diff --git a/ecdsa.cpp b/ecdsa.cpp
index fa6f9a6d074..0011907d72f 100644
--- a/ecdsa.cpp
+++ b/ecdsa.cpp
@@ -79,6 +79,7 @@ bool Signature::RecomputeR(Number &r2, const GroupElemJac &pubkey, const Number
     if (pr.IsInfinity())
         return false;
     FieldElem xr; pr.GetX(xr);
+    xr.Normalize();
     unsigned char xrb[32]; xr.GetBytes(xrb);
     r2.SetBytes(xrb,32); r2.SetMod(r2,c.order);
     return true;
@@ -99,6 +100,7 @@ bool Signature::Sign(const Number &seckey, const Number &message, const Number &
     FieldElem rx;
     rp.GetX(rx);
     unsigned char b[32];
+    rx.Normalize();
     rx.GetBytes(b);
     r.SetBytes(b, 32);
     r.SetMod(r, c.order);
diff --git a/field.cpp b/field.cpp
index 5f7714cd8b0..61710c34b47 100644
--- a/field.cpp
+++ b/field.cpp
@@ -24,6 +24,7 @@ FieldElem::FieldElem(int x) {
     n[1] = n[2] = n[3] = n[4] = 0;
 #ifdef VERIFY_MAGNITUDE
     magnitude = 1;
+    normalized = true;
 #endif
 }
 
@@ -69,22 +70,29 @@ void FieldElem::Normalize() {
     }
 #ifdef VERIFY_MAGNITUDE
     magnitude = 1;
+    normalized = true;
 #endif
 }
 
-bool inline FieldElem::IsZero() {
-    Normalize();
+bool inline FieldElem::IsZero() const {
+#ifdef VERIFY_MAGNITUDE
+    assert(normalized);
+#endif
     return (n[0] == 0 && n[1] == 0 && n[2] == 0 && n[3] == 0 && n[4] == 0);
 }
 
-bool inline operator==(FieldElem &a, FieldElem &b) {
-    a.Normalize();
-    b.Normalize();
+bool inline operator==(const FieldElem &a, const FieldElem &b) {
+#ifdef VERIFY_MAGNITUDE
+    assert(a.normalized);
+    assert(b.normalized);
+#endif
     return (a.n[0] == b.n[0] && a.n[1] == b.n[1] && a.n[2] == b.n[2] && a.n[3] == b.n[3] && a.n[4] == b.n[4]);
 }
 
 void FieldElem::GetBytes(unsigned char *o) {
-    Normalize();
+#ifdef VERIFY_MAGNITUDE
+    assert(normalized);
+#endif
     for (int i=0; i<32; i++) {
         int c = 0;
         for (int j=0; j<2; j++) {
@@ -107,6 +115,7 @@ void FieldElem::SetBytes(const unsigned char *in) {
     }
 #ifdef VERIFY_MAGNITUDE
     magnitude = 1;
+    normalized = true;
 #endif
 }
 
@@ -114,6 +123,7 @@ void inline FieldElem::SetNeg(const FieldElem &a, int magnitudeIn) {
 #ifdef VERIFY_MAGNITUDE
     assert(a.magnitude <= magnitudeIn);
     magnitude = magnitudeIn + 1;
+    normalized = false;
 #endif
     n[0] = 0xFFFFEFFFFFC2FULL * (magnitudeIn + 1) - a.n[0];
     n[1] = 0xFFFFFFFFFFFFFULL * (magnitudeIn + 1) - a.n[1];
@@ -125,6 +135,7 @@ void inline FieldElem::SetNeg(const FieldElem &a, int magnitudeIn) {
 void inline FieldElem::operator*=(int v) {
 #ifdef VERIFY_MAGNITUDE
     magnitude *= v;
+    normalized = false;
 #endif
     n[0] *= v;
     n[1] *= v;
@@ -136,6 +147,7 @@ void inline FieldElem::operator*=(int v) {
 void inline FieldElem::operator+=(const FieldElem &a) {
 #ifdef VERIFY_MAGNITUDE
     magnitude += a.magnitude;
+    normalized = false;
 #endif
     n[0] += a.n[0];
     n[1] += a.n[1];
@@ -200,6 +212,7 @@ void FieldElem::SetMult(const FieldElem &a, const FieldElem &b) {
     n[1] = t1 + c;
 #ifdef VERIFY_MAGNITUDE
     magnitude = 1;
+    normalized = false;
 #endif
 }
 
@@ -247,6 +260,7 @@ void FieldElem::SetSquare(const FieldElem &a) {
     n[1] = t1 + c;
 #ifdef VERIFY_MAGNITUDE
     assert(a.magnitude <= 8);
+    normalized = false;
 #endif
 }
 
@@ -283,13 +297,16 @@ void FieldElem::SetSquareRoot(const FieldElem &a) {
     SetMult(x,a780);
 }
 
-bool FieldElem::IsOdd() {
-    Normalize();
+bool FieldElem::IsOdd() const {
+#ifdef VERIFY_MAGNITUDE
+    assert(normalized);
+#endif
     return n[0] & 1;
 }
 
 std::string FieldElem::ToString() {
     unsigned char tmp[32];
+    Normalize();
     GetBytes(tmp);
     std::string ret;
     for (int i=0; i<32; i++) {
@@ -371,6 +388,7 @@ void FieldElem::SetInverse(FieldElem &a) {
     SetMult(x,a45);
 #else
     unsigned char b[32];
+    a.Normalize();
     a.GetBytes(b);
     {
         const Number &p = GetFieldConst().field_p;
diff --git a/field.h b/field.h
index 0584dbabda6..d74e2f98ff5 100644
--- a/field.h
+++ b/field.h
@@ -25,6 +25,7 @@ private:
     uint64_t n[5];
 #ifdef VERIFY_MAGNITUDE
     int magnitude;
+    bool normalized;
 #endif
 
 public:
@@ -37,9 +38,9 @@ public:
     /** Normalizes the internal representation entries. Magnitude=1 */
     void Normalize();
 
-    bool IsZero();
+    bool IsZero() const;
 
-    bool friend operator==(FieldElem &a, FieldElem &b);
+    bool friend operator==(const FieldElem &a, const FieldElem &b);
 
     /** extract as 32-byte big endian array */
     void GetBytes(unsigned char *o);
@@ -64,7 +65,7 @@ public:
     /** Set this to be the (modular) square root of another FieldElem. Magnitude=1 */
     void SetSquareRoot(const FieldElem &a);
 
-    bool IsOdd();
+    bool IsOdd() const;
 
     /** Set this to be the (modular) inverse of another FieldElem. Magnitude=1 */
     void SetInverse(FieldElem &a);
diff --git a/group.cpp b/group.cpp
index 3a66174e206..a22a9a8258a 100644
--- a/group.cpp
+++ b/group.cpp
@@ -71,6 +71,8 @@ bool GroupElemJac::IsValid() const {
     FieldElem z6; z6.SetSquare(z2); z6.SetMult(z6,z2);
     z6 *= 7;
     x3 += z6;
+    y2.Normalize();
+    x3.Normalize();
     return y2 == x3;
 }
 
@@ -122,12 +124,14 @@ void GroupElemJac::SetCompressed(const FieldElem &xin, bool fOdd) {
     c += x3;
     y.SetSquareRoot(c);
     z = FieldElem(1);
+    y.Normalize();
     if (y.IsOdd() != fOdd)
         y.SetNeg(y,1);
 }
 
 void GroupElemJac::SetDouble(const GroupElemJac &p) {
     FieldElem t5 = p.y;
+    t5.Normalize();
     if (p.fInfinity || t5.IsZero()) {
         fInfinity = true;
         return;
@@ -174,7 +178,11 @@ void GroupElemJac::SetAdd(const GroupElemJac &p, const GroupElemJac &q) {
     FieldElem u2; u2.SetMult(x2, z12);
     FieldElem s1; s1.SetMult(y1, z22); s1.SetMult(s1, z2);
     FieldElem s2; s2.SetMult(y2, z12); s2.SetMult(s2, z1);
+    u1.Normalize();
+    u2.Normalize();
     if (u1 == u2) {
+        s1.Normalize();
+        s2.Normalize();
         if (s1 == s2) {
             SetDouble(p);
         } else {
@@ -214,7 +222,11 @@ void GroupElemJac::SetAdd(const GroupElemJac &p, const GroupElem &q) {
     FieldElem u2; u2.SetMult(x2, z12);
     FieldElem s1 = y1; s1.Normalize(); 
     FieldElem s2; s2.SetMult(y2, z12); s2.SetMult(s2, z1);
+    u1.Normalize();
+    u2.Normalize();
     if (u1 == u2) {
+        s1.Normalize();
+        s2.Normalize();
         if (s1 == s2) {
             SetDouble(p);
         } else {