diff --git a/CHANGELOG.md b/CHANGELOG.md
index d5df23a..768dde2 100644
--- a/CHANGELOG.md
+++ b/CHANGELOG.md
@@ -21,6 +21,7 @@ and this project adheres to Rust's notion of
   - `FieldExt::from_u64` (use `From<u64> for ff::PrimeField` instead).
   - `FieldExt::{from_bytes, read, to_bytes, write}`
     (use `ff::PrimeField::{from_repr, to_repr}` instead).
+  - `FieldExt::rand` (use `ff::Field::random` instead).
 
 ## [0.2.1] - 2021-09-17
 ### Changed
diff --git a/Cargo.toml b/Cargo.toml
index 50b86c2..bbbe635 100644
--- a/Cargo.toml
+++ b/Cargo.toml
@@ -52,6 +52,8 @@ subtle = { version = "2.3", default-features = false }
 lazy_static = { version = "1.4.0", optional = true }
 
 [features]
-default = ["bits", "std"]
+default = ["bits", "sqrt-table", "std"]
+alloc = ["group/alloc"]
 bits = ["ff/bits"]
-std = ["group/alloc", "lazy_static", "rand/getrandom"]
+sqrt-table = ["alloc", "lazy_static"]
+std = ["alloc"]
diff --git a/src/arithmetic.rs b/src/arithmetic.rs
index 0fef7f5..e96a074 100644
--- a/src/arithmetic.rs
+++ b/src/arithmetic.rs
@@ -10,13 +10,11 @@ mod fields;
 pub(crate) use fields::*;
 
 pub use curves::*;
-#[cfg(feature = "std")]
 pub use fields::*;
 
 /// This represents an element of a group with basic operations that can be
 /// performed. This allows an FFT implementation (for example) to operate
 /// generically over either a field or elliptic curve group.
-#[cfg(feature = "std")]
 pub trait Group: Copy + Clone + Send + Sync + 'static {
     /// The group is assumed to be of prime order $p$. `Scalar` is the
     /// associated scalar field of size $p$.
diff --git a/src/arithmetic/curves.rs b/src/arithmetic/curves.rs
index d954893..4bd128f 100644
--- a/src/arithmetic/curves.rs
+++ b/src/arithmetic/curves.rs
@@ -23,6 +23,7 @@ use std::{
 /// Currently requires the `std` feature flag because of `hash_to_curve`, and
 /// `CurveAffine::{read, write}`.
 #[cfg(feature = "std")]
+#[cfg_attr(docsrs, doc(cfg(feature = "std")))]
 pub trait CurveExt:
     PrimeCurve<Affine = <Self as CurveExt>::AffineExt>
     + group::Group<Scalar = <Self as CurveExt>::ScalarExt>
@@ -90,6 +91,7 @@ pub trait CurveExt:
 /// This trait is the affine counterpart to `Curve` and is used for
 /// serialization, storage in memory, and inspection of $x$ and $y$ coordinates.
 #[cfg(feature = "std")]
+#[cfg_attr(docsrs, doc(cfg(feature = "std")))]
 pub trait CurveAffine:
     PrimeCurveAffine<
         Scalar = <Self as CurveAffine>::ScalarExt,
@@ -145,6 +147,7 @@ pub trait CurveAffine:
 
 /// The affine coordinates of a point on an elliptic curve.
 #[cfg(feature = "std")]
+#[cfg_attr(docsrs, doc(cfg(feature = "std")))]
 #[derive(Clone, Copy, Debug, Default)]
 pub struct Coordinates<C: CurveAffine> {
     pub(crate) x: C::Base,
diff --git a/src/arithmetic/fields.rs b/src/arithmetic/fields.rs
index dbea4f8..3f47a89 100644
--- a/src/arithmetic/fields.rs
+++ b/src/arithmetic/fields.rs
@@ -4,22 +4,21 @@
 use core::mem::size_of;
 
 use static_assertions::const_assert;
-use subtle::Choice;
+use subtle::{Choice, ConditionallySelectable, CtOption};
 
-#[cfg(not(feature = "std"))]
-use subtle::CtOption;
-
-#[cfg(feature = "std")]
 use super::Group;
 
-#[cfg(feature = "std")]
-use std::{assert, boxed::Box, convert::TryInto, marker::PhantomData, vec::Vec};
+use core::assert;
+
+#[cfg(feature = "sqrt-table")]
+use alloc::{boxed::Box, vec::Vec};
+#[cfg(feature = "sqrt-table")]
+use core::{convert::TryInto, marker::PhantomData};
 
 const_assert!(size_of::<usize>() >= 4);
 
 /// A trait that exposes additional operations related to calculating square roots of
 /// prime-order finite fields.
-#[cfg(feature = "std")]
 pub trait SqrtRatio: ff::PrimeField {
     /// The value $(T-1)/2$ such that $2^S \cdot T = p - 1$ with $T$ odd.
     const T_MINUS1_OVER2: [u64; 4];
@@ -54,7 +53,40 @@ pub trait SqrtRatio: ff::PrimeField {
     /// implementation of the SSWU hash-to-curve algorithm.
     ///
     /// The choice of root from sqrt is unspecified.
-    fn sqrt_ratio(num: &Self, div: &Self) -> (Choice, Self);
+    fn sqrt_ratio(num: &Self, div: &Self) -> (Choice, Self) {
+        // General implementation:
+        //
+        // a = num * inv0(div)
+        //   = {    0    if div is zero
+        //     { num/div otherwise
+        //
+        // b = G_S * a
+        //   = {      0      if div is zero
+        //     { G_S*num/div otherwise
+        //
+        // Since G_S is non-square, a and b are either both zero (and both square), or
+        // only one of them is square. We can therefore choose the square root to return
+        // based on whether a is square, but for the boolean output we need to handle the
+        // num != 0 && div == 0 case specifically.
+
+        let a = div.invert().unwrap_or_else(Self::zero) * num;
+        let b = a * Self::root_of_unity();
+        let sqrt_a = a.sqrt();
+        let sqrt_b = b.sqrt();
+
+        let num_is_zero = num.is_zero();
+        let div_is_zero = div.is_zero();
+        let is_square = sqrt_a.is_some();
+        let is_nonsquare = sqrt_b.is_some();
+        assert!(bool::from(
+            num_is_zero | div_is_zero | (is_square ^ is_nonsquare)
+        ));
+
+        (
+            is_square & !(!num_is_zero & div_is_zero),
+            CtOption::conditional_select(&sqrt_b, &sqrt_a, is_square).unwrap(),
+        )
+    }
 
     /// Equivalent to `Self::sqrt_ratio(self, one())`.
     fn sqrt_alt(&self) -> (Choice, Self) {
@@ -64,7 +96,6 @@ pub trait SqrtRatio: ff::PrimeField {
 
 /// This trait is a common interface for dealing with elements of a finite
 /// field.
-#[cfg(feature = "std")]
 pub trait FieldExt: SqrtRatio + From<bool> + Ord + Group<Scalar = Self> {
     /// Modulus of the field written as a string for display purposes
     const MODULUS: &'static str;
@@ -81,11 +112,6 @@ pub trait FieldExt: SqrtRatio + From<bool> + Ord + Group<Scalar = Self> {
     /// Element of multiplicative order $3$.
     const ZETA: Self;
 
-    /// This computes a random element of the field using system randomness.
-    fn rand() -> Self {
-        Self::random(rand::rngs::OsRng)
-    }
-
     /// Obtains a field element congruent to the integer `v`.
     fn from_u128(v: u128) -> Self;
 
@@ -118,12 +144,13 @@ pub trait FieldExt: SqrtRatio + From<bool> + Ord + Group<Scalar = Self> {
 /// https://eprint.iacr.org/2012/685.pdf (page 12, algorithm 5)
 ///
 /// `tm1d2` should be set to `(t - 1) // 2`, where `t = (modulus - 1) >> F::S`.
-#[cfg(not(feature = "std"))]
+#[cfg(not(feature = "sqrt-table"))]
+#[cfg_attr(docsrs, doc(cfg(not(feature = "sqrt-table"))))]
 pub(crate) fn sqrt_tonelli_shanks<F: ff::PrimeField, S: AsRef<[u64]>>(
     f: &F,
     tm1d2: S,
 ) -> CtOption<F> {
-    use subtle::{ConditionallySelectable, ConstantTimeEq};
+    use subtle::ConstantTimeEq;
 
     // w = self^((t - 1) // 2)
     let w = f.pow_vartime(tm1d2);
@@ -164,7 +191,8 @@ pub(crate) fn sqrt_tonelli_shanks<F: ff::PrimeField, S: AsRef<[u64]>>(
 }
 
 /// Parameters for a perfect hash function used in square root computation.
-#[cfg(feature = "std")]
+#[cfg(feature = "sqrt-table")]
+#[cfg_attr(docsrs, doc(cfg(feature = "sqrt-table")))]
 #[derive(Debug)]
 struct SqrtHasher<F: FieldExt> {
     hash_xor: u32,
@@ -172,7 +200,7 @@ struct SqrtHasher<F: FieldExt> {
     marker: PhantomData<F>,
 }
 
-#[cfg(feature = "std")]
+#[cfg(feature = "sqrt-table")]
 impl<F: FieldExt> SqrtHasher<F> {
     /// Returns a perfect hash of x for use with SqrtTables::inv.
     fn hash(&self, x: &F) -> usize {
@@ -185,7 +213,8 @@ impl<F: FieldExt> SqrtHasher<F> {
 }
 
 /// Tables used for square root computation.
-#[cfg(feature = "std")]
+#[cfg(feature = "sqrt-table")]
+#[cfg_attr(docsrs, doc(cfg(feature = "sqrt-table")))]
 #[derive(Debug)]
 pub struct SqrtTables<F: FieldExt> {
     hasher: SqrtHasher<F>,
@@ -196,11 +225,11 @@ pub struct SqrtTables<F: FieldExt> {
     g3: Box<[F; 129]>,
 }
 
-#[cfg(feature = "std")]
+#[cfg(feature = "sqrt-table")]
 impl<F: FieldExt> SqrtTables<F> {
     /// Build tables given parameters for the perfect hash.
     pub fn new(hash_xor: u32, hash_mod: usize) -> Self {
-        use std::vec;
+        use alloc::vec;
 
         let hasher = SqrtHasher {
             hash_xor,
diff --git a/src/curves.rs b/src/curves.rs
index 8661033..7a0db20 100644
--- a/src/curves.rs
+++ b/src/curves.rs
@@ -103,6 +103,7 @@ macro_rules! new_curve_impl {
         }
 
         #[cfg(feature = "std")]
+        #[cfg_attr(docsrs, doc(cfg(feature = "std")))]
         impl group::WnafGroup for $name {
             fn recommended_wnaf_for_num_scalars(num_scalars: usize) -> usize {
                 // Copied from bls12_381::g1, should be updated.
diff --git a/src/fields/fp.rs b/src/fields/fp.rs
index d27928f..17be5d6 100644
--- a/src/fields/fp.rs
+++ b/src/fields/fp.rs
@@ -6,16 +6,16 @@ use ff::PrimeField;
 use rand::RngCore;
 use subtle::{Choice, ConditionallySelectable, ConstantTimeEq, CtOption};
 
-#[cfg(feature = "std")]
+#[cfg(feature = "sqrt-table")]
 use lazy_static::lazy_static;
 
 #[cfg(feature = "bits")]
 use ff::{FieldBits, PrimeFieldBits};
 
-use crate::arithmetic::{adc, mac, sbb};
+use crate::arithmetic::{adc, mac, sbb, FieldExt, Group, SqrtRatio};
 
-#[cfg(feature = "std")]
-use crate::arithmetic::{FieldExt, Group, SqrtRatio, SqrtTables};
+#[cfg(feature = "sqrt-table")]
+use crate::arithmetic::SqrtTables;
 
 /// This represents an element of $\mathbb{F}_p$ where
 ///
@@ -224,7 +224,6 @@ const ROOT_OF_UNITY: Fp = Fp::from_raw([
 /// GENERATOR^{2^s} where t * 2^s + 1 = p
 /// with t odd. In other words, this
 /// is a t root of unity.
-#[cfg(feature = "std")]
 const DELTA: Fp = Fp::from_raw([
     0x6a6ccd20dd7b9ba2,
     0xf5e4f3f13eee5636,
@@ -463,7 +462,6 @@ impl<'a> From<&'a Fp> for [u8; 32] {
     }
 }
 
-#[cfg(feature = "std")]
 impl Group for Fp {
     type Scalar = Fp;
 
@@ -514,13 +512,13 @@ impl ff::Field for Fp {
 
     /// Computes the square root of this element, if it exists.
     fn sqrt(&self) -> CtOption<Self> {
-        #[cfg(feature = "std")]
+        #[cfg(feature = "sqrt-table")]
         {
             let (is_square, res) = FP_TABLES.sqrt_alt(self);
             CtOption::new(res, is_square)
         }
 
-        #[cfg(not(feature = "std"))]
+        #[cfg(not(feature = "sqrt-table"))]
         crate::arithmetic::sqrt_tonelli_shanks(self, &T_MINUS1_OVER2)
     }
 
@@ -623,6 +621,7 @@ type ReprBits = [u32; 8];
 type ReprBits = [u64; 4];
 
 #[cfg(feature = "bits")]
+#[cfg_attr(docsrs, doc(cfg(feature = "bits")))]
 impl PrimeFieldBits for Fp {
     type ReprBits = ReprBits;
 
@@ -663,13 +662,13 @@ impl PrimeFieldBits for Fp {
     }
 }
 
-#[cfg(feature = "std")]
+#[cfg(feature = "sqrt-table")]
 lazy_static! {
     // The perfect hash parameters are found by `squareroottab.sage` in zcash/pasta.
+    #[cfg_attr(docsrs, doc(cfg(feature = "sqrt-table")))]
     static ref FP_TABLES: SqrtTables<Fp> = SqrtTables::new(0x11BE, 1098);
 }
 
-#[cfg(feature = "std")]
 impl SqrtRatio for Fp {
     const T_MINUS1_OVER2: [u64; 4] = T_MINUS1_OVER2;
 
@@ -711,16 +710,17 @@ impl SqrtRatio for Fp {
         tmp.0[0] as u32
     }
 
+    #[cfg(feature = "sqrt-table")]
     fn sqrt_ratio(num: &Self, div: &Self) -> (Choice, Self) {
         FP_TABLES.sqrt_ratio(num, div)
     }
 
+    #[cfg(feature = "sqrt-table")]
     fn sqrt_alt(&self) -> (Choice, Self) {
         FP_TABLES.sqrt_alt(self)
     }
 }
 
-#[cfg(feature = "std")]
 impl FieldExt for Fp {
     const MODULUS: &'static str =
         "0x40000000000000000000000000000000224698fc094cf91b992d30ed00000001";
@@ -770,7 +770,7 @@ impl FieldExt for Fp {
     }
 }
 
-#[cfg(all(test, feature = "std"))]
+#[cfg(test)]
 use ff::Field;
 
 #[test]
@@ -788,7 +788,6 @@ fn test_inv() {
     assert_eq!(inv, INV);
 }
 
-#[cfg(feature = "std")]
 #[test]
 fn test_sqrt() {
     // NB: TWO_INV is standing in as a "random" field element
@@ -796,7 +795,6 @@ fn test_sqrt() {
     assert!(v == Fp::TWO_INV || (-v) == Fp::TWO_INV);
 }
 
-#[cfg(feature = "std")]
 #[test]
 fn test_pow_by_t_minus1_over2() {
     // NB: TWO_INV is standing in as a "random" field element
@@ -804,7 +802,6 @@ fn test_pow_by_t_minus1_over2() {
     assert!(v == ff::Field::pow_vartime(&Fp::TWO_INV, &T_MINUS1_OVER2));
 }
 
-#[cfg(feature = "std")]
 #[test]
 fn test_sqrt_ratio_and_alt() {
     // (true, sqrt(num/div)), if num and div are nonzero and num/div is a square in the field
@@ -851,7 +848,6 @@ fn test_sqrt_ratio_and_alt() {
     assert!(v == expected);
 }
 
-#[cfg(feature = "std")]
 #[test]
 fn test_zeta() {
     assert_eq!(
@@ -867,7 +863,6 @@ fn test_zeta() {
     assert!(c == Fp::one());
 }
 
-#[cfg(feature = "std")]
 #[test]
 fn test_root_of_unity() {
     assert_eq!(
@@ -876,19 +871,16 @@ fn test_root_of_unity() {
     );
 }
 
-#[cfg(feature = "std")]
 #[test]
 fn test_inv_root_of_unity() {
     assert_eq!(Fp::ROOT_OF_UNITY_INV, Fp::root_of_unity().invert().unwrap());
 }
 
-#[cfg(feature = "std")]
 #[test]
 fn test_inv_2() {
     assert_eq!(Fp::TWO_INV, Fp::from(2).invert().unwrap());
 }
 
-#[cfg(feature = "std")]
 #[test]
 fn test_delta() {
     assert_eq!(Fp::DELTA, GENERATOR.pow(&[1u64 << Fp::S, 0, 0, 0]));
diff --git a/src/fields/fq.rs b/src/fields/fq.rs
index f7111be..618bf54 100644
--- a/src/fields/fq.rs
+++ b/src/fields/fq.rs
@@ -6,16 +6,16 @@ use ff::PrimeField;
 use rand::RngCore;
 use subtle::{Choice, ConditionallySelectable, ConstantTimeEq, CtOption};
 
-#[cfg(feature = "std")]
+#[cfg(feature = "sqrt-table")]
 use lazy_static::lazy_static;
 
 #[cfg(feature = "bits")]
 use ff::{FieldBits, PrimeFieldBits};
 
-use crate::arithmetic::{adc, mac, sbb};
+use crate::arithmetic::{adc, mac, sbb, FieldExt, Group, SqrtRatio};
 
-#[cfg(feature = "std")]
-use crate::arithmetic::{FieldExt, Group, SqrtRatio, SqrtTables};
+#[cfg(feature = "sqrt-table")]
+use crate::arithmetic::SqrtTables;
 
 /// This represents an element of $\mathbb{F}_q$ where
 ///
@@ -224,7 +224,6 @@ const ROOT_OF_UNITY: Fq = Fq::from_raw([
 /// GENERATOR^{2^s} where t * 2^s + 1 = q
 /// with t odd. In other words, this
 /// is a t root of unity.
-#[cfg(feature = "std")]
 const DELTA: Fq = Fq::from_raw([
     0x8494392472d1683c,
     0xe3ac3376541d1140,
@@ -463,7 +462,6 @@ impl<'a> From<&'a Fq> for [u8; 32] {
     }
 }
 
-#[cfg(feature = "std")]
 impl Group for Fq {
     type Scalar = Fq;
 
@@ -514,13 +512,13 @@ impl ff::Field for Fq {
 
     /// Computes the square root of this element, if it exists.
     fn sqrt(&self) -> CtOption<Self> {
-        #[cfg(feature = "std")]
+        #[cfg(feature = "sqrt-table")]
         {
             let (is_square, res) = FQ_TABLES.sqrt_alt(self);
             CtOption::new(res, is_square)
         }
 
-        #[cfg(not(feature = "std"))]
+        #[cfg(not(feature = "sqrt-table"))]
         crate::arithmetic::sqrt_tonelli_shanks(self, &T_MINUS1_OVER2)
     }
 
@@ -663,13 +661,13 @@ impl PrimeFieldBits for Fq {
     }
 }
 
-#[cfg(feature = "std")]
+#[cfg(feature = "sqrt-table")]
 lazy_static! {
     // The perfect hash parameters are found by `squareroottab.sage` in zcash/pasta.
+    #[cfg_attr(docsrs, doc(cfg(feature = "sqrt-table")))]
     static ref FQ_TABLES: SqrtTables<Fq> = SqrtTables::new(0x116A9E, 1206);
 }
 
-#[cfg(feature = "std")]
 impl SqrtRatio for Fq {
     const T_MINUS1_OVER2: [u64; 4] = T_MINUS1_OVER2;
 
@@ -711,16 +709,17 @@ impl SqrtRatio for Fq {
         tmp.0[0] as u32
     }
 
+    #[cfg(feature = "sqrt-table")]
     fn sqrt_ratio(num: &Self, div: &Self) -> (Choice, Self) {
         FQ_TABLES.sqrt_ratio(num, div)
     }
 
+    #[cfg(feature = "sqrt-table")]
     fn sqrt_alt(&self) -> (Choice, Self) {
         FQ_TABLES.sqrt_alt(self)
     }
 }
 
-#[cfg(feature = "std")]
 impl FieldExt for Fq {
     const MODULUS: &'static str =
         "0x40000000000000000000000000000000224698fc0994a8dd8c46eb2100000001";
@@ -770,7 +769,7 @@ impl FieldExt for Fq {
     }
 }
 
-#[cfg(all(test, feature = "std"))]
+#[cfg(test)]
 use ff::Field;
 
 #[test]
@@ -788,7 +787,6 @@ fn test_inv() {
     assert_eq!(inv, INV);
 }
 
-#[cfg(feature = "std")]
 #[test]
 fn test_sqrt() {
     // NB: TWO_INV is standing in as a "random" field element
@@ -796,7 +794,6 @@ fn test_sqrt() {
     assert!(v == Fq::TWO_INV || (-v) == Fq::TWO_INV);
 }
 
-#[cfg(feature = "std")]
 #[test]
 fn test_pow_by_t_minus1_over2() {
     // NB: TWO_INV is standing in as a "random" field element
@@ -804,7 +801,6 @@ fn test_pow_by_t_minus1_over2() {
     assert!(v == ff::Field::pow_vartime(&Fq::TWO_INV, &T_MINUS1_OVER2));
 }
 
-#[cfg(feature = "std")]
 #[test]
 fn test_sqrt_ratio_and_alt() {
     // (true, sqrt(num/div)), if num and div are nonzero and num/div is a square in the field
@@ -851,7 +847,6 @@ fn test_sqrt_ratio_and_alt() {
     assert!(v == expected);
 }
 
-#[cfg(feature = "std")]
 #[test]
 fn test_zeta() {
     assert_eq!(
@@ -866,7 +861,6 @@ fn test_zeta() {
     assert!(c == Fq::one());
 }
 
-#[cfg(feature = "std")]
 #[test]
 fn test_root_of_unity() {
     assert_eq!(
@@ -875,19 +869,16 @@ fn test_root_of_unity() {
     );
 }
 
-#[cfg(feature = "std")]
 #[test]
 fn test_inv_root_of_unity() {
     assert_eq!(Fq::ROOT_OF_UNITY_INV, Fq::root_of_unity().invert().unwrap());
 }
 
-#[cfg(feature = "std")]
 #[test]
 fn test_inv_2() {
     assert_eq!(Fq::TWO_INV, Fq::from(2).invert().unwrap());
 }
 
-#[cfg(feature = "std")]
 #[test]
 fn test_delta() {
     assert_eq!(Fq::DELTA, GENERATOR.pow(&[1u64 << Fq::S, 0, 0, 0]));
diff --git a/src/lib.rs b/src/lib.rs
index add7d5d..c87d047 100644
--- a/src/lib.rs
+++ b/src/lib.rs
@@ -9,6 +9,9 @@
 #![deny(missing_docs)]
 #![deny(unsafe_code)]
 
+#[cfg(feature = "alloc")]
+extern crate alloc;
+
 #[cfg(any(feature = "std", test))]
 #[macro_use]
 extern crate std;