diff --git a/src/group.h b/src/group.h
index 77ad7435f8..877c3eaeed 100644
--- a/src/group.h
+++ b/src/group.h
@@ -51,6 +51,12 @@ static void secp256k1_ge_set_xy(secp256k1_ge *r, const secp256k1_fe *x, const se
  *  for Y. Return value indicates whether the result is valid. */
 static int secp256k1_ge_set_xo_var(secp256k1_ge *r, const secp256k1_fe *x, int odd);
 
+/** Determine whether x is a valid X coordinate on the curve. */
+static int secp256k1_ge_x_on_curve_var(const secp256k1_fe *x);
+
+/** Determine whether fraction xn/xd is a valid X coordinate on the curve (xd != 0). */
+static int secp256k1_ge_x_frac_on_curve_var(const secp256k1_fe *xn, const secp256k1_fe *xd);
+
 /** Check whether a group element is the point at infinity. */
 static int secp256k1_ge_is_infinity(const secp256k1_ge *a);
 
diff --git a/src/group_impl.h b/src/group_impl.h
index 44d98434ca..dcd171f574 100644
--- a/src/group_impl.h
+++ b/src/group_impl.h
@@ -823,4 +823,32 @@ static int secp256k1_ge_is_in_correct_subgroup(const secp256k1_ge* ge) {
 #endif
 }
 
+static int secp256k1_ge_x_on_curve_var(const secp256k1_fe *x) {
+    secp256k1_fe c;
+    secp256k1_fe_sqr(&c, x);
+    secp256k1_fe_mul(&c, &c, x);
+    secp256k1_fe_add_int(&c, SECP256K1_B);
+    return secp256k1_fe_is_square_var(&c);
+}
+
+static int secp256k1_ge_x_frac_on_curve_var(const secp256k1_fe *xn, const secp256k1_fe *xd) {
+    /* We want to determine whether (xn/xd) is on the curve.
+     *
+     * (xn/xd)^3 + 7 is square <=> xd*xn^3 + 7*xd^4 is square (multiplying by xd^4, a square).
+     */
+     secp256k1_fe r, t;
+#ifdef VERIFY
+     VERIFY_CHECK(!secp256k1_fe_normalizes_to_zero_var(xd));
+#endif
+     secp256k1_fe_mul(&r, xd, xn); /* r = xd*xn */
+     secp256k1_fe_sqr(&t, xn); /* t = xn^2 */
+     secp256k1_fe_mul(&r, &r, &t); /* r = xd*xn^3 */
+     secp256k1_fe_sqr(&t, xd); /* t = xd^2 */
+     secp256k1_fe_sqr(&t, &t); /* t = xd^4 */
+     VERIFY_CHECK(SECP256K1_B <= 31);
+     secp256k1_fe_mul_int(&t, SECP256K1_B); /* t = 7*xd^4 */
+     secp256k1_fe_add(&r, &t); /* r = xd*xn^3 + 7*xd^4 */
+     return secp256k1_fe_is_square_var(&r);
+}
+
 #endif /* SECP256K1_GROUP_IMPL_H */
diff --git a/src/tests.c b/src/tests.c
index 9be2b7a33e..4b93bd29aa 100644
--- a/src/tests.c
+++ b/src/tests.c
@@ -3775,7 +3775,7 @@ static void test_ge(void) {
      */
     secp256k1_ge *ge = (secp256k1_ge *)checked_malloc(&CTX->error_callback, sizeof(secp256k1_ge) * (1 + 4 * runs));
     secp256k1_gej *gej = (secp256k1_gej *)checked_malloc(&CTX->error_callback, sizeof(secp256k1_gej) * (1 + 4 * runs));
-    secp256k1_fe zf;
+    secp256k1_fe zf, r;
     secp256k1_fe zfi2, zfi3;
 
     secp256k1_gej_set_infinity(&gej[0]);
@@ -3817,6 +3817,11 @@ static void test_ge(void) {
     secp256k1_fe_sqr(&zfi2, &zfi3);
     secp256k1_fe_mul(&zfi3, &zfi3, &zfi2);
 
+    /* Generate random r */
+    do {
+        random_field_element_test(&r);
+    } while(secp256k1_fe_is_zero(&r));
+
     for (i1 = 0; i1 < 1 + 4 * runs; i1++) {
         int i2;
         for (i2 = 0; i2 < 1 + 4 * runs; i2++) {
@@ -3929,6 +3934,29 @@ static void test_ge(void) {
         free(ge_set_all);
     }
 
+    /* Test that all elements have X coordinates on the curve. */
+    for (i = 1; i < 4 * runs + 1; i++) {
+        secp256k1_fe n;
+        CHECK(secp256k1_ge_x_on_curve_var(&ge[i].x));
+        /* And the same holds after random rescaling. */
+        secp256k1_fe_mul(&n, &zf, &ge[i].x);
+        CHECK(secp256k1_ge_x_frac_on_curve_var(&n, &zf));
+    }
+
+    /* Test correspondence of secp256k1_ge_x{,_frac}_on_curve_var with ge_set_xo. */
+    {
+        secp256k1_fe n;
+        secp256k1_ge q;
+        int ret_on_curve, ret_frac_on_curve, ret_set_xo;
+        secp256k1_fe_mul(&n, &zf, &r);
+        ret_on_curve = secp256k1_ge_x_on_curve_var(&r);
+        ret_frac_on_curve = secp256k1_ge_x_frac_on_curve_var(&n, &zf);
+        ret_set_xo = secp256k1_ge_set_xo_var(&q, &r, 0);
+        CHECK(ret_on_curve == ret_frac_on_curve);
+        CHECK(ret_on_curve == ret_set_xo);
+        if (ret_set_xo) CHECK(secp256k1_fe_equal_var(&r, &q.x));
+    }
+
     /* Test batch gej -> ge conversion with many infinities. */
     for (i = 0; i < 4 * runs + 1; i++) {
         int odd;