diff --git a/arrabiata/src/constraints.rs b/arrabiata/src/constraints.rs
index 9ee1962263..5f37192220 100644
--- a/arrabiata/src/constraints.rs
+++ b/arrabiata/src/constraints.rs
@@ -322,7 +322,7 @@ impl<Fp: PrimeField> InterpreterEnv for Env<Fp> {
         unimplemented!("TODO")
     }
 
-    fn load_value(&mut self, _pos: Self::Position) -> Self::Variable {
+    fn load_value(&mut self, _row: usize, _pos: Self::Position) -> Self::Variable {
         unimplemented!("TODO")
     }
 }
diff --git a/arrabiata/src/interpreter.rs b/arrabiata/src/interpreter.rs
index e6bb674e23..35541786ca 100644
--- a/arrabiata/src/interpreter.rs
+++ b/arrabiata/src/interpreter.rs
@@ -681,7 +681,7 @@ pub trait InterpreterEnv {
     fn save_value(&mut self, pos: Self::Position) -> Self::Variable;
 
     /// Load the latest value saved in the CPU cache for the position [pos].
-    fn load_value(&mut self, pos: Self::Position) -> Self::Variable;
+    fn load_value(&mut self, row: usize, pos: Self::Position) -> Self::Variable;
 }
 
 /// Run the application
diff --git a/arrabiata/src/witness.rs b/arrabiata/src/witness.rs
index 6d00b47924..2cc438007a 100644
--- a/arrabiata/src/witness.rs
+++ b/arrabiata/src/witness.rs
@@ -8,15 +8,18 @@ use num_integer::Integer;
 use o1_utils::field_helpers::FieldHelpers;
 use poly_commitment::{commitment::CommitmentCurve, srs::SRS, PolyComm, SRS as _};
 use rayon::iter::{IntoParallelRefIterator, ParallelIterator};
-use std::time::Instant;
+use std::{
+    collections::{hash_map::Entry, HashMap},
+    time::Instant,
+};
 
 use crate::{
     columns::{Column, Gadget},
     interpreter::{Instruction, InterpreterEnv, Side},
     poseidon_3_60_0_5_5_fp, poseidon_3_60_0_5_5_fq, BIT_DECOMPOSITION_NUMBER_OF_CHUNKS,
-    MAXIMUM_FIELD_SIZE_IN_BITS, NUMBER_OF_COLUMNS, NUMBER_OF_PUBLIC_INPUTS, NUMBER_OF_SELECTORS,
-    NUMBER_OF_VALUES_TO_ABSORB_PUBLIC_IO, POSEIDON_ALPHA, POSEIDON_ROUNDS_FULL,
-    POSEIDON_STATE_SIZE,
+    MAXIMUM_FIELD_SIZE_IN_BITS, NUMBER_OF_COLUMNS, NUMBER_OF_PERMUTATION_COLUMNS,
+    NUMBER_OF_PUBLIC_INPUTS, NUMBER_OF_SELECTORS, NUMBER_OF_VALUES_TO_ABSORB_PUBLIC_IO,
+    POSEIDON_ALPHA, POSEIDON_ROUNDS_FULL, POSEIDON_STATE_SIZE,
 };
 
 pub const IVC_STARTING_INSTRUCTION: Instruction = Instruction::Poseidon(0);
@@ -93,6 +96,20 @@ pub struct Env<
     // FIXME: I don't like this design. Feel free to suggest a better solution
     pub public_state: [BigInt; NUMBER_OF_PUBLIC_INPUTS],
 
+    /// Keep track of the permutations
+    /// Each value of the array represents a mapping i -> x_i, where i in [n] and
+    /// x_i is the value at the previous step i.
+    ///
+    /// When a value is "saved" at step `i`, the mapping from `i` is created,
+    /// but set to `None`, meaning it has to be "loaded" later.
+    ///
+    /// TODO: The permutation itself should be part of a setup phase, where we
+    /// keep track of the the time we accessed the value.
+    ///
+    /// The permutations are supposed to be on the first
+    /// [NUMBER_OF_PERMUTATION_COLUMNS], and are only valid on the same column.
+    pub permutations: [HashMap<usize, (BigInt, Option<usize>)>; NUMBER_OF_PERMUTATION_COLUMNS],
+
     /// Selectors to activate the gadgets.
     /// The size of the outer vector must be equal to the number of gadgets in
     /// the circuit.
@@ -800,12 +817,45 @@ where
         (x3, y3)
     }
 
-    fn save_value(&mut self, _pos: Self::Position) -> Self::Variable {
-        unimplemented!("TODO")
+    fn save_value(&mut self, pos: Self::Position) -> Self::Variable {
+        let (Column::X(idx), _) = pos else {
+            panic!("The 'cache' can only be used on private columns. To load public values, use a lookup argument")
+        };
+        assert!(idx < NUMBER_OF_PERMUTATION_COLUMNS, "The permutation argument only works on the first {NUMBER_OF_PERMUTATION_COLUMNS} columns");
+        let permutation = &mut self.permutations[idx];
+        // This should never happen as we build from top to bottom, but we never
+        // know...
+        if let Entry::Vacant(e) = permutation.entry(idx) {
+            let v = self.state[idx].clone();
+            e.insert((v, None));
+        } else {
+            panic!("It seems that the index {idx} is already in the permutation, which should never happen has the execution trace is built from top to bottom")
+        };
+        self.state[idx].clone()
     }
 
-    fn load_value(&mut self, _pos: Self::Position) -> Self::Variable {
-        unimplemented!("TODO")
+    fn load_value(&mut self, row: usize, pos: Self::Position) -> Self::Variable {
+        let (Column::X(idx), _) = pos else {
+            panic!("The 'cache' can only be used on private columns. To load public values, use a lookup argument")
+        };
+        assert!(idx < NUMBER_OF_PERMUTATION_COLUMNS, "The permutation argument only works on the first {NUMBER_OF_PERMUTATION_COLUMNS} columns");
+        if let Some(v) = self.permutations[idx].get_mut(&row) {
+            match v {
+                (_, Some(row_prime)) => {
+                    panic!("It seems that the value saved at row {row} has been already read at step {row_prime}")
+                }
+                (res, None) => {
+                    let output = (*res).clone();
+                    *v = (output, Some(row))
+                }
+            }
+        } else {
+            panic!(
+                "No value has been saved at row {row} for the position {:?}",
+                pos
+            )
+        };
+        self.permutations[idx].get(&row).unwrap().0.clone()
     }
 }
 
@@ -918,6 +968,7 @@ impl<
             state: std::array::from_fn(|_| BigInt::from(0_usize)),
             next_state: std::array::from_fn(|_| BigInt::from(0_usize)),
             public_state: std::array::from_fn(|_| BigInt::from(0_usize)),
+            permutations: std::array::from_fn(|_| HashMap::new()),
             selectors,
             challenges,
             current_instruction: IVC_STARTING_INSTRUCTION,