diff --git a/air/src/virtual_column.rs b/air/src/virtual_column.rs
index 0ccc9835..899207fa 100644
--- a/air/src/virtual_column.rs
+++ b/air/src/virtual_column.rs
@@ -1,3 +1,4 @@
+use alloc::borrow::Cow;
 use alloc::vec;
 use alloc::vec::Vec;
 use core::ops::Mul;
@@ -6,8 +7,8 @@ use p3_field::{AbstractField, Field};
 
 /// An affine function over columns in a PAIR.
 #[derive(Clone, Debug)]
-pub struct VirtualPairCol<F: Field> {
-    column_weights: Vec<(PairCol, F)>,
+pub struct VirtualPairCol<'a, F: Field> {
+    column_weights: Cow<'a, [(PairCol, F)]>,
     constant: F,
 }
 
@@ -19,7 +20,7 @@ pub enum PairCol {
 }
 
 impl PairCol {
-    fn get<T: Copy>(&self, preprocessed: &[T], main: &[T]) -> T {
+    pub const fn get<T: Copy>(&self, preprocessed: &[T], main: &[T]) -> T {
         match self {
             PairCol::Preprocessed(i) => preprocessed[*i],
             PairCol::Main(i) => main[*i],
@@ -27,14 +28,28 @@ impl PairCol {
     }
 }
 
-impl<F: Field> VirtualPairCol<F> {
-    pub fn new(column_weights: Vec<(PairCol, F)>, constant: F) -> Self {
+impl<'a, F: Field> VirtualPairCol<'a, F> {
+    pub const fn new(column_weights: Cow<'a, [(PairCol, F)]>, constant: F) -> Self {
         Self {
             column_weights,
             constant,
         }
     }
 
+    pub const fn new_owned(column_weights: Vec<(PairCol, F)>, constant: F) -> Self {
+        Self {
+            column_weights: Cow::Owned(column_weights),
+            constant,
+        }
+    }
+
+    pub const fn new_borrowed(column_weights: &'a [(PairCol, F)], constant: F) -> Self {
+        Self {
+            column_weights: Cow::Borrowed(column_weights),
+            constant,
+        }
+    }
+
     pub fn new_preprocessed(column_weights: Vec<(usize, F)>, constant: F) -> Self {
         Self::new(
             column_weights
@@ -55,6 +70,14 @@ impl<F: Field> VirtualPairCol<F> {
         )
     }
 
+    pub fn get_column_weights(&self) -> &[(PairCol, F)] {
+        &self.column_weights
+    }
+
+    pub const fn get_constant(&self) -> F {
+        self.constant
+    }
+
     #[must_use]
     pub fn one() -> Self {
         Self::constant(F::one())
@@ -63,7 +86,7 @@ impl<F: Field> VirtualPairCol<F> {
     #[must_use]
     pub fn constant(x: F) -> Self {
         Self {
-            column_weights: vec![],
+            column_weights: Cow::Owned(vec![]),
             constant: x,
         }
     }
@@ -71,7 +94,7 @@ impl<F: Field> VirtualPairCol<F> {
     #[must_use]
     pub fn single(column: PairCol) -> Self {
         Self {
-            column_weights: vec![(column, F::one())],
+            column_weights: Cow::Owned(vec![(column, F::one())]),
             constant: F::zero(),
         }
     }
@@ -117,7 +140,7 @@ impl<F: Field> VirtualPairCol<F> {
         Var: Into<Expr> + Copy,
     {
         let mut result = self.constant.into();
-        for (column, weight) in &self.column_weights {
+        for (column, weight) in self.column_weights.iter() {
             result += column.get(preprocessed, main).into() * *weight;
         }
         result
diff --git a/baby-bear/src/baby_bear.rs b/baby-bear/src/baby_bear.rs
index 88f03180..30981483 100644
--- a/baby-bear/src/baby_bear.rs
+++ b/baby-bear/src/baby_bear.rs
@@ -3,8 +3,8 @@ use core::iter::{Product, Sum};
 use core::ops::{Add, AddAssign, Div, Mul, MulAssign, Neg, Sub, SubAssign};
 
 use p3_field::{
-    exp_1725656503, exp_u64_by_squaring, AbstractField, Field, Packable, PrimeField, PrimeField32,
-    PrimeField64, TwoAdicField,
+    exp_1725656503, exp_u64_by_squaring, halve_u32, AbstractField, Field, Packable, PrimeField,
+    PrimeField32, PrimeField64, TwoAdicField,
 };
 use rand::distributions::{Distribution, Standard};
 use rand::Rng;
@@ -45,7 +45,7 @@ pub struct BabyBear {
 impl BabyBear {
     /// create a new `BabyBear` from a canonical `u32`.
     #[inline]
-    pub(crate) const fn new(n: u32) -> Self {
+    pub const fn new(n: u32) -> Self {
         Self { value: to_monty(n) }
     }
 }
@@ -238,6 +238,13 @@ impl Field for BabyBear {
 
         Some(p1110111111111111111111111111111)
     }
+
+    #[inline]
+    fn halve(&self) -> Self {
+        BabyBear {
+            value: halve_u32::<P>(self.value),
+        }
+    }
 }
 
 impl PrimeField for BabyBear {}
diff --git a/commit/src/adapters/extension_mmcs.rs b/commit/src/adapters/extension_mmcs.rs
index 9377586a..319eb431 100644
--- a/commit/src/adapters/extension_mmcs.rs
+++ b/commit/src/adapters/extension_mmcs.rs
@@ -14,7 +14,7 @@ pub struct ExtensionMmcs<F, EF, InnerMmcs> {
 }
 
 impl<F, EF, InnerMmcs> ExtensionMmcs<F, EF, InnerMmcs> {
-    pub fn new(inner: InnerMmcs) -> Self {
+    pub const fn new(inner: InnerMmcs) -> Self {
         Self {
             inner,
             _phantom: PhantomData,
diff --git a/field-testing/src/lib.rs b/field-testing/src/lib.rs
index 107c0db4..0b6959d4 100644
--- a/field-testing/src/lib.rs
+++ b/field-testing/src/lib.rs
@@ -26,6 +26,7 @@ where
     assert_eq!(x + (-x), F::zero());
     assert_eq!(-x, F::zero() - x);
     assert_eq!(x + x, x * F::two());
+    assert_eq!(x, x.halve() * F::two());
     assert_eq!(x * (-x), -x.square());
     assert_eq!(x + y, y + x);
     assert_eq!(x * y, y * x);
diff --git a/field/src/extension/binomial_extension.rs b/field/src/extension/binomial_extension.rs
index b773ad82..d44cfb30 100644
--- a/field/src/extension/binomial_extension.rs
+++ b/field/src/extension/binomial_extension.rs
@@ -218,6 +218,12 @@ impl<F: BinomiallyExtendable<D>, const D: usize> Field for BinomialExtensionFiel
             _ => Some(self.frobenius_inv()),
         }
     }
+
+    fn halve(&self) -> Self {
+        Self {
+            value: self.value.map(|x| x.halve()),
+        }
+    }
 }
 
 impl<F, const D: usize> Display for BinomialExtensionField<F, D>
diff --git a/field/src/field.rs b/field/src/field.rs
index d547b55a..72a374da 100644
--- a/field/src/field.rs
+++ b/field/src/field.rs
@@ -207,6 +207,17 @@ pub trait Field:
     fn inverse(&self) -> Self {
         self.try_inverse().expect("Tried to invert zero")
     }
+
+    /// Computes input/2.
+    /// Should be overwritten by most field implementations to use bitshifts.
+    /// Will error if the field characteristic is 2.
+    #[must_use]
+    fn halve(&self) -> Self {
+        let half = Self::two()
+            .try_inverse()
+            .expect("Cannot divide by 2 in fields with characteristic 2");
+        *self * half
+    }
 }
 
 pub trait PrimeField: Field + Ord {}
diff --git a/field/src/helpers.rs b/field/src/helpers.rs
index 3167c506..1d0747f2 100644
--- a/field/src/helpers.rs
+++ b/field/src/helpers.rs
@@ -99,3 +99,31 @@ pub fn eval_poly<AF: AbstractField>(poly: &[AF], x: AF) -> AF {
     }
     acc
 }
+
+/// Given an element x from a 32 bit field F_P compute x/2.
+#[inline]
+pub fn halve_u32<const P: u32>(input: u32) -> u32 {
+    let shift = (P + 1) >> 1;
+    let shr = input >> 1;
+    let lo_bit = input & 1;
+    let shr_corr = shr + shift;
+    if lo_bit == 0 {
+        shr
+    } else {
+        shr_corr
+    }
+}
+
+/// Given an element x from a 64 bit field F_P compute x/2.
+#[inline]
+pub fn halve_u64<const P: u64>(input: u64) -> u64 {
+    let shift = (P + 1) >> 1;
+    let shr = input >> 1;
+    let lo_bit = input & 1;
+    let shr_corr = shr + shift;
+    if lo_bit == 0 {
+        shr
+    } else {
+        shr_corr
+    }
+}
diff --git a/fri/src/two_adic_pcs.rs b/fri/src/two_adic_pcs.rs
index 1845bef0..5eeea655 100644
--- a/fri/src/two_adic_pcs.rs
+++ b/fri/src/two_adic_pcs.rs
@@ -77,7 +77,7 @@ pub struct TwoAdicFriPcs<C: TwoAdicFriPcsGenericConfig> {
 }
 
 impl<C: TwoAdicFriPcsGenericConfig> TwoAdicFriPcs<C> {
-    pub fn new(fri: FriConfig<C::FriMmcs>, dft: C::Dft, mmcs: C::InputMmcs) -> Self {
+    pub const fn new(fri: FriConfig<C::FriMmcs>, dft: C::Dft, mmcs: C::InputMmcs) -> Self {
         Self { fri, dft, mmcs }
     }
 }
diff --git a/goldilocks/src/lib.rs b/goldilocks/src/lib.rs
index e5f6557b..c53c01c7 100644
--- a/goldilocks/src/lib.rs
+++ b/goldilocks/src/lib.rs
@@ -11,14 +11,17 @@ use core::iter::{Product, Sum};
 use core::ops::{Add, AddAssign, Div, Mul, MulAssign, Neg, Sub, SubAssign};
 
 use p3_field::{
-    exp_10540996611094048183, exp_u64_by_squaring, AbstractField, Field, Packable, PrimeField,
-    PrimeField64, TwoAdicField,
+    exp_10540996611094048183, exp_u64_by_squaring, halve_u64, AbstractField, Field, Packable,
+    PrimeField, PrimeField64, TwoAdicField,
 };
 use p3_util::{assume, branch_hint};
 use rand::distributions::{Distribution, Standard};
 use rand::Rng;
 use serde::{Deserialize, Serialize};
 
+/// The Goldilocks prime
+const P: u64 = 0xFFFF_FFFF_0000_0001;
+
 /// The prime field known as Goldilocks, defined as `F_p` where `p = 2^64 - 2^32 + 1`.
 #[derive(Copy, Clone, Default, Serialize, Deserialize)]
 pub struct Goldilocks {
@@ -210,12 +213,17 @@ impl Field for Goldilocks {
         // compute base^1111111111111111111111111111111011111111111111111111111111111111
         Some(t63.square() * *self)
     }
+
+    #[inline]
+    fn halve(&self) -> Self {
+        Goldilocks::new(halve_u64::<P>(self.value))
+    }
 }
 
 impl PrimeField for Goldilocks {}
 
 impl PrimeField64 for Goldilocks {
-    const ORDER_U64: u64 = 0xFFFF_FFFF_0000_0001;
+    const ORDER_U64: u64 = P;
 
     #[inline]
     fn as_canonical_u64(&self) -> u64 {
diff --git a/keccak-air/Cargo.toml b/keccak-air/Cargo.toml
index c1b5f97d..5d4d95b5 100644
--- a/keccak-air/Cargo.toml
+++ b/keccak-air/Cargo.toml
@@ -22,6 +22,7 @@ p3-keccak = { path = "../keccak" }
 p3-maybe-rayon = { path = "../maybe-rayon" }
 p3-mds = { path = "../mds" }
 p3-merkle-tree = { path = "../merkle-tree" }
+p3-poseidon = {path = "../poseidon"}
 p3-poseidon2 = { path = "../poseidon2" }
 p3-symmetric = { path = "../symmetric" }
 p3-uni-stark = { path = "../uni-stark" }
@@ -38,6 +39,9 @@ name = "prove_baby_bear_poseidon2"
 [[example]]
 name = "prove_goldilocks_keccak"
 
+[[example]]
+name = "prove_goldilocks_poseidon"
+
 [features]
 # TODO: Consider removing, at least when this gets split off into another repository.
 # We should be able to enable p3-maybe-rayon/parallel directly; this just doesn't
diff --git a/keccak-air/examples/prove_goldilocks_poseidon.rs b/keccak-air/examples/prove_goldilocks_poseidon.rs
new file mode 100644
index 00000000..395eb01b
--- /dev/null
+++ b/keccak-air/examples/prove_goldilocks_poseidon.rs
@@ -0,0 +1,83 @@
+use p3_challenger::DuplexChallenger;
+use p3_commit::ExtensionMmcs;
+use p3_dft::Radix2DitParallel;
+use p3_field::extension::BinomialExtensionField;
+use p3_field::Field;
+use p3_fri::{FriConfig, TwoAdicFriPcs, TwoAdicFriPcsConfig};
+use p3_goldilocks::Goldilocks;
+use p3_keccak_air::{generate_trace_rows, KeccakAir};
+use p3_mds::goldilocks::MdsMatrixGoldilocks;
+use p3_merkle_tree::FieldMerkleTreeMmcs;
+use p3_poseidon::Poseidon;
+use p3_symmetric::{PaddingFreeSponge, TruncatedPermutation};
+use p3_uni_stark::{prove, verify, StarkConfig, VerificationError};
+use rand::{random, thread_rng};
+use tracing_forest::util::LevelFilter;
+use tracing_forest::ForestLayer;
+use tracing_subscriber::layer::SubscriberExt;
+use tracing_subscriber::util::SubscriberInitExt;
+use tracing_subscriber::{EnvFilter, Registry};
+
+const NUM_HASHES: usize = 680;
+
+fn main() -> Result<(), VerificationError> {
+    let env_filter = EnvFilter::builder()
+        .with_default_directive(LevelFilter::INFO.into())
+        .from_env_lossy();
+
+    Registry::default()
+        .with(env_filter)
+        .with(ForestLayer::default())
+        .init();
+
+    type Val = Goldilocks;
+    type Challenge = BinomialExtensionField<Val, 2>;
+
+    type Perm = Poseidon<Val, MdsMatrixGoldilocks, 8, 7>;
+    let perm = Perm::new_from_rng(4, 22, MdsMatrixGoldilocks, &mut thread_rng());
+
+    type MyHash = PaddingFreeSponge<Perm, 8, 4, 4>;
+    let hash = MyHash::new(perm.clone());
+
+    type MyCompress = TruncatedPermutation<Perm, 2, 4, 8>;
+    let compress = MyCompress::new(perm.clone());
+
+    type ValMmcs = FieldMerkleTreeMmcs<
+        <Val as Field>::Packing,
+        <Val as Field>::Packing,
+        MyHash,
+        MyCompress,
+        4,
+    >;
+    let val_mmcs = ValMmcs::new(hash, compress);
+
+    type ChallengeMmcs = ExtensionMmcs<Val, Challenge, ValMmcs>;
+    let challenge_mmcs = ChallengeMmcs::new(val_mmcs.clone());
+
+    type Dft = Radix2DitParallel;
+    let dft = Dft {};
+
+    type Challenger = DuplexChallenger<Val, Perm, 8>;
+
+    let fri_config = FriConfig {
+        log_blowup: 1,
+        num_queries: 100,
+        proof_of_work_bits: 16,
+        mmcs: challenge_mmcs,
+    };
+    type Pcs =
+        TwoAdicFriPcs<TwoAdicFriPcsConfig<Val, Challenge, Challenger, Dft, ValMmcs, ChallengeMmcs>>;
+    let pcs = Pcs::new(fri_config, dft, val_mmcs);
+
+    type MyConfig = StarkConfig<Val, Challenge, Pcs, Challenger>;
+    let config = StarkConfig::new(pcs);
+
+    let mut challenger = Challenger::new(perm.clone());
+
+    let inputs = (0..NUM_HASHES).map(|_| random()).collect::<Vec<_>>();
+    let trace = generate_trace_rows::<Val>(inputs);
+    let proof = prove::<MyConfig, _>(&config, &KeccakAir {}, &mut challenger, trace);
+
+    let mut challenger = Challenger::new(perm);
+    verify(&config, &KeccakAir {}, &mut challenger, &proof)
+}
diff --git a/merkle-tree/src/mmcs.rs b/merkle-tree/src/mmcs.rs
index b3ad012a..7556ccc1 100644
--- a/merkle-tree/src/mmcs.rs
+++ b/merkle-tree/src/mmcs.rs
@@ -27,7 +27,7 @@ pub struct FieldMerkleTreeMmcs<P, PW, H, C, const DIGEST_ELEMS: usize> {
 }
 
 impl<P, PW, H, C, const DIGEST_ELEMS: usize> FieldMerkleTreeMmcs<P, PW, H, C, DIGEST_ELEMS> {
-    pub fn new(hash: H, compress: C) -> Self {
+    pub const fn new(hash: H, compress: C) -> Self {
         Self {
             hash,
             compress,
diff --git a/mersenne-31/src/mersenne_31.rs b/mersenne-31/src/mersenne_31.rs
index febed03a..e973f0e6 100644
--- a/mersenne-31/src/mersenne_31.rs
+++ b/mersenne-31/src/mersenne_31.rs
@@ -5,13 +5,16 @@ use core::iter::{Product, Sum};
 use core::ops::{Add, AddAssign, Div, Mul, MulAssign, Neg, Sub, SubAssign};
 
 use p3_field::{
-    exp_1717986917, exp_u64_by_squaring, AbstractField, Field, Packable, PrimeField, PrimeField32,
-    PrimeField64,
+    exp_1717986917, exp_u64_by_squaring, halve_u32, AbstractField, Field, Packable, PrimeField,
+    PrimeField32, PrimeField64,
 };
 use rand::distributions::{Distribution, Standard};
 use rand::Rng;
 use serde::{Deserialize, Serialize};
 
+/// The Mersenne31 prime
+const P: u32 = (1 << 31) - 1;
+
 /// The prime field `F_p` where `p = 2^31 - 1`.
 #[derive(Copy, Clone, Default, Serialize, Deserialize)]
 pub struct Mersenne31 {
@@ -225,12 +228,17 @@ impl Field for Mersenne31 {
             p1111111111111111111111111111.exp_power_of_2(3) * p101;
         Some(p1111111111111111111111111111101)
     }
+
+    #[inline]
+    fn halve(&self) -> Self {
+        Mersenne31::new(halve_u32::<P>(self.value))
+    }
 }
 
 impl PrimeField for Mersenne31 {}
 
 impl PrimeField32 for Mersenne31 {
-    const ORDER_U32: u32 = (1 << 31) - 1;
+    const ORDER_U32: u32 = P;
 
     #[inline]
     fn as_canonical_u32(&self) -> u32 {
diff --git a/symmetric/src/compression.rs b/symmetric/src/compression.rs
index 1a3ad7a0..3347a7ca 100644
--- a/symmetric/src/compression.rs
+++ b/symmetric/src/compression.rs
@@ -58,7 +58,7 @@ where
     T: Clone,
     H: CryptographicHasher<T, [T; CHUNK]>,
 {
-    pub fn new(hasher: H) -> Self {
+    pub const fn new(hasher: H) -> Self {
         Self {
             hasher,
             _phantom: PhantomData,
diff --git a/symmetric/src/serializing_hasher.rs b/symmetric/src/serializing_hasher.rs
index d2e88a22..61e87c57 100644
--- a/symmetric/src/serializing_hasher.rs
+++ b/symmetric/src/serializing_hasher.rs
@@ -15,13 +15,13 @@ pub struct SerializingHasher64<Inner> {
 }
 
 impl<Inner> SerializingHasher32<Inner> {
-    pub fn new(inner: Inner) -> Self {
+    pub const fn new(inner: Inner) -> Self {
         Self { inner }
     }
 }
 
 impl<Inner> SerializingHasher64<Inner> {
-    pub fn new(inner: Inner) -> Self {
+    pub const fn new(inner: Inner) -> Self {
         Self { inner }
     }
 }
diff --git a/uni-stark/src/decompose.rs b/uni-stark/src/decompose.rs
index 1858d614..c6a3a9a4 100644
--- a/uni-stark/src/decompose.rs
+++ b/uni-stark/src/decompose.rs
@@ -53,7 +53,6 @@ fn decompose<F: TwoAdicField>(poly: Vec<F>, shift: F, log_chunks: usize) -> Vec<
     let half_n = poly.len() / 2;
     let g_inv = F::two_adic_generator(log_n).inverse();
 
-    let one_half = F::two().inverse();
     let (first, second) = poly.split_at(half_n);
 
     // Note that
@@ -73,7 +72,7 @@ fn decompose<F: TwoAdicField>(poly: Vec<F>, shift: F, log_chunks: usize) -> Vec<
         .map(|((&a, &b), g_inv_power)| {
             let sum = a + b;
             let diff = a - b;
-            (sum * one_half, diff * one_half * *g_inv_power)
+            (sum.halve(), diff.halve() * *g_inv_power)
         })
         .unzip();