diff --git a/crypto/crypto/Cargo.toml b/crypto/crypto/Cargo.toml
index 6b78f81e7..23e5e949b 100644
--- a/crypto/crypto/Cargo.toml
+++ b/crypto/crypto/Cargo.toml
@@ -17,8 +17,6 @@ serde = { version = "1.0", default-features = false, features = [
     "alloc",
 ], optional = true }
 rayon = { version = "1.8.0", optional = true }
-rand = { version = "0.8.5", default-features = false }
-rand_chacha = { version = "0.3.1", default-features = false }
 memmap2 = { version = "0.9", optional = true }
 tempfile = { version = "3", optional = true }
 libc = { version = "0.2", optional = true }
diff --git a/crypto/crypto/src/fiat_shamir/default_transcript.rs b/crypto/crypto/src/fiat_shamir/default_transcript.rs
index 819b0f761..d64f805a2 100644
--- a/crypto/crypto/src/fiat_shamir/default_transcript.rs
+++ b/crypto/crypto/src/fiat_shamir/default_transcript.rs
@@ -10,10 +10,33 @@ use math::{
     },
     traits::AsBytes,
 };
-use rand_chacha::{ChaCha20Rng, rand_core::SeedableRng};
 
+/// Bytes produced by one Keccak squeeze; the duplex output buffer holds this
+/// many bytes and hands them out `8` at a time (`SQUEEZE_LEN / 8` u64 candidates
+/// per squeeze).
+const SQUEEZE_LEN: usize = 32;
+
+/// Keccak-sponge Fiat-Shamir transcript with a Plonky3-style duplex output
+/// buffer.
+///
+/// Challenges are derived by squeezing the sponge and rejection-sampling field
+/// coordinates directly from those bytes — there is **no CSPRNG**. Earlier this
+/// type seeded a `ChaCha20Rng` from every squeeze and pulled the field element
+/// from the keystream; on the recursion guest that ChaCha block was pure
+/// software (Keccak is a precompile, ChaCha is not), so it dominated the
+/// challenge-sampling cost while producing bytes the sponge already gives for
+/// free. The output buffer amortizes one squeeze across up to `SQUEEZE_LEN / 8`
+/// 64-bit candidates, so a cubic-extension element (3 coordinates) usually costs
+/// a single squeeze.
 pub struct DefaultTranscript<F: HasDefaultTranscript> {
     hasher: Keccak256,
+    /// Duplex output buffer: bytes squeezed from the sponge, consumed 8 at a
+    /// time by field/`u64` sampling. Positions `[out_pos, SQUEEZE_LEN)` are the
+    /// bytes not yet handed out; `out_pos == SQUEEZE_LEN` means "empty, squeeze
+    /// to refill". Absorbing new data invalidates it (see `append_bytes`) so a
+    /// squeeze can never reflect input appended after it was produced.
+    out_buf: [u8; SQUEEZE_LEN],
+    out_pos: usize,
     phantom: PhantomData<F>,
 }
 
@@ -21,6 +44,8 @@ impl<F: HasDefaultTranscript> Clone for DefaultTranscript<F> {
     fn clone(&self) -> Self {
         Self {
             hasher: self.hasher.clone(),
+            out_buf: self.out_buf,
+            out_pos: self.out_pos,
             phantom: PhantomData,
         }
     }
@@ -34,18 +59,40 @@ where
     pub fn new(data: &[u8]) -> Self {
         let mut res = Self {
             hasher: Keccak256::new(),
+            out_buf: [0u8; SQUEEZE_LEN],
+            // Empty: the first sample forces a squeeze.
+            out_pos: SQUEEZE_LEN,
             phantom: PhantomData,
         };
         res.append_bytes(data);
         res
     }
 
+    /// Raw squeeze: finalize the current sponge state, advance the hash chain by
+    /// absorbing the (reversed) output, and return it. Also invalidates the
+    /// duplex output buffer, so interleaving raw `sample()` calls with buffered
+    /// field/`u64` sampling can never reuse stale squeeze bytes.
     pub fn sample(&mut self) -> [u8; 32] {
         let mut result_hash: [u8; 32] = self.hasher.finalize_reset().into();
         result_hash.reverse();
         self.hasher.update(result_hash);
+        self.out_pos = SQUEEZE_LEN;
         result_hash
     }
+
+    /// Next 64-bit candidate from the duplex output buffer, refilling with one
+    /// squeeze when fewer than 8 bytes remain. Big-endian, matching the byte
+    /// order `sample_u64` used when it read directly from `sample()`.
+    fn next_sample_u64(&mut self) -> u64 {
+        if self.out_pos + 8 > SQUEEZE_LEN {
+            self.out_buf = self.sample();
+            self.out_pos = 0;
+        }
+        let mut bytes = [0u8; 8];
+        bytes.copy_from_slice(&self.out_buf[self.out_pos..self.out_pos + 8]);
+        self.out_pos += 8;
+        u64::from_be_bytes(bytes)
+    }
 }
 
 impl<F> Default for DefaultTranscript<F>
@@ -64,10 +111,17 @@ where
     FieldElement<F>: AsBytes,
 {
     fn append_bytes(&mut self, new_bytes: &[u8]) {
+        // Absorbing new input invalidates any buffered squeeze output: a
+        // subsequent challenge must depend on this input, so drop the bytes
+        // squeezed before it.
+        self.out_pos = SQUEEZE_LEN;
         self.hasher.update(new_bytes);
     }
 
     fn append_field_element(&mut self, element: &FieldElement<F>) {
+        // Absorb, same invalidation as `append_bytes` (the field element's bytes
+        // are streamed straight into the sponge with no intermediate `Vec`).
+        self.out_pos = SQUEEZE_LEN;
         element.stream_bytes(&mut |b| self.hasher.update(b));
     }
 
@@ -76,15 +130,14 @@ where
     }
 
     fn sample_field_element(&mut self) -> FieldElement<F> {
-        let mut rng = <ChaCha20Rng as SeedableRng>::from_seed(self.sample());
-        F::get_random_field_element_from_rng(&mut rng)
+        F::sample_field_element_from(|| self.next_sample_u64())
     }
 
     fn sample_u64(&mut self, upper_bound: u64) -> u64 {
         assert!(upper_bound > 0, "upper_bound must be greater than 0");
         let threshold = upper_bound.wrapping_neg() % upper_bound;
         loop {
-            let candidate = u64::from_be_bytes(self.sample()[..8].try_into().unwrap());
+            let candidate = self.next_sample_u64();
             if candidate >= threshold {
                 return candidate % upper_bound;
             }
diff --git a/crypto/math/src/field/extensions_goldilocks.rs b/crypto/math/src/field/extensions_goldilocks.rs
index 246a3cb87..3bf19b70f 100644
--- a/crypto/math/src/field/extensions_goldilocks.rs
+++ b/crypto/math/src/field/extensions_goldilocks.rs
@@ -568,16 +568,14 @@ impl AsBytes for FieldElement<Degree3GoldilocksExtensionField> {
 }
 
 impl HasDefaultTranscript for Degree3GoldilocksExtensionField {
-    fn get_random_field_element_from_rng(rng: &mut impl rand::Rng) -> FieldElement<Self> {
-        let mut sample = [0u8; 8];
+    fn sample_field_element_from(mut next_u64: impl FnMut() -> u64) -> FieldElement<Self> {
         let mut coeffs = [FpE::zero(), FpE::zero(), FpE::zero()];
 
         for coeff in &mut coeffs {
             loop {
-                rng.fill(&mut sample);
-                let int_sample = u64::from_be_bytes(sample);
-                if int_sample < GOLDILOCKS_PRIME {
-                    *coeff = FpE::from(int_sample);
+                let candidate = next_u64();
+                if candidate < GOLDILOCKS_PRIME {
+                    *coeff = FpE::from(candidate);
                     break;
                 }
             }
diff --git a/crypto/math/src/field/goldilocks.rs b/crypto/math/src/field/goldilocks.rs
index 1d60ee5b2..39fd707b7 100644
--- a/crypto/math/src/field/goldilocks.rs
+++ b/crypto/math/src/field/goldilocks.rs
@@ -545,13 +545,11 @@ impl IsFFTField for GoldilocksField {
 }
 
 impl HasDefaultTranscript for GoldilocksField {
-    fn get_random_field_element_from_rng(rng: &mut impl rand::Rng) -> FieldElement<Self> {
-        let mut sample = [0u8; 8];
+    fn sample_field_element_from(mut next_u64: impl FnMut() -> u64) -> FieldElement<Self> {
         loop {
-            rng.fill(&mut sample);
-            let int_sample = u64::from_be_bytes(sample);
-            if int_sample < GOLDILOCKS_PRIME {
-                return FieldElement::from(int_sample);
+            let candidate = next_u64();
+            if candidate < GOLDILOCKS_PRIME {
+                return FieldElement::from(candidate);
             }
         }
     }
diff --git a/crypto/math/src/field/traits.rs b/crypto/math/src/field/traits.rs
index 04dcc410d..a0e0a7fbc 100644
--- a/crypto/math/src/field/traits.rs
+++ b/crypto/math/src/field/traits.rs
@@ -298,7 +298,11 @@ pub trait IsPrimeField: IsField {
 
 /// This trait is necessary for sampling a random field element with a uniform distribution.
 pub trait HasDefaultTranscript: IsField {
-    /// This function should truncates the sampled bits to the quantity required to represent the order of the base field
-    /// and returns a field element.
-    fn get_random_field_element_from_rng(rng: &mut impl rand::Rng) -> FieldElement<Self>;
+    /// Sample a uniform field element by pulling 64-bit candidates from `next_u64`
+    /// — a transcript squeeze stream — and rejection-sampling each field
+    /// coordinate into its canonical range. Rejection (rather than modular
+    /// reduction) keeps the distribution exactly uniform. The caller feeds bytes
+    /// straight from the Fiat-Shamir sponge, so no separate CSPRNG keystream is
+    /// generated (see `DefaultTranscript`).
+    fn sample_field_element_from(next_u64: impl FnMut() -> u64) -> FieldElement<Self>;
 }