diff --git a/.github/workflows/test.yml b/.github/workflows/test.yml
index 03d8ff6..1b5664d 100644
--- a/.github/workflows/test.yml
+++ b/.github/workflows/test.yml
@@ -32,7 +32,9 @@ jobs:
 
       - name: Run fuzzy field tests
         run: |
-          go test -tags=fuzz -fuzz=Fuzz -fuzztime=30s github.com/elliottech/poseidon_crypto/field
+          for test in $(go test -list='Fuzz.*' github.com/elliottech/poseidon_crypto/field | grep ^Fuzz); do
+              go test -fuzz="^${test}$" -fuzztime=30s github.com/elliottech/poseidon_crypto/field
+          done
       - name: Run fuzzy signature tests
         run: |
           go test -tags=fuzz -fuzz=Fuzz -fuzztime=30s github.com/elliottech/poseidon_crypto/signature/schnorr
@@ -41,6 +43,11 @@ jobs:
           for test in $(go test -list='Fuzz.*' github.com/elliottech/poseidon_crypto/int | grep ^Fuzz); do
               go test -fuzz="^${test}$" -fuzztime=30s github.com/elliottech/poseidon_crypto/int
           done
+      - name: Run fuzzy ecgfp5 tests
+        run: |
+          for test in $(go test -list='Fuzz.*' github.com/elliottech/poseidon_crypto/curve/ecgfp5 | grep ^Fuzz); do
+              go test -fuzz="^${test}$" -fuzztime=30s github.com/elliottech/poseidon_crypto/curve/ecgfp5
+          done
 
   lint:
     name: Lint
diff --git a/curve/ecgfp5/scalar_field.go b/curve/ecgfp5/scalar_field.go
index b14b29a..ea2a763 100644
--- a/curve/ecgfp5/scalar_field.go
+++ b/curve/ecgfp5/scalar_field.go
@@ -51,11 +51,12 @@ func ScalarElementFromLittleEndianBytes(data []byte) ECgFp5Scalar {
 		value[i] = binary.LittleEndian.Uint64(data[i*8:])
 	}
 
-	if !value.IsCanonical() {
-		panic("trying to deserialize non-canonical bytes")
+	bigValue := ToNonCanonicalBigInt(value)
+	if bigValue.Cmp(ORDER) < 0 {
+		return value
 	}
 
-	return value
+	return FromNonCanonicalBigInt(bigValue)
 }
 
 func (s ECgFp5Scalar) SplitTo4BitLimbs() [80]uint8 {
@@ -163,12 +164,26 @@ func Select(c uint64, a0, a1 ECgFp5Scalar) ECgFp5Scalar {
 }
 
 func (s ECgFp5Scalar) Add(rhs ECgFp5Scalar) ECgFp5Scalar {
+	if !s.IsCanonical() {
+		panic("Add: first operand 's' must be canonical (< n)")
+	}
+	if !rhs.IsCanonical() {
+		panic("Add: second operand 'rhs' must be canonical (< n)")
+	}
+
 	r0 := s.AddInner(rhs)
 	r1, c := r0.SubInner(N) // one reduce is enough if s < n and rhs < n
 	return Select(c, r1, r0)
 }
 
 func (s *ECgFp5Scalar) Sub(rhs ECgFp5Scalar) ECgFp5Scalar {
+	if !s.IsCanonical() {
+		panic("Sub: first operand 's' must be canonical (< n)")
+	}
+	if !rhs.IsCanonical() {
+		panic("Sub: second operand 'rhs' must be canonical (< n)")
+	}
+
 	r0, c := s.SubInner(rhs)
 	r1 := r0.AddInner(N) // one add is enough if s < n and rhs < n
 	return Select(c, r0, r1)
@@ -242,7 +257,7 @@ func FromGfp5(fp5 gFp5.Element) ECgFp5Scalar {
 	result := new(big.Int)
 	for i := 4; i >= 0; i-- {
 		result.Lsh(result, 64)
-		result.Or(result, new(big.Int).SetUint64(fp5[i].ToCanonicalUint64())) // it always fit to
+		result.Or(result, new(big.Int).SetUint64(fp5[i].ToCanonicalUint64()))
 	}
 
 	return FromNonCanonicalBigInt(result)
diff --git a/curve/ecgfp5/scalar_field_test.go b/curve/ecgfp5/scalar_field_test.go
index 38d99ea..68c648a 100644
--- a/curve/ecgfp5/scalar_field_test.go
+++ b/curve/ecgfp5/scalar_field_test.go
@@ -1,6 +1,7 @@
 package ecgfp5
 
 import (
+	"math/big"
 	"testing"
 
 	g "github.com/elliottech/poseidon_crypto/field/goldilocks"
@@ -24,6 +25,40 @@ func TestSerdes(t *testing.T) {
 	}
 }
 
+func TestScalarElementFromLittleEndianBytesReduces(t *testing.T) {
+	// Create a byte array that represents a scalar larger than the order
+	bigScalar := new(big.Int).Add(ORDER, big.NewInt(1234567890))
+	leBytes := bigScalar.Bytes()
+	s := ScalarElementFromLittleEndianBytes(leBytes)
+
+	if ToNonCanonicalBigInt(s).Cmp(ORDER) != -1 {
+		t.Fatalf("Expected scalar to be reduced modulo order, but got %v", ToNonCanonicalBigInt(s))
+	}
+}
+
+func FuzzSerdes(f *testing.F) {
+	f.Add([]byte{1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40})
+	f.Add(ORDER.Bytes())
+
+	f.Fuzz(func(t *testing.T, a []byte) {
+		// take 40 bytes only
+		if len(a) > 40 {
+			a = a[:40]
+		} else if len(a) < 40 {
+			// pad with zeros
+			a = append(a, make([]byte, 40-len(a))...)
+		}
+		s := ScalarElementFromLittleEndianBytes(a)
+
+		b := s.ToLittleEndianBytes()
+		ss := ScalarElementFromLittleEndianBytes(b)
+
+		if !s.Equals(ss) {
+			t.Fatalf("Serdes mismatch: %v != %v", s, ss)
+		}
+	})
+}
+
 func TestSplitTo4LimbBits(t *testing.T) {
 	scalar := ECgFp5Scalar{
 		6950590877883398434,
@@ -120,12 +155,31 @@ func TestAddScalar(t *testing.T) {
 	}
 }
 
+func FuzzAddScalar(f *testing.F) {
+	f.Add([]byte{1, 2, 3, 4}, []byte{5, 6, 7, 8})
+	f.Add(ORDER.Bytes(), ORDER.Bytes())
+
+	f.Fuzz(func(t *testing.T, a []byte, b []byte) {
+		aBig := new(big.Int).SetBytes(a)
+		scalar1 := FromNonCanonicalBigInt(aBig)
+		bBig := new(big.Int).SetBytes(b)
+		scalar2 := FromNonCanonicalBigInt(bBig)
+
+		result := scalar1.Add(scalar2)
+		resultBig := FromNonCanonicalBigInt(new(big.Int).Add(aBig, bBig))
+
+		if !result.Equals(resultBig) {
+			t.Fatalf("Addition mismatch: %v + %v != %v", scalar1, scalar2, result)
+		}
+	})
+}
+
 func TestSub(t *testing.T) {
 	scalar1 := ECgFp5Scalar{1, 2, 0, 0, 0}
-	scalar2 := ECgFp5Scalar{0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF}
+	scalar2 := ECgFp5Scalar{0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0x0FFFFFFFFFFFFFFF}
 
 	result := scalar1.Sub(scalar2)
-	expectedValues := ECgFp5Scalar{0xe80fd996948bffe3, 0xe8885c39d724a09e, 0x7fffffe6cfb80639, 0x7ffffff100000016, 0x7ffffffd80000007}
+	expectedValues := ECgFp5Scalar{0xe80fd996948bffe3, 0xe8885c39d724a09e, 0x7fffffe6cfb80639, 0x7ffffff100000016, 8070450521510510599}
 
 	for i := 0; i < 5; i++ {
 		if result[i] != expectedValues[i] {
@@ -134,6 +188,25 @@ func TestSub(t *testing.T) {
 	}
 }
 
+func FuzzSubScalar(f *testing.F) {
+	f.Add([]byte{1, 2, 3, 4}, []byte{5, 6, 7, 8})
+	f.Add(ORDER.Bytes(), ORDER.Bytes())
+
+	f.Fuzz(func(t *testing.T, a []byte, b []byte) {
+		aBig := new(big.Int).SetBytes(a)
+		scalar1 := FromNonCanonicalBigInt(aBig)
+		bBig := new(big.Int).SetBytes(b)
+		scalar2 := FromNonCanonicalBigInt(bBig)
+
+		result := scalar1.Sub(scalar2)
+		resultBig := FromNonCanonicalBigInt(new(big.Int).Sub(aBig, bBig))
+
+		if !result.Equals(resultBig) {
+			t.Fatalf("Subtraction mismatch: %v - %v != %v", scalar1, scalar2, result)
+		}
+	})
+}
+
 func TestSelect(t *testing.T) {
 	a0 := ECgFp5Scalar{1, 2, 3, 4, 5}
 	a1 := ECgFp5Scalar{0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFE, 0xFFFFFFFFFFFFFFFD, 0xFFFFFFFFFFFFFFFC, 0xFFFFFFFFFFFFFFFB}
@@ -155,7 +228,7 @@ func TestSelect(t *testing.T) {
 
 func TestMontyMul(t *testing.T) {
 	scalar1 := ECgFp5Scalar{1, 2, 3, 4, 5}
-	scalar2 := ECgFp5Scalar{0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF}
+	scalar2 := ECgFp5Scalar{0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF, 0xFFFFFFFFFFFFFFFF} // montymul can work with non-canonical inputs
 
 	result := scalar1.MontyMul(scalar2)
 	expectedValues := ECgFp5Scalar{10974894505036100890, 7458803775930281466, 744239893213209819, 3396127080529349464, 5979369289905897562}
@@ -192,6 +265,25 @@ func TestMul(t *testing.T) {
 	}
 }
 
+func FuzzMulScalar(f *testing.F) {
+	f.Add([]byte{1, 2, 3, 4}, []byte{5, 6, 7, 8})
+	f.Add(ORDER.Bytes(), ORDER.Bytes())
+
+	f.Fuzz(func(t *testing.T, a []byte, b []byte) {
+		aBig := new(big.Int).SetBytes(a)
+		scalar1 := FromNonCanonicalBigInt(aBig)
+		bBig := new(big.Int).SetBytes(b)
+		scalar2 := FromNonCanonicalBigInt(bBig)
+
+		result := scalar1.Mul(scalar2)
+		resultBig := FromNonCanonicalBigInt(new(big.Int).Mul(aBig, bBig))
+
+		if !result.Equals(resultBig) {
+			t.Fatalf("Multiplication mismatch: %v * %v != %v", scalar1, scalar2, result)
+		}
+	})
+}
+
 func TestRecodeSigned(t *testing.T) {
 	var ss [50]int32
 	scalar := ECgFp5Scalar{
diff --git a/field/field_test.go b/field/field_test.go
index b89ee06..a8c8085 100644
--- a/field/field_test.go
+++ b/field/field_test.go
@@ -72,7 +72,8 @@ var inputs = []uint64{
 	9223372036854775798, 9223372036854775799, 9223372036854775800, 9223372036854775801, 9223372036854775802, 9223372036854775803, 9223372036854775804, 9223372036854775805,
 	9223372036854775806, 9223372036854775807, 9223372036854775808, 9223372036854775809, 9223372036854775810, 9223372036854775811, 9223372036854775812, 9223372036854775813,
 	9223372036854775814, 9223372036854775815, 9223372036854775816, 9223372036854775817, 18446744069414584311, 18446744069414584312, 18446744069414584313, 18446744069414584314,
-	18446744069414584315, 18446744069414584316, 18446744069414584317, 18446744069414584318, 18446744069414584319, 18446744069414584320,
+	18446744069414584315, 18446744069414584316, 18446744069414584317, 18446744069414584318, 18446744069414584319, 18446744069414584320, 18446744069414584321, 18446744069414584323,
+	math.MaxUint64,
 }
 
 func NewBigInt(x uint64) *big.Int {
@@ -427,6 +428,27 @@ func FuzzTestF(f *testing.F) {
 	})
 }
 
+func FuzzEquivalenceF(f *testing.F) {
+	f.Add(uint64(0))
+	f.Add(uint64(1))
+	f.Add(g.ORDER - 1)
+	f.Add(g.ORDER + 1)
+	f.Add(uint64(math.MaxUint64))
+
+	f.Fuzz(func(t *testing.T, val uint64) {
+		fVal := g.GoldilocksField(val)
+		gVal := g.FromUint64(val)
+
+		if fVal.ToCanonicalUint64() != gVal.Uint64() {
+			t.Fatalf("FromUint64: Expected %d to be equal, but got %d and %d", val, fVal.ToCanonicalUint64(), gVal.Uint64())
+		}
+
+		if g.SquareF(fVal).ToCanonicalUint64() != new(g.Element).Square(&gVal).Uint64() {
+			t.Fatalf("Square: Expected square of %d to be equal, but got %d and %d", val, g.SquareF(fVal).ToCanonicalUint64(), new(g.Element).Square(&gVal).Uint64())
+		}
+	})
+}
+
 // Quintic extension tests
 
 func TestQuinticExtensionAddSubMulSquare(t *testing.T) {
diff --git a/signature/schnorr/schnorr_test.go b/signature/schnorr/schnorr_test.go
index 4b03588..0eab7a5 100644
--- a/signature/schnorr/schnorr_test.go
+++ b/signature/schnorr/schnorr_test.go
@@ -192,6 +192,14 @@ func TestBytes(t *testing.T) {
 	if err := Validate(pk.ToLittleEndianBytes(), hashedMsg.ToLittleEndianBytes(), sig2.ToBytes()); err != nil {
 		t.Fatalf("Signature is invalid")
 	}
+
+	// Works with non-canonical inputs
+	sig3 := sig2
+	sig3.S = sig3.S.AddInner(curve.N)
+	sig3.E = sig3.E.AddInner(curve.N)
+	if err := Validate(pk.ToLittleEndianBytes(), hashedMsg.ToLittleEndianBytes(), sig3.ToBytes()); err != nil {
+		t.Fatalf("Signature is invalid")
+	}
 }
 
 func BenchmarkSignatureVerify(b *testing.B) {