
+        K: Mul<F, Output = K>,
+    {
+        self.reduce(eval).evaluate(joint_combiner)
     }
 }
 
@@ -158,7 +189,7 @@ pub enum GateType {
 pub struct LookupInfo<F> {
     /// A single lookup constraint is a vector of lookup constraints to be applied at a row.
     /// This is a vector of all the kinds of lookup constraints in this configuration.
-    pub kinds: Vec<Vec<JointLookup<F>>>,
+    pub kinds: Vec<Vec<JointLookupSpec<F>>>,
     /// A map from the kind of gate (and whether it is the current row or next row) to the lookup
     /// constraint (given as an index into `kinds`) that should be applied there, if any.
     pub kinds_map: HashMap<(GateType, CurrOrNext), usize>,
@@ -170,10 +201,10 @@ pub struct LookupInfo<F> {
     /// The maximum joint size of any joint lookup in a constraint in `kinds`. This can be computed from `kinds`.
     pub max_joint_size: u32,
     /// An empty vector.
-    empty: Vec<JointLookup<F>>,
+    empty: Vec<JointLookupSpec<F>>,
 }
 
-fn max_lookups_per_row<F>(kinds: &[Vec<JointLookup<F>>]) -> usize {
+fn max_lookups_per_row<F>(kinds: &[Vec<JointLookupSpec<F>>]) -> usize {
     kinds.iter().fold(0, |acc, x| std::cmp::max(x.len(), acc))
 }
 
@@ -286,7 +317,7 @@ impl<F: FftField> LookupInfo<F> {
     }
 
     /// For each row in the circuit, which lookup-constraints should be enforced at that row.
-    pub fn by_row<'a>(&'a self, gates: &[CircuitGate<F>]) -> Vec<&'a Vec<JointLookup<F>>> {
+    pub fn by_row<'a>(&'a self, gates: &[CircuitGate<F>]) -> Vec<&'a Vec<JointLookupSpec<F>>> {
         let mut kinds = vec![&self.empty; gates.len() + 1];
         for i in 0..gates.len() {
             let typ = gates[i].typ;
@@ -327,7 +358,7 @@ impl GateType {
     ///
     /// See circuits/kimchi/src/polynomials/chacha.rs for an explanation of
     /// how these work.
-    pub fn lookup_kinds<F: Field>() -> (Vec<Vec<JointLookup<F>>>, Vec<GatesLookupSpec>) {
+    pub fn lookup_kinds<F: Field>() -> (Vec<Vec<JointLookupSpec<F>>>, Vec<GatesLookupSpec>) {
         let curr_row = |column| LocalPosition {
             row: CurrOrNext::Curr,
             column,
diff --git a/kimchi/src/circuits/polynomials/lookup.rs b/kimchi/src/circuits/polynomials/lookup.rs
index b3aea32c0e..f10a328494 100644
--- a/kimchi/src/circuits/polynomials/lookup.rs
+++ b/kimchi/src/circuits/polynomials/lookup.rs
@@ -122,11 +122,8 @@
 
 use crate::{
     circuits::{
-        expr::{prologue::*, Column, ConstantExpr, Variable},
-        gate::{
-            CircuitGate, CurrOrNext, JointLookup, LocalPosition, LookupInfo, LookupsUsed,
-            SingleLookup,
-        },
+        expr::{prologue::*, Column, ConstantExpr},
+        gate::{CircuitGate, CurrOrNext, JointLookupSpec, LocalPosition, LookupInfo, LookupsUsed},
         wires::COLUMNS,
     },
     error::ProofError,
@@ -142,29 +139,6 @@ use CurrOrNext::*;
 /// Number of constraints produced by the argument.
 pub const CONSTRAINTS: u32 = 7;
 
-// TODO: Update for multiple tables
-fn single_lookup<F: FftField>(s: &SingleLookup<F>) -> E<F> {
-    // Combine the linear combination.
-    s.value
-        .iter()
-        .map(|(c, pos)| {
-            E::literal(*c)
-                * E::Cell(Variable {
-                    col: Column::Witness(pos.column),
-                    row: pos.row,
-                })
-        })
-        .fold(E::zero(), |acc, e| acc + e)
-}
-
-fn joint_lookup<F: FftField>(j: &JointLookup<F>) -> E<F> {
-    j.entry
-        .iter()
-        .enumerate()
-        .map(|(i, s)| E::constant(ConstantExpr::JointCombiner.pow(i as u64)) * single_lookup(s))
-        .fold(E::zero(), |acc, x| acc + x)
-}
-
 struct AdjacentPairs<A, I: Iterator<Item = A>> {
     prev_second_component: Option<A>,
     i: I,
@@ -356,7 +330,7 @@ pub trait Entry {
 
     fn evaluate(
         p: &Self::Params,
-        j: &JointLookup<Self::Field>,
+        j: &JointLookupSpec<Self::Field>,
         witness: &[Vec<Self::Field>; COLUMNS],
         row: usize,
     ) -> Self;
@@ -370,7 +344,7 @@ impl<F: Field> Entry for CombinedEntry<F> {
 
     fn evaluate(
         joint_combiner: &F,
-        j: &JointLookup<F>,
+        j: &JointLookupSpec<F>,
         witness: &[Vec<F>; COLUMNS],
         row: usize,
     ) -> CombinedEntry<F> {
@@ -395,7 +369,7 @@ impl<F: Field> Entry for UncombinedEntry<F> {
 
     fn evaluate(
         _: &(),
-        j: &JointLookup<F>,
+        j: &JointLookupSpec<F>,
         witness: &[Vec<F>; COLUMNS],
         row: usize,
     ) -> UncombinedEntry<F> {
@@ -661,16 +635,21 @@ pub fn constraints<F: FftField>(configuration: &LookupConfiguration<F>, d1: D<F>
             .collect()
     };
 
+    let eval = |pos: LocalPosition| witness(pos.column, pos.row);
+
     // This is set up so that on rows that have lookups, chunk will be equal
     // to the product over all lookups `f` in that row of `gamma + f`
     // and
     // on non-lookup rows, will be equal to 1.
-    let f_term = |spec: &Vec<_>| {
+    let f_term = |spec: &Vec<JointLookupSpec<_>>| {
         assert!(spec.len() <= lookup_info.max_per_row);
         let padding = complements_with_beta_term[lookup_info.max_per_row - spec.len()].clone();
 
         spec.iter()
-            .map(|j| E::Constant(ConstantExpr::Gamma) + joint_lookup(j))
+            .map(|j| {
+                E::Constant(ConstantExpr::Gamma)
+                    + j.evaluate(E::constant(ConstantExpr::JointCombiner), &eval)
+            })
             .fold(E::Constant(padding), |acc: E<F>, x| acc * x)
     };
     let f_chunk = lookup_info