diff --git a/ec/src/hashing/curve_maps/swu/mod.rs b/ec/src/hashing/curve_maps/swu/mod.rs
index 47330c52c..e3a16a729 100644
--- a/ec/src/hashing/curve_maps/swu/mod.rs
+++ b/ec/src/hashing/curve_maps/swu/mod.rs
@@ -206,6 +206,9 @@ mod test {
 
         /// AFFINE_GENERATOR_COEFFS = (G1_GENERATOR_X, G1_GENERATOR_Y)
         const GENERATOR: Affine<Self> = Affine::new_unchecked(MontFp!("62"), MontFp!("70"));
+
+        /// We use `bool` because the point (0, 0) could be on the curve.
+        type ZeroIndicator = bool;
     }
 
     impl SWUConfig for TestSWUMapToCurveConfig {
diff --git a/ec/src/hashing/curve_maps/wb/mod.rs b/ec/src/hashing/curve_maps/wb/mod.rs
index 4e2644009..6d32c51b7 100644
--- a/ec/src/hashing/curve_maps/wb/mod.rs
+++ b/ec/src/hashing/curve_maps/wb/mod.rs
@@ -165,6 +165,9 @@ mod test {
 
         /// AFFINE_GENERATOR_COEFFS = (G1_GENERATOR_X, G1_GENERATOR_Y)
         const GENERATOR: Affine<Self> = Affine::new_unchecked(MontFp!("62"), MontFp!("70"));
+
+        /// We use `()` because the point (0, 0) is not on the curve.
+        type ZeroIndicator = ();
     }
 
     /// Testing WB19 hashing on a small curve
@@ -193,11 +196,14 @@ mod test {
         const COEFF_A: F127 = MontFp!("109");
 
         /// COEFF_B = 124
-    #[rustfmt::skip]
+        #[rustfmt::skip]
         const COEFF_B: F127 = MontFp!("124");
 
         /// AFFINE_GENERATOR_COEFFS = (G1_GENERATOR_X, G1_GENERATOR_Y)
         const GENERATOR: Affine<Self> = Affine::new_unchecked(MontFp!("84"), MontFp!("2"));
+
+        /// We use `bool because the point (0, 0) could be on the curve.
+        type ZeroIndicator = bool;
     }
 
     /// SWU parameters for E_isogenous
diff --git a/ec/src/models/bn/g1.rs b/ec/src/models/bn/g1.rs
index 4a6ea65b8..fdfb3b9e9 100644
--- a/ec/src/models/bn/g1.rs
+++ b/ec/src/models/bn/g1.rs
@@ -44,7 +44,7 @@ impl<'a, P: BnConfig> From<&'a G1Projective<P>> for G1Prepared<P> {
 
 impl<P: BnConfig> G1Prepared<P> {
     pub fn is_zero(&self) -> bool {
-        self.0.infinity
+        self.0.is_zero()
     }
 }
 
diff --git a/ec/src/models/bn/g2.rs b/ec/src/models/bn/g2.rs
index 434624520..5b7276813 100644
--- a/ec/src/models/bn/g2.rs
+++ b/ec/src/models/bn/g2.rs
@@ -103,7 +103,7 @@ impl<P: BnConfig> Default for G2Prepared<P> {
 
 impl<P: BnConfig> From<G2Affine<P>> for G2Prepared<P> {
     fn from(q: G2Affine<P>) -> Self {
-        if q.infinity {
+        if q.is_zero() {
             G2Prepared {
                 ell_coeffs: vec![],
                 infinity: true,
diff --git a/ec/src/models/bw6/g1.rs b/ec/src/models/bw6/g1.rs
index 6960d19d7..7f09b4094 100644
--- a/ec/src/models/bw6/g1.rs
+++ b/ec/src/models/bw6/g1.rs
@@ -45,7 +45,7 @@ impl<'a, P: BW6Config> From<&'a G1Projective<P>> for G1Prepared<P> {
 
 impl<P: BW6Config> G1Prepared<P> {
     pub fn is_zero(&self) -> bool {
-        self.0.infinity
+        self.0.is_zero()
     }
 }
 
diff --git a/ec/src/models/bw6/g2.rs b/ec/src/models/bw6/g2.rs
index 8c826efcd..0dac90407 100644
--- a/ec/src/models/bw6/g2.rs
+++ b/ec/src/models/bw6/g2.rs
@@ -48,7 +48,7 @@ impl<P: BW6Config> Default for G2Prepared<P> {
 
 impl<P: BW6Config> From<G2Affine<P>> for G2Prepared<P> {
     fn from(q: G2Affine<P>) -> Self {
-        if q.infinity {
+        if q.is_zero() {
             return Self {
                 ell_coeffs_1: vec![],
                 ell_coeffs_2: vec![],
diff --git a/ec/src/models/short_weierstrass/affine.rs b/ec/src/models/short_weierstrass/affine.rs
index b1340de3a..87ac6cf07 100644
--- a/ec/src/models/short_weierstrass/affine.rs
+++ b/ec/src/models/short_weierstrass/affine.rs
@@ -18,7 +18,7 @@ use ark_ff::{fields::Field, PrimeField, ToConstraintField, UniformRand};
 
 use zeroize::Zeroize;
 
-use super::{Projective, SWCurveConfig, SWFlags};
+use super::{Projective, SWCurveConfig, SWFlags, ZeroInd};
 use crate::AffineRepr;
 
 /// Affine coordinates for a point on an elliptic curve in short Weierstrass
@@ -38,7 +38,7 @@ pub struct Affine<P: SWCurveConfig> {
     #[doc(hidden)]
     pub y: P::BaseField,
     #[doc(hidden)]
-    pub infinity: bool,
+    pub infinity: P::ZeroIndicator,
 }
 
 impl<P: SWCurveConfig> PartialEq<Projective<P>> for Affine<P> {
@@ -49,7 +49,7 @@ impl<P: SWCurveConfig> PartialEq<Projective<P>> for Affine<P> {
 
 impl<P: SWCurveConfig> Display for Affine<P> {
     fn fmt(&self, f: &mut Formatter<'_>) -> FmtResult {
-        match self.infinity {
+        match self.is_zero() {
             true => write!(f, "infinity"),
             false => write!(f, "({}, {})", self.x, self.y),
         }
@@ -58,7 +58,7 @@ impl<P: SWCurveConfig> Display for Affine<P> {
 
 impl<P: SWCurveConfig> Debug for Affine<P> {
     fn fmt(&self, f: &mut Formatter<'_>) -> FmtResult {
-        match self.infinity {
+        match self.is_zero() {
             true => write!(f, "infinity"),
             false => write!(f, "({}, {})", self.x, self.y),
         }
@@ -72,7 +72,7 @@ impl<P: SWCurveConfig> Affine<P> {
         let point = Self {
             x,
             y,
-            infinity: false,
+            infinity: P::ZeroIndicator::IS_NOT_ZERO,
         };
         assert!(point.is_on_curve());
         assert!(point.is_in_correct_subgroup_assuming_on_curve());
@@ -89,15 +89,18 @@ impl<P: SWCurveConfig> Affine<P> {
         Self {
             x,
             y,
-            infinity: false,
+            infinity: P::ZeroIndicator::IS_NOT_ZERO,
         }
     }
 
     pub const fn identity() -> Self {
+        // Setting these to zero is *load-bearing* and important.
+        // These are the values that represent the identity element
+        // when `P::ZeroIndicator` is `()`.
         Self {
             x: P::BaseField::ZERO,
             y: P::BaseField::ZERO,
-            infinity: true,
+            infinity: P::ZeroIndicator::IS_ZERO,
         }
     }
 
@@ -142,7 +145,7 @@ impl<P: SWCurveConfig> Affine<P> {
 
     /// Checks if `self` is a valid point on the curve.
     pub fn is_on_curve(&self) -> bool {
-        if !self.infinity {
+        if !self.is_zero() {
             // Rust does not optimise away addition with zero
             let mut x3b = P::add_b(self.x.square() * self.x);
             if !P::COEFF_A.is_zero() {
@@ -155,7 +158,7 @@ impl<P: SWCurveConfig> Affine<P> {
     }
 
     pub fn to_flags(&self) -> SWFlags {
-        if self.infinity {
+        if self.is_zero() {
             SWFlags::PointAtInfinity
         } else if self.y <= -self.y {
             SWFlags::YIsPositive
@@ -206,7 +209,7 @@ impl<P: SWCurveConfig> AffineRepr for Affine<P> {
     type Group = Projective<P>;
 
     fn xy(&self) -> Option<(&Self::BaseField, &Self::BaseField)> {
-        (!self.infinity).then(|| (&self.x, &self.y))
+        (!self.is_zero()).then(|| (&self.x, &self.y))
     }
 
     #[inline]
@@ -215,11 +218,7 @@ impl<P: SWCurveConfig> AffineRepr for Affine<P> {
     }
 
     fn zero() -> Self {
-        Self {
-            x: P::BaseField::ZERO,
-            y: P::BaseField::ZERO,
-            infinity: true,
-        }
+        Self::identity()
     }
 
     fn from_random_bytes(bytes: &[u8]) -> Option<Self> {
@@ -382,6 +381,7 @@ impl<P: SWCurveConfig> CanonicalDeserialize for Affine<P> {
 impl<M: SWCurveConfig, ConstraintF: Field> ToConstraintField<ConstraintF> for Affine<M>
 where
     M::BaseField: ToConstraintField<ConstraintF>,
+    M::ZeroIndicator: ToConstraintField<ConstraintF>,
 {
     #[inline]
     fn to_field_elements(&self) -> Option<Vec<ConstraintF>> {
diff --git a/ec/src/models/short_weierstrass/group.rs b/ec/src/models/short_weierstrass/group.rs
index 0513229f2..aa3dad1ae 100644
--- a/ec/src/models/short_weierstrass/group.rs
+++ b/ec/src/models/short_weierstrass/group.rs
@@ -620,6 +620,7 @@ impl<P: SWCurveConfig> CanonicalDeserialize for Projective<P> {
 impl<M: SWCurveConfig, ConstraintF: Field> ToConstraintField<ConstraintF> for Projective<M>
 where
     M::BaseField: ToConstraintField<ConstraintF>,
+    M::ZeroIndicator: ToConstraintField<ConstraintF>,
 {
     #[inline]
     fn to_field_elements(&self) -> Option<Vec<ConstraintF>> {
diff --git a/ec/src/models/short_weierstrass/mod.rs b/ec/src/models/short_weierstrass/mod.rs
index 965cbb83f..86ea0367b 100644
--- a/ec/src/models/short_weierstrass/mod.rs
+++ b/ec/src/models/short_weierstrass/mod.rs
@@ -2,7 +2,10 @@ use ark_serialize::{
     CanonicalDeserialize, CanonicalDeserializeWithFlags, CanonicalSerialize,
     CanonicalSerializeWithFlags, Compress, SerializationError, Valid, Validate,
 };
-use ark_std::io::{Read, Write};
+use ark_std::{
+    hash::Hash,
+    io::{Read, Write},
+};
 
 use ark_ff::fields::Field;
 
@@ -30,6 +33,9 @@ pub trait SWCurveConfig: super::CurveConfig {
     /// Generator of the prime-order subgroup.
     const GENERATOR: Affine<Self>;
 
+    /// A type that is stored in `Affine<Self>` to indicate whether the point is at infinity.
+    type ZeroIndicator: ZeroInd<Self>;
+
     /// Helper method for computing `elem * Self::COEFF_A`.
     ///
     /// The default implementation should be overridden only if
@@ -124,7 +130,7 @@ pub trait SWCurveConfig: super::CurveConfig {
         mut writer: W,
         compress: ark_serialize::Compress,
     ) -> Result<(), SerializationError> {
-        let (x, y, flags) = match item.infinity {
+        let (x, y, flags) = match item.is_zero() {
             true => (
                 Self::BaseField::zero(),
                 Self::BaseField::zero(),
@@ -198,3 +204,30 @@ pub trait SWCurveConfig: super::CurveConfig {
         }
     }
 }
+
+pub trait ZeroInd<C: SWCurveConfig>:
+    Hash + Ord + Eq + Copy + Sync + Send + Sized + 'static
+{
+    const IS_ZERO: Self;
+    const IS_NOT_ZERO: Self;
+    fn is_zero(point: &Affine<C>) -> bool;
+    fn zeroize(&mut self) {
+        *self = Self::IS_NOT_ZERO;
+    }
+}
+
+impl<C: SWCurveConfig<ZeroIndicator = bool>> ZeroInd<C> for bool {
+    const IS_ZERO: Self = true;
+    const IS_NOT_ZERO: Self = false;
+    fn is_zero(point: &Affine<C>) -> bool {
+        point.infinity
+    }
+}
+
+impl<C: SWCurveConfig> ZeroInd<C> for () {
+    const IS_ZERO: Self = ();
+    const IS_NOT_ZERO: Self = ();
+    fn is_zero(point: &Affine<C>) -> bool {
+        point.x.is_zero() & point.y.is_zero()
+    }
+}
diff --git a/test-curves/src/bls12_381/g1.rs b/test-curves/src/bls12_381/g1.rs
index f653e6f7a..78098b573 100644
--- a/test-curves/src/bls12_381/g1.rs
+++ b/test-curves/src/bls12_381/g1.rs
@@ -36,6 +36,8 @@ impl short_weierstrass::SWCurveConfig for Config {
     /// AFFINE_GENERATOR_COEFFS = (G1_GENERATOR_X, G1_GENERATOR_Y)
     const GENERATOR: G1Affine = G1Affine::new_unchecked(G1_GENERATOR_X, G1_GENERATOR_Y);
 
+    type ZeroIndicator = ();
+
     #[inline(always)]
     fn mul_by_a(_: Self::BaseField) -> Self::BaseField {
         Self::BaseField::zero()
diff --git a/test-curves/src/bls12_381/g1_swu_iso.rs b/test-curves/src/bls12_381/g1_swu_iso.rs
index 8532422eb..012e1b17f 100644
--- a/test-curves/src/bls12_381/g1_swu_iso.rs
+++ b/test-curves/src/bls12_381/g1_swu_iso.rs
@@ -43,6 +43,7 @@ impl SWCurveConfig for SwuIsoConfig {
     const COEFF_B: Fq = MontFp!("2906670324641927570491258158026293881577086121416628140204402091718288198173574630967936031029026176254968826637280");
 
     const GENERATOR: G1Affine = G1Affine::new_unchecked(G1_GENERATOR_X, G1_GENERATOR_Y);
+    type ZeroIndicator = ();
 }
 
 /// Lexicographically smallest, valid x-coordinate of a point P on the curve (with its corresponding y) multiplied by the cofactor.
diff --git a/test-curves/src/bls12_381/g2.rs b/test-curves/src/bls12_381/g2.rs
index f4c970910..f8ba1b874 100644
--- a/test-curves/src/bls12_381/g2.rs
+++ b/test-curves/src/bls12_381/g2.rs
@@ -53,6 +53,8 @@ impl short_weierstrass::SWCurveConfig for Config {
     /// AFFINE_GENERATOR_COEFFS = (G2_GENERATOR_X, G2_GENERATOR_Y)
     const GENERATOR: G2Affine = G2Affine::new_unchecked(G2_GENERATOR_X, G2_GENERATOR_Y);
 
+    type ZeroIndicator = bool;
+
     #[inline(always)]
     fn mul_by_a(_: Self::BaseField) -> Self::BaseField {
         Self::BaseField::zero()
diff --git a/test-curves/src/bls12_381/g2_swu_iso.rs b/test-curves/src/bls12_381/g2_swu_iso.rs
index 0790cf99f..a4e431ea2 100644
--- a/test-curves/src/bls12_381/g2_swu_iso.rs
+++ b/test-curves/src/bls12_381/g2_swu_iso.rs
@@ -56,6 +56,8 @@ impl SWCurveConfig for SwuIsoConfig {
     const COEFF_B: Fq2 = Fq2::new(MontFp!("1012"), MontFp!("1012"));
 
     const GENERATOR: G2Affine = G2Affine::new_unchecked(G2_GENERATOR_X, G2_GENERATOR_Y);
+
+    type ZeroIndicator = bool;
 }
 
 /// Lexicographically smallest, valid x-coordinate of a point P on the curve (with its corresponding y) multiplied by the cofactor.
diff --git a/test-curves/src/bn384_small_two_adicity/g1.rs b/test-curves/src/bn384_small_two_adicity/g1.rs
index 82704be8e..12e351dab 100644
--- a/test-curves/src/bn384_small_two_adicity/g1.rs
+++ b/test-curves/src/bn384_small_two_adicity/g1.rs
@@ -33,6 +33,9 @@ impl short_weierstrass::SWCurveConfig for Config {
     /// AFFINE_GENERATOR_COEFFS = (G1_GENERATOR_X, G1_GENERATOR_Y)
     const GENERATOR: G1Affine = G1Affine::new_unchecked(G1_GENERATOR_X, G1_GENERATOR_Y);
 
+    /// We can use `()`, because the point `(0, 0)` is not on the curve.
+    type ZeroIndicator = ();
+
     #[inline(always)]
     fn mul_by_a(_: Self::BaseField) -> Self::BaseField {
         Self::BaseField::zero()
diff --git a/test-curves/src/mnt4_753/g1.rs b/test-curves/src/mnt4_753/g1.rs
index 2a329a56c..204eee263 100644
--- a/test-curves/src/mnt4_753/g1.rs
+++ b/test-curves/src/mnt4_753/g1.rs
@@ -36,6 +36,9 @@ impl short_weierstrass::SWCurveConfig for Config {
 
     /// AFFINE_GENERATOR_COEFFS = (G1_GENERATOR_X, G1_GENERATOR_Y)
     const GENERATOR: G1Affine = G1Affine::new_unchecked(G1_GENERATOR_X, G1_GENERATOR_Y);
+
+    /// We use `bool because `(0, 0)` could be on the curve.
+    type ZeroIndicator = bool;
 }
 
 // Generator of G1
diff --git a/test-curves/src/secp256k1/g1.rs b/test-curves/src/secp256k1/g1.rs
index 9e7e4de24..b1f216992 100644
--- a/test-curves/src/secp256k1/g1.rs
+++ b/test-curves/src/secp256k1/g1.rs
@@ -34,6 +34,9 @@ impl SWCurveConfig for Config {
     fn mul_by_a(_: Self::BaseField) -> Self::BaseField {
         Self::BaseField::zero()
     }
+
+    /// We use `()` because `(0, 0)` cannot be on the curve.
+    type ZeroIndicator = ();
 }
 
 /// G_GENERATOR_X = 55066263022277343669578718895168534326250603453777594175500187360389116729240