diff --git a/src/group.h b/src/group.h
index b79ba597db..14cdb4062d 100644
--- a/src/group.h
+++ b/src/group.h
@@ -164,4 +164,12 @@ static void secp256k1_gej_rescale(secp256k1_gej *r, const secp256k1_fe *b);
  */
 static int secp256k1_ge_is_in_correct_subgroup(const secp256k1_ge* ge);
 
+#ifdef VERIFY
+/** Check invariants on an affine group element. */
+static void secp256k1_ge_verify(const secp256k1_ge *a);
+
+/** Check invariants on a Jacobian group element. */
+static void secp256k1_gej_verify(const secp256k1_gej *a);
+#endif
+
 #endif /* SECP256K1_GROUP_H */
diff --git a/src/group_impl.h b/src/group_impl.h
index d6031e58cf..86be92d59d 100644
--- a/src/group_impl.h
+++ b/src/group_impl.h
@@ -73,35 +73,78 @@ static const secp256k1_ge secp256k1_ge_const_g = SECP256K1_G;
 #endif
 /* End of section generated by sage/gen_exhaustive_groups.sage. */
 
+#ifdef VERIFY
+static void secp256k1_ge_verify(const secp256k1_ge *a) {
+    secp256k1_fe_verify(&a->x);
+    secp256k1_fe_verify(&a->y);
+    VERIFY_CHECK(a->infinity == 0 || a->infinity == 1);
+}
+
+static void secp256k1_gej_verify(const secp256k1_gej *a) {
+    secp256k1_fe_verify(&a->x);
+    secp256k1_fe_verify(&a->y);
+    secp256k1_fe_verify(&a->z);
+    VERIFY_CHECK(a->infinity == 0 || a->infinity == 1);
+}
+#endif
+
 static void secp256k1_ge_set_gej_zinv(secp256k1_ge *r, const secp256k1_gej *a, const secp256k1_fe *zi) {
     secp256k1_fe zi2;
     secp256k1_fe zi3;
+#ifdef VERIFY
+    /* Do not call secp256k1_ge_verify, as we do not require a->z to be initialized. */
+    secp256k1_fe_verify(&a->x);
+    secp256k1_fe_verify(&a->y);
+    secp256k1_fe_verify(zi);
+#endif
     VERIFY_CHECK(!a->infinity);
     secp256k1_fe_sqr(&zi2, zi);
     secp256k1_fe_mul(&zi3, &zi2, zi);
     secp256k1_fe_mul(&r->x, &a->x, &zi2);
     secp256k1_fe_mul(&r->y, &a->y, &zi3);
     r->infinity = a->infinity;
+#ifdef VERIFY
+    secp256k1_ge_verify(r);
+#endif
 }
 
 static void secp256k1_ge_set_xy(secp256k1_ge *r, const secp256k1_fe *x, const secp256k1_fe *y) {
+#ifdef VERIFY
+    secp256k1_fe_verify(x);
+    secp256k1_fe_verify(y);
+#endif
     r->infinity = 0;
     r->x = *x;
     r->y = *y;
+#ifdef VERIFY
+    secp256k1_ge_verify(r);
+#endif
 }
 
 static int secp256k1_ge_is_infinity(const secp256k1_ge *a) {
+#ifdef VERIFY
+    secp256k1_ge_verify(a);
+#endif
     return a->infinity;
 }
 
 static void secp256k1_ge_neg(secp256k1_ge *r, const secp256k1_ge *a) {
+#ifdef VERIFY
+    secp256k1_ge_verify(a);
+#endif
     *r = *a;
     secp256k1_fe_normalize_weak(&r->y);
     secp256k1_fe_negate(&r->y, &r->y, 1);
+#ifdef VERIFY
+    secp256k1_ge_verify(r);
+#endif
 }
 
 static void secp256k1_ge_set_gej(secp256k1_ge *r, secp256k1_gej *a) {
     secp256k1_fe z2, z3;
+#ifdef VERIFY
+    secp256k1_gej_verify(a);
+#endif
     r->infinity = a->infinity;
     secp256k1_fe_inv(&a->z, &a->z);
     secp256k1_fe_sqr(&z2, &a->z);
@@ -111,10 +154,16 @@ static void secp256k1_ge_set_gej(secp256k1_ge *r, secp256k1_gej *a) {
     secp256k1_fe_set_int(&a->z, 1);
     r->x = a->x;
     r->y = a->y;
+#ifdef VERIFY
+    secp256k1_ge_verify(r);
+#endif
 }
 
 static void secp256k1_ge_set_gej_var(secp256k1_ge *r, secp256k1_gej *a) {
     secp256k1_fe z2, z3;
+#ifdef VERIFY
+    secp256k1_gej_verify(a);
+#endif
     if (secp256k1_gej_is_infinity(a)) {
         secp256k1_ge_set_infinity(r);
         return;
@@ -127,6 +176,9 @@ static void secp256k1_ge_set_gej_var(secp256k1_ge *r, secp256k1_gej *a) {
     secp256k1_fe_mul(&a->y, &a->y, &z3);
     secp256k1_fe_set_int(&a->z, 1);
     secp256k1_ge_set_xy(r, &a->x, &a->y);
+#ifdef VERIFY
+    secp256k1_ge_verify(r);
+#endif
 }
 
 static void secp256k1_ge_set_all_gej_var(secp256k1_ge *r, const secp256k1_gej *a, size_t len) {
@@ -135,6 +187,9 @@ static void secp256k1_ge_set_all_gej_var(secp256k1_ge *r, const secp256k1_gej *a
     size_t last_i = SIZE_MAX;
 
     for (i = 0; i < len; i++) {
+#ifdef VERIFY
+        secp256k1_gej_verify(&a[i]);
+#endif
         if (a[i].infinity) {
             secp256k1_ge_set_infinity(&r[i]);
         } else {
@@ -168,6 +223,9 @@ static void secp256k1_ge_set_all_gej_var(secp256k1_ge *r, const secp256k1_gej *a
         if (!a[i].infinity) {
             secp256k1_ge_set_gej_zinv(&r[i], &a[i], &r[i].x);
         }
+#ifdef VERIFY
+        secp256k1_ge_verify(&r[i]);
+#endif
     }
 }
 
@@ -176,6 +234,11 @@ static void secp256k1_ge_table_set_globalz(size_t len, secp256k1_ge *a, const se
     secp256k1_fe zs;
 
     if (len > 0) {
+#ifdef VERIFY
+        /* Verify inputs a[len-1] and zr[len-1]. */
+        secp256k1_ge_verify(&a[i]);
+        secp256k1_fe_verify(&zr[i]);
+#endif
         /* Ensure all y values are in weak normal form for fast negation of points */
         secp256k1_fe_normalize_weak(&a[i].y);
         zs = zr[i];
@@ -183,6 +246,11 @@ static void secp256k1_ge_table_set_globalz(size_t len, secp256k1_ge *a, const se
         /* Work our way backwards, using the z-ratios to scale the x/y values. */
         while (i > 0) {
             secp256k1_gej tmpa;
+#ifdef VERIFY
+            /* Verify all inputs a[i] and zr[i]. */
+            secp256k1_fe_verify(&zr[i]);
+            secp256k1_ge_verify(&a[i]);
+#endif
             if (i != len - 1) {
                 secp256k1_fe_mul(&zs, &zs, &zr[i]);
             }
@@ -191,6 +259,10 @@ static void secp256k1_ge_table_set_globalz(size_t len, secp256k1_ge *a, const se
             tmpa.y = a[i].y;
             tmpa.infinity = 0;
             secp256k1_ge_set_gej_zinv(&a[i], &tmpa, &zs);
+#ifdef VERIFY
+            /* Verify the output a[i]. */
+            secp256k1_ge_verify(&a[i]);
+#endif
         }
     }
 }
@@ -200,12 +272,18 @@ static void secp256k1_gej_set_infinity(secp256k1_gej *r) {
     secp256k1_fe_clear(&r->x);
     secp256k1_fe_clear(&r->y);
     secp256k1_fe_clear(&r->z);
+#ifdef VERIFY
+    secp256k1_gej_verify(r);
+#endif
 }
 
 static void secp256k1_ge_set_infinity(secp256k1_ge *r) {
     r->infinity = 1;
     secp256k1_fe_clear(&r->x);
     secp256k1_fe_clear(&r->y);
+#ifdef VERIFY
+    secp256k1_ge_verify(r);
+#endif
 }
 
 static void secp256k1_gej_clear(secp256k1_gej *r) {
@@ -223,31 +301,45 @@ static void secp256k1_ge_clear(secp256k1_ge *r) {
 
 static int secp256k1_ge_set_xo_var(secp256k1_ge *r, const secp256k1_fe *x, int odd) {
     secp256k1_fe x2, x3;
+    int ret;
+#ifdef VERIFY
+    secp256k1_fe_verify(x);
+#endif
     r->x = *x;
     secp256k1_fe_sqr(&x2, x);
     secp256k1_fe_mul(&x3, x, &x2);
     r->infinity = 0;
     secp256k1_fe_add_int(&x3, SECP256K1_B);
-    if (!secp256k1_fe_sqrt(&r->y, &x3)) {
-        return 0;
-    }
+    ret = secp256k1_fe_sqrt(&r->y, &x3);
     secp256k1_fe_normalize_var(&r->y);
     if (secp256k1_fe_is_odd(&r->y) != odd) {
         secp256k1_fe_negate(&r->y, &r->y, 1);
     }
-    return 1;
-
+#ifdef VERIFY
+    secp256k1_ge_verify(r);
+#endif
+    return ret;
 }
 
 static void secp256k1_gej_set_ge(secp256k1_gej *r, const secp256k1_ge *a) {
+#ifdef VERIFY
+   secp256k1_ge_verify(a);
+#endif
    r->infinity = a->infinity;
    r->x = a->x;
    r->y = a->y;
    secp256k1_fe_set_int(&r->z, 1);
+#ifdef VERIFY
+   secp256k1_gej_verify(r);
+#endif
 }
 
 static int secp256k1_gej_eq_var(const secp256k1_gej *a, const secp256k1_gej *b) {
     secp256k1_gej tmp;
+#ifdef VERIFY
+    secp256k1_gej_verify(b);
+    secp256k1_gej_verify(a);
+#endif
     secp256k1_gej_neg(&tmp, a);
     secp256k1_gej_add_var(&tmp, &tmp, b, NULL);
     return secp256k1_gej_is_infinity(&tmp);
@@ -255,6 +347,10 @@ static int secp256k1_gej_eq_var(const secp256k1_gej *a, const secp256k1_gej *b)
 
 static int secp256k1_gej_eq_x_var(const secp256k1_fe *x, const secp256k1_gej *a) {
     secp256k1_fe r, r2;
+#ifdef VERIFY
+    secp256k1_fe_verify(x);
+    secp256k1_gej_verify(a);
+#endif
     VERIFY_CHECK(!a->infinity);
     secp256k1_fe_sqr(&r, &a->z); secp256k1_fe_mul(&r, &r, x);
     r2 = a->x; secp256k1_fe_normalize_weak(&r2);
@@ -262,20 +358,32 @@ static int secp256k1_gej_eq_x_var(const secp256k1_fe *x, const secp256k1_gej *a)
 }
 
 static void secp256k1_gej_neg(secp256k1_gej *r, const secp256k1_gej *a) {
+#ifdef VERIFY
+    secp256k1_gej_verify(a);
+#endif
     r->infinity = a->infinity;
     r->x = a->x;
     r->y = a->y;
     r->z = a->z;
     secp256k1_fe_normalize_weak(&r->y);
     secp256k1_fe_negate(&r->y, &r->y, 1);
+#ifdef VERIFY
+    secp256k1_gej_verify(r);
+#endif
 }
 
 static int secp256k1_gej_is_infinity(const secp256k1_gej *a) {
+#ifdef VERIFY
+    secp256k1_gej_verify(a);
+#endif
     return a->infinity;
 }
 
 static int secp256k1_ge_is_valid_var(const secp256k1_ge *a) {
     secp256k1_fe y2, x3;
+#ifdef VERIFY
+    secp256k1_ge_verify(a);
+#endif
     if (a->infinity) {
         return 0;
     }
@@ -291,6 +399,9 @@ static SECP256K1_INLINE void secp256k1_gej_double(secp256k1_gej *r, const secp25
     /* Operations: 3 mul, 4 sqr, 8 add/half/mul_int/negate */
     secp256k1_fe l, s, t;
 
+#ifdef VERIFY
+    secp256k1_gej_verify(a);
+#endif
     r->infinity = a->infinity;
 
     /* Formula used:
@@ -317,6 +428,9 @@ static SECP256K1_INLINE void secp256k1_gej_double(secp256k1_gej *r, const secp25
     secp256k1_fe_mul(&r->y, &t, &l);       /* Y3 = L*(X3 + T) (1) */
     secp256k1_fe_add(&r->y, &s);           /* Y3 = L*(X3 + T) + S^2 (2) */
     secp256k1_fe_negate(&r->y, &r->y, 2);  /* Y3 = -(L*(X3 + T) + S^2) (3) */
+#ifdef VERIFY
+    secp256k1_gej_verify(r);
+#endif
 }
 
 static void secp256k1_gej_double_var(secp256k1_gej *r, const secp256k1_gej *a, secp256k1_fe *rzr) {
@@ -330,6 +444,9 @@ static void secp256k1_gej_double_var(secp256k1_gej *r, const secp256k1_gej *a, s
      *  the infinity flag even though the point doubles to infinity, and the result
      *  point will be gibberish (z = 0 but infinity = 0).
      */
+#ifdef VERIFY
+    secp256k1_gej_verify(a);
+#endif
     if (a->infinity) {
         secp256k1_gej_set_infinity(r);
         if (rzr != NULL) {
@@ -344,12 +461,19 @@ static void secp256k1_gej_double_var(secp256k1_gej *r, const secp256k1_gej *a, s
     }
 
     secp256k1_gej_double(r, a);
+#ifdef VERIFY
+    secp256k1_gej_verify(r);
+#endif
 }
 
 static void secp256k1_gej_add_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_gej *b, secp256k1_fe *rzr) {
     /* 12 mul, 4 sqr, 11 add/negate/normalizes_to_zero (ignoring special cases) */
     secp256k1_fe z22, z12, u1, u2, s1, s2, h, i, h2, h3, t;
 
+#ifdef VERIFY
+    secp256k1_gej_verify(a);
+    secp256k1_gej_verify(b);
+#endif
     if (a->infinity) {
         VERIFY_CHECK(rzr == NULL);
         *r = *b;
@@ -404,11 +528,18 @@ static void secp256k1_gej_add_var(secp256k1_gej *r, const secp256k1_gej *a, cons
     secp256k1_fe_mul(&r->y, &t, &i);
     secp256k1_fe_mul(&h3, &h3, &s1);
     secp256k1_fe_add(&r->y, &h3);
+#ifdef VERIFY
+    secp256k1_gej_verify(r);
+#endif
 }
 
 static void secp256k1_gej_add_ge_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b, secp256k1_fe *rzr) {
     /* 8 mul, 3 sqr, 13 add/negate/normalize_weak/normalizes_to_zero (ignoring special cases) */
     secp256k1_fe z12, u1, u2, s1, s2, h, i, h2, h3, t;
+#ifdef VERIFY
+    secp256k1_gej_verify(a);
+    secp256k1_ge_verify(b);
+#endif
     if (a->infinity) {
         VERIFY_CHECK(rzr == NULL);
         secp256k1_gej_set_ge(r, b);
@@ -461,12 +592,20 @@ static void secp256k1_gej_add_ge_var(secp256k1_gej *r, const secp256k1_gej *a, c
     secp256k1_fe_mul(&r->y, &t, &i);
     secp256k1_fe_mul(&h3, &h3, &s1);
     secp256k1_fe_add(&r->y, &h3);
+#ifdef VERIFY
+    secp256k1_gej_verify(r);
+    if (rzr != NULL) secp256k1_fe_verify(rzr);
+#endif
 }
 
 static void secp256k1_gej_add_zinv_var(secp256k1_gej *r, const secp256k1_gej *a, const secp256k1_ge *b, const secp256k1_fe *bzinv) {
     /* 9 mul, 3 sqr, 13 add/negate/normalize_weak/normalizes_to_zero (ignoring special cases) */
     secp256k1_fe az, z12, u1, u2, s1, s2, h, i, h2, h3, t;
 
+#ifdef VERIFY
+    secp256k1_ge_verify(b);
+    secp256k1_fe_verify(bzinv);
+#endif
     if (a->infinity) {
         secp256k1_fe bzinv2, bzinv3;
         r->infinity = b->infinity;
@@ -525,6 +664,9 @@ static void secp256k1_gej_add_zinv_var(secp256k1_gej *r, const secp256k1_gej *a,
     secp256k1_fe_mul(&r->y, &t, &i);
     secp256k1_fe_mul(&h3, &h3, &s1);
     secp256k1_fe_add(&r->y, &h3);
+#ifdef VERIFY
+    secp256k1_gej_verify(r);
+#endif
 }
 
 
@@ -533,6 +675,10 @@ static void secp256k1_gej_add_ge(secp256k1_gej *r, const secp256k1_gej *a, const
     secp256k1_fe zz, u1, u2, s1, s2, t, tt, m, n, q, rr;
     secp256k1_fe m_alt, rr_alt;
     int degenerate;
+#ifdef VERIFY
+    secp256k1_gej_verify(a);
+    secp256k1_ge_verify(b);
+#endif
     VERIFY_CHECK(!b->infinity);
     VERIFY_CHECK(a->infinity == 0 || a->infinity == 1);
 
@@ -658,21 +804,34 @@ static void secp256k1_gej_add_ge(secp256k1_gej *r, const secp256k1_gej *a, const
      * We have degenerate = false, r->z = (y1 + y2) * Z.
      * Then r->infinity = ((y1 + y2)Z == 0) = (y1 == -y2) = false. */
     r->infinity = secp256k1_fe_normalizes_to_zero(&r->z);
+#ifdef VERIFY
+    secp256k1_gej_verify(r);
+#endif
 }
 
 static void secp256k1_gej_rescale(secp256k1_gej *r, const secp256k1_fe *s) {
     /* Operations: 4 mul, 1 sqr */
     secp256k1_fe zz;
+#ifdef VERIFY
+    secp256k1_gej_verify(r);
+    secp256k1_fe_verify(s);
+#endif
     VERIFY_CHECK(!secp256k1_fe_is_zero(s));
     secp256k1_fe_sqr(&zz, s);
     secp256k1_fe_mul(&r->x, &r->x, &zz);                /* r->x *= s^2 */
     secp256k1_fe_mul(&r->y, &r->y, &zz);
     secp256k1_fe_mul(&r->y, &r->y, s);                  /* r->y *= s^3 */
     secp256k1_fe_mul(&r->z, &r->z, s);                  /* r->z *= s   */
+#ifdef VERIFY
+    secp256k1_gej_verify(r);
+#endif
 }
 
 static void secp256k1_ge_to_storage(secp256k1_ge_storage *r, const secp256k1_ge *a) {
     secp256k1_fe x, y;
+#ifdef VERIFY
+    secp256k1_ge_verify(a);
+#endif
     VERIFY_CHECK(!a->infinity);
     x = a->x;
     secp256k1_fe_normalize(&x);
@@ -686,14 +845,24 @@ static void secp256k1_ge_from_storage(secp256k1_ge *r, const secp256k1_ge_storag
     secp256k1_fe_from_storage(&r->x, &a->x);
     secp256k1_fe_from_storage(&r->y, &a->y);
     r->infinity = 0;
+#ifdef VERIFY
+    secp256k1_ge_verify(r);
+#endif
 }
 
 static SECP256K1_INLINE void secp256k1_gej_cmov(secp256k1_gej *r, const secp256k1_gej *a, int flag) {
+#ifdef VERIFY
+    secp256k1_gej_verify(r);
+    secp256k1_gej_verify(a);
+#endif
     secp256k1_fe_cmov(&r->x, &a->x, flag);
     secp256k1_fe_cmov(&r->y, &a->y, flag);
     secp256k1_fe_cmov(&r->z, &a->z, flag);
 
     r->infinity ^= (r->infinity ^ a->infinity) & flag;
+#ifdef VERIFY
+    secp256k1_gej_verify(r);
+#endif
 }
 
 static SECP256K1_INLINE void secp256k1_ge_storage_cmov(secp256k1_ge_storage *r, const secp256k1_ge_storage *a, int flag) {
@@ -703,7 +872,13 @@ static SECP256K1_INLINE void secp256k1_ge_storage_cmov(secp256k1_ge_storage *r,
 
 static void secp256k1_ge_mul_lambda(secp256k1_ge *r, const secp256k1_ge *a) {
     *r = *a;
+#ifdef VERIFY
+    secp256k1_ge_verify(a);
+#endif
     secp256k1_fe_mul(&r->x, &r->x, &secp256k1_const_beta);
+#ifdef VERIFY
+    secp256k1_ge_verify(r);
+#endif
 }
 
 static int secp256k1_ge_is_in_correct_subgroup(const secp256k1_ge* ge) {
@@ -711,6 +886,9 @@ static int secp256k1_ge_is_in_correct_subgroup(const secp256k1_ge* ge) {
     secp256k1_gej out;
     int i;
 
+#ifdef VERIFY
+    secp256k1_ge_verify(ge);
+#endif
     /* A very simple EC multiplication ladder that avoids a dependency on ecmult. */
     secp256k1_gej_set_infinity(&out);
     for (i = 0; i < 32; ++i) {