diff --git a/src/ecmult_const_impl.h b/src/ecmult_const_impl.h
index 3d198dcedead8..12dbcc6c5b692 100644
--- a/src/ecmult_const_impl.h
+++ b/src/ecmult_const_impl.h
@@ -19,13 +19,10 @@
  *  It only operates on tables sized for WINDOW_A wnaf multiples.
  */
 static void secp256k1_ecmult_odd_multiples_table_globalz_windowa(secp256k1_ge *pre, secp256k1_fe *globalz, const secp256k1_gej *a) {
-    secp256k1_gej prej[ECMULT_TABLE_SIZE(WINDOW_A)];
     secp256k1_fe zr[ECMULT_TABLE_SIZE(WINDOW_A)];
 
-    /* Compute the odd multiples in Jacobian form. */
-    secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), prej, zr, a);
-    /* Bring them to the same Z denominator. */
-    secp256k1_ge_globalz_set_table_gej(ECMULT_TABLE_SIZE(WINDOW_A), pre, globalz, prej, zr);
+    secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), pre, zr, globalz, a);
+    secp256k1_ge_table_set_globalz(ECMULT_TABLE_SIZE(WINDOW_A), pre, zr);
 }
 
 /* This is like `ECMULT_TABLE_GET_GE` but is constant time */
diff --git a/src/ecmult_impl.h b/src/ecmult_impl.h
index 52af941fb1062..f3b581395660c 100644
--- a/src/ecmult_impl.h
+++ b/src/ecmult_impl.h
@@ -56,14 +56,23 @@
 
 #define ECMULT_MAX_POINTS_PER_BATCH 5000000
 
-/** Fill a table 'prej' with precomputed odd multiples of a. Prej will contain
- *  the values [1*a,3*a,...,(2*n-1)*a], so it space for n values. zr[0] will
- *  contain prej[0].z / a.z. The other zr[i] values = prej[i].z / prej[i-1].z.
- *  Prej's Z values are undefined, except for the last value.
+/** Fill a table 'pre_a' with precomputed odd multiples of a.
+ *  pre_a will contain [1*a,3*a,...,(2*n-1)*a], so it needs space for n group elements.
+ *  zr needs space for n field elements.
+ *
+ *  Although pre_a is an array of _ge rather than _gej, it actually represents elements
+ *  in Jacobian coordinates with their z coordinates omitted. The omitted z-coordinates
+ *  can be recovered using z and zr. Using the notation z(b) to represent the omitted
+ *  z coordinate of b:
+ *  - z(pre_a[n-1]) = 'z'
+ *  - z(pre_a[i-1]) = z(pre_a[i]) / zr[i] for n > i > 0
+ *
+ *  Lastly the zr[0] value, which isn't used above, is set so that:
+ *  - a.z = z(pre_a[0]) / zr[0]
  */
-static void secp256k1_ecmult_odd_multiples_table(int n, secp256k1_gej *prej, secp256k1_fe *zr, const secp256k1_gej *a) {
-    secp256k1_gej d;
-    secp256k1_ge a_ge, d_ge;
+static void secp256k1_ecmult_odd_multiples_table(int n, secp256k1_ge *pre_a, secp256k1_fe *zr, secp256k1_fe *z, const secp256k1_gej *a) {
+    secp256k1_gej d, ai;
+    secp256k1_ge d_ge;
     int i;
 
     VERIFY_CHECK(!a->infinity);
@@ -71,29 +80,38 @@ static void secp256k1_ecmult_odd_multiples_table(int n, secp256k1_gej *prej, sec
     secp256k1_gej_double_var(&d, a, NULL);
 
     /*
-     * Perform the additions on an isomorphism where 'd' is affine: drop the z coordinate
-     * of 'd', and scale the 1P starting value's x/y coordinates without changing its z.
+     * Perform the additions using an isomorphic curve Y^2 = X^3 + 7*C^6 where C := d.z.
+     * The isomorphism, phi, maps a secp256k1 point (x, y) to the point (x*C^2, y*C^3) on the other curve.
+     * In Jacobian coordinates phi maps (x, y, z) to (x*C^2, y*C^3, z) or, equivalently to (x, y, z/C).
+     *
+     *     phi(x, y, z) = (x*C^2, y*C^3, z) = (x, y, z/C)
+     *   d_ge := phi(d) = (d.x, d.y, 1)
+     *     ai := phi(a) = (a.x*C^2, a.y*C^3, a.z)
+     *
+     * The group addition functions work correctly on these isomorphic curves.
+     * In particular phi(d) is easy to represent in affine coordinates under this isomorphism.
+     * This lets us use the faster secp256k1_gej_add_ge_var group addition function that we wouldn't be able to use otherwise.
      */
-    d_ge.x = d.x;
-    d_ge.y = d.y;
-    d_ge.infinity = 0;
-
-    secp256k1_ge_set_gej_zinv(&a_ge, a, &d.z);
-    prej[0].x = a_ge.x;
-    prej[0].y = a_ge.y;
-    prej[0].z = a->z;
-    prej[0].infinity = 0;
+    secp256k1_ge_set_xy(&d_ge, &d.x, &d.y);
+    secp256k1_ge_set_gej_zinv(&pre_a[0], a, &d.z);
+    secp256k1_gej_set_ge(&ai, &pre_a[0]);
+    ai.z = a->z;
 
+    /* pre_a[0] is the point (a.x*C^2, a.y*C^3, a.z*C) which is equvalent to a.
+     * Set zr[0] to C, which is the ratio between the omitted z(pre_a[0]) value and a.z.
+     */
     zr[0] = d.z;
+
     for (i = 1; i < n; i++) {
-        secp256k1_gej_add_ge_var(&prej[i], &prej[i-1], &d_ge, &zr[i]);
+        secp256k1_gej_add_ge_var(&ai, &ai, &d_ge, &zr[i]);
+        secp256k1_ge_set_xy(&pre_a[i], &ai.x, &ai.y);
     }
 
-    /*
-     * Each point in 'prej' has a z coordinate too small by a factor of 'd.z'. Only
-     * the final point's z coordinate is actually used though, so just update that.
+    /* Multiply the last z-coordinate by C to undo the isomorphism.
+     * Since the z-coordinates of the pre_a values are implied by the zr array of z-coordinate ratios,
+     * undoing the isomorphism here undoes the isomorphism for all pre_a values.
      */
-    secp256k1_fe_mul(&prej[n-1].z, &prej[n-1].z, &d.z);
+    secp256k1_fe_mul(z, &ai.z, &d.z);
 }
 
 /** The following two macro retrieves a particular odd multiple from a table
@@ -246,18 +264,18 @@ static void secp256k1_ecmult_strauss_wnaf(const struct secp256k1_strauss_state *
      */
     if (no > 0) {
         /* Compute the odd multiples in Jacobian form. */
-        secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), state->prej, state->zr, &a[state->ps[0].input_pos]);
+        secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), state->pre_a, state->zr, &Z, &a[state->ps[0].input_pos]);
         for (np = 1; np < no; ++np) {
             secp256k1_gej tmp = a[state->ps[np].input_pos];
 #ifdef VERIFY
-            secp256k1_fe_normalize_var(&(state->prej[(np - 1) * ECMULT_TABLE_SIZE(WINDOW_A) + ECMULT_TABLE_SIZE(WINDOW_A) - 1].z));
+            secp256k1_fe_normalize_var(&Z);
 #endif
-            secp256k1_gej_rescale(&tmp, &(state->prej[(np - 1) * ECMULT_TABLE_SIZE(WINDOW_A) + ECMULT_TABLE_SIZE(WINDOW_A) - 1].z));
-            secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), state->prej + np * ECMULT_TABLE_SIZE(WINDOW_A), state->zr + np * ECMULT_TABLE_SIZE(WINDOW_A), &tmp);
+            secp256k1_gej_rescale(&tmp, &Z);
+            secp256k1_ecmult_odd_multiples_table(ECMULT_TABLE_SIZE(WINDOW_A), state->pre_a + np * ECMULT_TABLE_SIZE(WINDOW_A), state->zr + np * ECMULT_TABLE_SIZE(WINDOW_A), &Z, &tmp);
             secp256k1_fe_mul(state->zr + np * ECMULT_TABLE_SIZE(WINDOW_A), state->zr + np * ECMULT_TABLE_SIZE(WINDOW_A), &(a[state->ps[np].input_pos].z));
         }
         /* Bring them to the same Z denominator. */
-        secp256k1_ge_globalz_set_table_gej(ECMULT_TABLE_SIZE(WINDOW_A) * no, state->pre_a, &Z, state->prej, state->zr);
+        secp256k1_ge_table_set_globalz(ECMULT_TABLE_SIZE(WINDOW_A) * no, state->pre_a, state->zr);
     } else {
         secp256k1_fe_set_int(&Z, 1);
     }
diff --git a/src/group.h b/src/group.h
index 0b55ad73c6d4a..bb7dae1cf781e 100644
--- a/src/group.h
+++ b/src/group.h
@@ -9,7 +9,10 @@
 
 #include "field.h"
 
-/** A group element of the secp256k1 curve, in affine coordinates. */
+/** A group element in affine coordinates on the secp256k1 curve,
+ *  or occasionally on an isomorphic curve of the form y^2 = x^3 + 7*t^6.
+ *  Note: For exhaustive test mode, secp256k1 is replaced by a small subgroup of a different curve.
+ */
 typedef struct {
     secp256k1_fe x;
     secp256k1_fe y;
@@ -19,7 +22,9 @@ typedef struct {
 #define SECP256K1_GE_CONST(a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p) {SECP256K1_FE_CONST((a),(b),(c),(d),(e),(f),(g),(h)), SECP256K1_FE_CONST((i),(j),(k),(l),(m),(n),(o),(p)), 0}
 #define SECP256K1_GE_CONST_INFINITY {SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 0), SECP256K1_FE_CONST(0, 0, 0, 0, 0, 0, 0, 0), 1}
 
-/** A group element of the secp256k1 curve, in jacobian coordinates. */
+/** A group element of the secp256k1 curve, in jacobian coordinates.
+ *  Note: For exhastive test mode, sepc256k1 is replaced by a small subgroup of a different curve.
+ */
 typedef struct {
     secp256k1_fe x; /* actual X: x/z^2 */
     secp256k1_fe y; /* actual Y: y/z^3 */
@@ -64,12 +69,24 @@ static void secp256k1_ge_set_gej_var(secp256k1_ge *r, secp256k1_gej *a);
 /** Set a batch of group elements equal to the inputs given in jacobian coordinates */
 static void secp256k1_ge_set_all_gej_var(secp256k1_ge *r, const secp256k1_gej *a, size_t len);
 
-/** Bring a batch inputs given in jacobian coordinates (with known z-ratios) to
- *  the same global z "denominator". zr must contain the known z-ratios such
- *  that mul(a[i].z, zr[i+1]) == a[i+1].z. zr[0] is ignored. The x and y
- *  coordinates of the result are stored in r, the common z coordinate is
- *  stored in globalz. */
-static void secp256k1_ge_globalz_set_table_gej(size_t len, secp256k1_ge *r, secp256k1_fe *globalz, const secp256k1_gej *a, const secp256k1_fe *zr);
+/** Bring a batch of inputs to the same global z "denominator", based on ratios between
+ *  (omitted) z coordinates of adjacent elements.
+ *
+ *  Although the elements a[i] are _ge rather than _gej, they actually represent elements
+ *  in Jacobian coordinates with their z coordinates omitted.
+ *
+ *  Using the notation z(b) to represent the omitted z coordinate of b, the array zr of
+ *  z coordinate ratios must satisfy zr[i] == z(a[i]) / z(a[i-1]) for 0 < 'i' < len.
+ *  The zr[0] value is unused.
+ *
+ *  This function adjusts the coordinates of 'a' in place so that for all 'i', z(a[i]) == z(a[len-1]).
+ *  In other words, the initial value of z(a[len-1]) becomes the global z "denominator". Only the
+ *  a[i].x and a[i].y coordinates are explicitly modified; the adjustment of the omitted z coordinate is
+ *  implicit.
+ *
+ *  The coordinates of the final element a[len-1] are not changed.
+ */
+static void secp256k1_ge_table_set_globalz(size_t len, secp256k1_ge *a, const secp256k1_fe *zr);
 
 /** Set a group element (affine) equal to the point at infinity. */
 static void secp256k1_ge_set_infinity(secp256k1_ge *r);
diff --git a/src/group_impl.h b/src/group_impl.h
index 1bb859efbf7e4..1d5cfa2805940 100644
--- a/src/group_impl.h
+++ b/src/group_impl.h
@@ -161,27 +161,26 @@ static void secp256k1_ge_set_all_gej_var(secp256k1_ge *r, const secp256k1_gej *a
     }
 }
 
-static void secp256k1_ge_globalz_set_table_gej(size_t len, secp256k1_ge *r, secp256k1_fe *globalz, const secp256k1_gej *a, const secp256k1_fe *zr) {
+static void secp256k1_ge_table_set_globalz(size_t len, secp256k1_ge *a, const secp256k1_fe *zr) {
     size_t i = len - 1;
     secp256k1_fe zs;
 
     if (len > 0) {
-        /* The z of the final point gives us the "global Z" for the table. */
-        r[i].x = a[i].x;
-        r[i].y = a[i].y;
         /* Ensure all y values are in weak normal form for fast negation of points */
-        secp256k1_fe_normalize_weak(&r[i].y);
-        *globalz = a[i].z;
-        r[i].infinity = 0;
+        secp256k1_fe_normalize_weak(&a[i].y);
         zs = zr[i];
 
         /* Work our way backwards, using the z-ratios to scale the x/y values. */
         while (i > 0) {
+            secp256k1_gej tmpa;
             if (i != len - 1) {
                 secp256k1_fe_mul(&zs, &zs, &zr[i]);
             }
             i--;
-            secp256k1_ge_set_gej_zinv(&r[i], &a[i], &zs);
+            tmpa.x = a[i].x;
+            tmpa.y = a[i].y;
+            tmpa.infinity = 0;
+            secp256k1_ge_set_gej_zinv(&a[i], &tmpa, &zs);
         }
     }
 }