diff --git a/elliptic-curve/src/arithmetic.rs b/elliptic-curve/src/arithmetic.rs
index f09e8d728..53654d57e 100644
--- a/elliptic-curve/src/arithmetic.rs
+++ b/elliptic-curve/src/arithmetic.rs
@@ -1,7 +1,7 @@
 //! Elliptic curve arithmetic traits.
 
 use crate::{
-    ops::{LinearCombination, MulByGenerator, ShrAssign},
+    ops::{LinearCombination, MulByGenerator, Reduce, ShrAssign},
     point::{AffineXCoordinate, AffineYIsOdd},
     scalar::FromUintUnchecked,
     scalar::IsHigh,
@@ -72,6 +72,7 @@ pub trait CurveArithmetic: Curve {
         + Into<Self::Uint>
         + IsHigh
         + PartialOrd
+        + Reduce<Self::Uint, Bytes = FieldBytes<Self>>
         + ShrAssign<usize>
         + ff::Field
         + ff::PrimeField<Repr = FieldBytes<Self>>;
diff --git a/elliptic-curve/src/dev.rs b/elliptic-curve/src/dev.rs
index f63d34b76..2d45bf32f 100644
--- a/elliptic-curve/src/dev.rs
+++ b/elliptic-curve/src/dev.rs
@@ -325,12 +325,18 @@ impl<'a> Product<&'a Scalar> for Scalar {
 }
 
 impl Reduce<U256> for Scalar {
+    type Bytes = FieldBytes;
+
     fn reduce(w: U256) -> Self {
         let (r, underflow) = w.sbb(&MockCurve::ORDER, Limb::ZERO);
         let underflow = Choice::from((underflow.0 >> (Limb::BITS - 1)) as u8);
         let reduced = U256::conditional_select(&w, &r, !underflow);
         Self(ScalarPrimitive::new(reduced).unwrap())
     }
+
+    fn reduce_bytes(_: &FieldBytes) -> Self {
+        todo!()
+    }
 }
 
 impl From<u64> for Scalar {
diff --git a/elliptic-curve/src/lib.rs b/elliptic-curve/src/lib.rs
index fb14e0df7..cda6ebad7 100644
--- a/elliptic-curve/src/lib.rs
+++ b/elliptic-curve/src/lib.rs
@@ -72,7 +72,6 @@ extern crate alloc;
 #[cfg(feature = "std")]
 extern crate std;
 
-pub mod ops;
 pub mod point;
 pub mod scalar;
 
@@ -82,6 +81,8 @@ pub mod dev;
 pub mod ecdh;
 #[cfg(feature = "hash2curve")]
 pub mod hash2curve;
+#[cfg(feature = "arithmetic")]
+pub mod ops;
 #[cfg(feature = "sec1")]
 pub mod sec1;
 #[cfg(feature = "arithmetic")]
diff --git a/elliptic-curve/src/ops.rs b/elliptic-curve/src/ops.rs
index aa16ee701..377558a15 100644
--- a/elliptic-curve/src/ops.rs
+++ b/elliptic-curve/src/ops.rs
@@ -3,8 +3,6 @@
 pub use core::ops::{Add, AddAssign, Mul, Neg, Shr, ShrAssign, Sub, SubAssign};
 
 use crypto_bigint::Integer;
-
-#[cfg(feature = "arithmetic")]
 use {group::Group, subtle::CtOption};
 
 /// Perform an inversion on a field element (i.e. base field element or scalar)
@@ -16,7 +14,6 @@ pub trait Invert {
     fn invert(&self) -> Self::Output;
 }
 
-#[cfg(feature = "arithmetic")]
 impl<F: ff::Field> Invert for F {
     type Output = CtOption<F>;
 
@@ -31,7 +28,6 @@ impl<F: ff::Field> Invert for F {
 /// linear combinations (e.g. Shamir's Trick), or otherwise provides a default
 /// non-optimized implementation.
 // TODO(tarcieri): replace this with a trait from the `group` crate? (see zkcrypto/group#25)
-#[cfg(feature = "arithmetic")]
 pub trait LinearCombination: Group {
     /// Calculates `x * k + y * l`.
     fn lincomb(x: &Self, k: &Self::Scalar, y: &Self, l: &Self::Scalar) -> Self {
@@ -43,7 +39,6 @@ pub trait LinearCombination: Group {
 ///
 /// May use optimizations (e.g. precomputed tables) when available.
 // TODO(tarcieri): replace this with `Group::mul_by_generator``? (see zkcrypto/group#44)
-#[cfg(feature = "arithmetic")]
 pub trait MulByGenerator: Group {
     /// Multiply by the generator of the prime-order subgroup.
     #[must_use]
@@ -54,8 +49,14 @@ pub trait MulByGenerator: Group {
 
 /// Modular reduction.
 pub trait Reduce<Uint: Integer>: Sized {
+    /// Bytes used as input to [`Reduce::reduce_bytes`].
+    type Bytes: AsRef<[u8]>;
+
     /// Perform a modular reduction, returning a field element.
     fn reduce(n: Uint) -> Self;
+
+    /// Interpret the given bytes as an integer and perform a modular reduction.
+    fn reduce_bytes(bytes: &Self::Bytes) -> Self;
 }
 
 /// Modular reduction to a non-zero output.
@@ -65,7 +66,11 @@ pub trait Reduce<Uint: Integer>: Sized {
 ///
 /// End users should use the [`Reduce`] impl on
 /// [`NonZeroScalar`][`crate::NonZeroScalar`] instead.
-pub trait ReduceNonZero<Uint: Integer>: Sized {
+pub trait ReduceNonZero<Uint: Integer>: Reduce<Uint> + Sized {
     /// Perform a modular reduction, returning a field element.
     fn reduce_nonzero(n: Uint) -> Self;
+
+    /// Interpret the given bytes as an integer and perform a modular reduction
+    /// to a non-zero output.
+    fn reduce_nonzero_bytes(bytes: &Self::Bytes) -> Self;
 }
diff --git a/elliptic-curve/src/scalar/nonzero.rs b/elliptic-curve/src/scalar/nonzero.rs
index a4a2dbeca..460d6fc5f 100644
--- a/elliptic-curve/src/scalar/nonzero.rs
+++ b/elliptic-curve/src/scalar/nonzero.rs
@@ -249,24 +249,36 @@ impl<C, I> Reduce<I> for NonZeroScalar<C>
 where
     C: CurveArithmetic,
     I: Integer + ArrayEncoding,
-    Scalar<C>: ReduceNonZero<I>,
+    Scalar<C>: Reduce<I, Bytes = FieldBytes<C>> + ReduceNonZero<I>,
 {
+    type Bytes = FieldBytes<C>;
+
     fn reduce(n: I) -> Self {
         Self::reduce_nonzero(n)
     }
+
+    fn reduce_bytes(bytes: &FieldBytes<C>) -> Self {
+        Self::reduce_nonzero_bytes(bytes)
+    }
 }
 
 impl<C, I> ReduceNonZero<I> for NonZeroScalar<C>
 where
     C: CurveArithmetic,
     I: Integer + ArrayEncoding,
-    Scalar<C>: ReduceNonZero<I>,
+    Scalar<C>: Reduce<I, Bytes = FieldBytes<C>> + ReduceNonZero<I>,
 {
     fn reduce_nonzero(n: I) -> Self {
         let scalar = Scalar::<C>::reduce_nonzero(n);
         debug_assert!(!bool::from(scalar.is_zero()));
         Self::new(scalar).unwrap()
     }
+
+    fn reduce_nonzero_bytes(bytes: &FieldBytes<C>) -> Self {
+        let scalar = Scalar::<C>::reduce_nonzero_bytes(bytes);
+        debug_assert!(!bool::from(scalar.is_zero()));
+        Self::new(scalar).unwrap()
+    }
 }
 
 impl<C> TryFrom<&[u8]> for NonZeroScalar<C>
diff --git a/elliptic-curve/src/scalar/primitive.rs b/elliptic-curve/src/scalar/primitive.rs
index d0a78f083..0da18f589 100644
--- a/elliptic-curve/src/scalar/primitive.rs
+++ b/elliptic-curve/src/scalar/primitive.rs
@@ -2,13 +2,17 @@
 
 use crate::{
     bigint::{prelude::*, Limb, NonZero},
-    ops::{Add, AddAssign, Neg, ShrAssign, Sub, SubAssign},
     scalar::FromUintUnchecked,
     scalar::IsHigh,
     Curve, Error, FieldBytes, Result,
 };
 use base16ct::HexDisplay;
-use core::{cmp::Ordering, fmt, str};
+use core::{
+    cmp::Ordering,
+    fmt,
+    ops::{Add, AddAssign, Neg, ShrAssign, Sub, SubAssign},
+    str,
+};
 use generic_array::GenericArray;
 use rand_core::CryptoRngCore;
 use subtle::{