diff --git a/ipa-core/src/ff/boolean.rs b/ipa-core/src/ff/boolean.rs
index 3296e3a4e..db166ed4b 100644
--- a/ipa-core/src/ff/boolean.rs
+++ b/ipa-core/src/ff/boolean.rs
@@ -108,15 +108,23 @@ impl rand::distributions::Distribution<Boolean> for rand::distributions::Standar
     }
 }
 
-impl std::ops::Add for Boolean {
+impl std::ops::Add<&Boolean> for Boolean {
     type Output = Self;
 
     #[allow(clippy::suspicious_arithmetic_impl)]
-    fn add(self, rhs: Self) -> Self::Output {
+    fn add(self, rhs: &Self) -> Self::Output {
         Self(self.0 ^ rhs.0)
     }
 }
 
+impl std::ops::Add for Boolean {
+    type Output = Self;
+
+    fn add(self, rhs: Self) -> Self::Output {
+        std::ops::Add::add(self, &rhs)
+    }
+}
+
 impl std::ops::AddAssign for Boolean {
     #[allow(clippy::assign_op_pattern)]
     fn add_assign(&mut self, rhs: Self) {
@@ -132,12 +140,20 @@ impl std::ops::Neg for Boolean {
     }
 }
 
-impl std::ops::Sub for Boolean {
+impl std::ops::Sub<&Self> for Boolean {
     type Output = Self;
 
     #[allow(clippy::suspicious_arithmetic_impl)]
+    fn sub(self, rhs: &Self) -> Self::Output {
+        std::ops::Add::add(self, rhs)
+    }
+}
+
+impl std::ops::Sub for Boolean {
+    type Output = Self;
+
     fn sub(self, rhs: Self) -> Self::Output {
-        self + rhs
+        std::ops::Sub::sub(self, &rhs)
     }
 }
 
diff --git a/ipa-core/src/ff/curve_points.rs b/ipa-core/src/ff/curve_points.rs
index 32d3077e3..78eccff15 100644
--- a/ipa-core/src/ff/curve_points.rs
+++ b/ipa-core/src/ff/curve_points.rs
@@ -78,11 +78,19 @@ impl Serializable for RP25519 {
 ///## Panics
 /// Panics when decompressing invalid curve point. This can happen when deserialize curve point
 /// from bit array that does not have a valid representation on the curve
+impl std::ops::Add<&Self> for RP25519 {
+    type Output = Self;
+
+    fn add(self, rhs: &Self) -> Self::Output {
+        Self((self.0.decompress().unwrap() + rhs.0.decompress().unwrap()).compress())
+    }
+}
+
 impl std::ops::Add for RP25519 {
     type Output = Self;
 
     fn add(self, rhs: Self) -> Self::Output {
-        Self((self.0.decompress().unwrap() + rhs.0.decompress().unwrap()).compress())
+        std::ops::Add::add(self, &rhs)
     }
 }
 
@@ -107,11 +115,19 @@ impl std::ops::Neg for RP25519 {
 ///## Panics
 /// Panics when decompressing invalid curve point. This can happen when deserialize curve point
 /// from bit array that does not have a valid representation on the curve
+impl std::ops::Sub<&Self> for RP25519 {
+    type Output = Self;
+
+    fn sub(self, rhs: &Self) -> Self::Output {
+        Self((self.0.decompress().unwrap() - rhs.0.decompress().unwrap()).compress())
+    }
+}
+
 impl std::ops::Sub for RP25519 {
     type Output = Self;
 
     fn sub(self, rhs: Self) -> Self::Output {
-        Self((self.0.decompress().unwrap() - rhs.0.decompress().unwrap()).compress())
+        std::ops::Sub::sub(self, &rhs)
     }
 }
 
diff --git a/ipa-core/src/ff/ec_prime_field.rs b/ipa-core/src/ff/ec_prime_field.rs
index 64de59f9e..e3dc3d971 100644
--- a/ipa-core/src/ff/ec_prime_field.rs
+++ b/ipa-core/src/ff/ec_prime_field.rs
@@ -84,11 +84,19 @@ impl rand::distributions::Distribution<Fp25519> for rand::distributions::Standar
     }
 }
 
+impl std::ops::Add<&Self> for Fp25519 {
+    type Output = Self;
+
+    fn add(self, rhs: &Self) -> Self::Output {
+        Self(self.0 + rhs.0)
+    }
+}
+
 impl std::ops::Add for Fp25519 {
     type Output = Self;
 
     fn add(self, rhs: Self) -> Self::Output {
-        Self(self.0 + rhs.0)
+        std::ops::Add::add(self, &rhs)
     }
 }
 
@@ -107,11 +115,19 @@ impl std::ops::Neg for Fp25519 {
     }
 }
 
+impl std::ops::Sub<&Self> for Fp25519 {
+    type Output = Self;
+
+    fn sub(self, rhs: &Self) -> Self::Output {
+        Self(self.0 - rhs.0)
+    }
+}
+
 impl std::ops::Sub for Fp25519 {
     type Output = Self;
 
     fn sub(self, rhs: Self) -> Self::Output {
-        Self(self.0 - rhs.0)
+        std::ops::Sub::sub(self, &rhs)
     }
 }
 
diff --git a/ipa-core/src/ff/galois_field.rs b/ipa-core/src/ff/galois_field.rs
index 0f4874dd0..293f282a8 100644
--- a/ipa-core/src/ff/galois_field.rs
+++ b/ipa-core/src/ff/galois_field.rs
@@ -287,10 +287,17 @@ macro_rules! bit_array_impl {
                 }
             }
 
+            impl std::ops::Sub<&$name> for $name {
+                type Output = Self;
+                fn sub(self, rhs: &Self) -> Self::Output {
+                    std::ops::Add::add(self, rhs)
+                }
+            }
+
             impl std::ops::Sub for $name {
                 type Output = Self;
                 fn sub(self, rhs: Self) -> Self::Output {
-                    self + rhs
+                    std::ops::Sub::sub(self, &rhs)
                 }
             }
 
diff --git a/ipa-core/src/ff/prime_field.rs b/ipa-core/src/ff/prime_field.rs
index cfe90bd28..10ec87694 100644
--- a/ipa-core/src/ff/prime_field.rs
+++ b/ipa-core/src/ff/prime_field.rs
@@ -117,10 +117,10 @@ macro_rules! field_impl {
             const PRIME: Self::PrimeInteger = $prime;
         }
 
-        impl std::ops::Add for $field {
+        impl std::ops::Add<&$field> for $field {
             type Output = Self;
 
-            fn add(self, rhs: Self) -> Self::Output {
+            fn add(self, rhs: &Self) -> Self::Output {
                 let c = u64::from;
                 debug_assert!(c(Self::PRIME) < (u64::MAX >> 1));
                 #[allow(clippy::cast_possible_truncation)]
@@ -128,6 +128,14 @@ macro_rules! field_impl {
             }
         }
 
+        impl std::ops::Add for $field {
+            type Output = Self;
+
+            fn add(self, rhs: Self) -> Self::Output {
+                std::ops::Add::add(self, &rhs)
+            }
+        }
+
         impl std::ops::AddAssign for $field {
             #[allow(clippy::assign_op_pattern)]
             fn add_assign(&mut self, rhs: Self) {
@@ -143,10 +151,10 @@ macro_rules! field_impl {
             }
         }
 
-        impl std::ops::Sub for $field {
+        impl std::ops::Sub<&$field> for $field {
             type Output = Self;
 
-            fn sub(self, rhs: Self) -> Self::Output {
+            fn sub(self, rhs: &Self) -> Self::Output {
                 let c = u64::from;
                 debug_assert!(c(Self::PRIME) < (u64::MAX >> 1));
                 // TODO(mt) - constant time?
@@ -158,6 +166,14 @@ macro_rules! field_impl {
             }
         }
 
+        impl std::ops::Sub for $field {
+            type Output = Self;
+
+            fn sub(self, rhs: Self) -> Self::Output {
+                std::ops::Sub::sub(self, &rhs)
+            }
+        }
+
         impl std::ops::SubAssign for $field {
             #[allow(clippy::assign_op_pattern)]
             fn sub_assign(&mut self, rhs: Self) {
diff --git a/ipa-core/src/protocol/basics/reshare.rs b/ipa-core/src/protocol/basics/reshare.rs
index cb33a8146..28d9169a8 100644
--- a/ipa-core/src/protocol/basics/reshare.rs
+++ b/ipa-core/src/protocol/basics/reshare.rs
@@ -74,7 +74,7 @@ impl<C: Context, F: Field> Reshare<C> for Replicated<F> {
                 .await?;
 
             // Sleep until `to_helper.right` sends us their part2 value
-            let part2 = ctx
+            let part2: F = ctx
                 .recv_channel(to_helper.peer(Direction::Right))
                 .receive(record_id)
                 .await?;
diff --git a/ipa-core/src/protocol/context/mod.rs b/ipa-core/src/protocol/context/mod.rs
index ecb569d63..83adb7d5f 100644
--- a/ipa-core/src/protocol/context/mod.rs
+++ b/ipa-core/src/protocol/context/mod.rs
@@ -690,7 +690,7 @@ mod tests {
             ctx.role().peer(Direction::Right),
         );
         let record_id = index.into();
-        let (l, r) = ctx.prss().generate_fields(record_id);
+        let (l, r): (F, F) = ctx.prss().generate_fields(record_id);
 
         let (seq_l, seq_r) = {
             let ctx = ctx.narrow(&format!("seq-prss-{record_id}"));
diff --git a/ipa-core/src/protocol/ipa_prf/shuffle/sharded.rs b/ipa-core/src/protocol/ipa_prf/shuffle/sharded.rs
index 6a37fd6f3..0f2351439 100644
--- a/ipa-core/src/protocol/ipa_prf/shuffle/sharded.rs
+++ b/ipa-core/src/protocol/ipa_prf/shuffle/sharded.rs
@@ -369,7 +369,7 @@ where
                 send_channel.send(record_id, c1),
                 recv_channel.receive(record_id),
             )
-            .map_ok(move |((), c2)| S::new(b, c1 + c2))
+            .map_ok(move |((), c2): ((), S::Share)| S::new(b, c1 + c2))
         }))
         .await?;
 
diff --git a/ipa-core/src/protocol/prss/mod.rs b/ipa-core/src/protocol/prss/mod.rs
index 3ec697ba9..46a4be17c 100644
--- a/ipa-core/src/protocol/prss/mod.rs
+++ b/ipa-core/src/protocol/prss/mod.rs
@@ -655,8 +655,8 @@ pub mod test {
         let s3 = p3.indexed(&step);
 
         let r1: Fp31 = random(&*s1, IDX1);
-        let r2 = random(&*s2, IDX1);
-        let r3 = random(&*s3, IDX1);
+        let r2: Fp31 = random(&*s2, IDX1);
+        let r3: Fp31 = random(&*s3, IDX1);
         let v1 = r1 + r2 + r3;
 
         // There isn't enough entropy in this field (~5 bits) to be sure that the test will pass.
@@ -664,8 +664,8 @@ pub mod test {
         let mut v2 = Fp31::truncate_from(0_u8);
         for i in IDX2..(IDX2 + 21) {
             let r1: Fp31 = random(&*s1, i);
-            let r2 = random(&*s2, i);
-            let r3 = random(&*s3, i);
+            let r2: Fp31 = random(&*s2, i);
+            let r3: Fp31 = random(&*s3, i);
 
             v2 = r1 + r2 + r3;
             if v1 != v2 {
diff --git a/ipa-core/src/secret_sharing/mod.rs b/ipa-core/src/secret_sharing/mod.rs
index 3e3085149..384f57387 100644
--- a/ipa-core/src/secret_sharing/mod.rs
+++ b/ipa-core/src/secret_sharing/mod.rs
@@ -36,12 +36,15 @@ use crate::{
 
 /// Operations supported for weak shared values.
 pub trait Additive<Rhs = Self, Output = Self>:
-    AddSub<Rhs, Output> + AddSubAssign<Rhs> + Neg<Output = Output>
+    AddSub<Rhs, Output> + for<'a> AddSub<&'a Rhs, Output> + AddSubAssign<Rhs> + Neg<Output = Output>
 {
 }
 
 impl<T, Rhs, Output> Additive<Rhs, Output> for T where
-    T: AddSub<Rhs, Output> + AddSubAssign<Rhs> + Neg<Output = Output>
+    T: AddSub<Rhs, Output>
+        + for<'a> AddSub<&'a Rhs, Output>
+        + AddSubAssign<Rhs>
+        + Neg<Output = Output>
 {
 }