fix: non-native arithmetic autoreduction for division, inversion and …

…sqrt (#870)

* test: add failing test cases

* fix: autoreduce Division if intermediate results overflow

* fix: autoreduce inversion if intermediate results overflow

* fix: autoreduce sqrt if intermediate results overflow

std/math/emulated/field_ops.go

@@ -11,6 +11,18 @@ import (
 
 // Div computes a/b and returns it. It uses [DivHint] as a hint function.
 func (f *Field[T]) Div(a, b *Element[T]) *Element[T] {
+	return f.reduceAndOp(f.div, f.divPreCond, a, b)
+}
+
+func (f *Field[T]) divPreCond(a, b *Element[T]) (nextOverflow uint, err error) {
+	mulOf, err := f.mulPreCond(&Element[T]{overflow: 0}, b)
+	if err != nil {
+		return mulOf, err
+	}
+	return f.subPreCond(a, &Element[T]{overflow: mulOf})
+}
+
+func (f *Field[T]) div(a, b *Element[T], _ uint) *Element[T] {
 	// omit width assertion as for a is done in AssertIsEqual and for b is done in Mul below
 	if !f.fParams.IsPrime() {
 		// TODO shouldn't we still try to do a classic int div in a hint, constraint the result, and let it fail?
@@ -29,6 +41,18 @@ func (f *Field[T]) Div(a, b *Element[T]) *Element[T] {
 
 // Inverse compute 1/a and returns it. It uses [InverseHint].
 func (f *Field[T]) Inverse(a *Element[T]) *Element[T] {
+	return f.reduceAndOp(f.inverse, f.inversePreCond, a, nil)
+}
+
+func (f *Field[T]) inversePreCond(a, _ *Element[T]) (nextOverflow uint, err error) {
+	mulOf, err := f.mulPreCond(a, &Element[T]{overflow: 0}) // order is important, we want that reduce left side
+	if err != nil {
+		return mulOf, err
+	}
+	return f.subPreCond(&Element[T]{overflow: 0}, &Element[T]{overflow: mulOf})
+}
+
+func (f *Field[T]) inverse(a, _ *Element[T], _ uint) *Element[T] {
 	// omit width assertion as is done in Mul below
 	if !f.fParams.IsPrime() {
 		panic("modulus not a prime")
@@ -46,6 +70,18 @@ func (f *Field[T]) Inverse(a *Element[T]) *Element[T] {
 
 // Sqrt computes square root of a and returns it. It uses [SqrtHint].
 func (f *Field[T]) Sqrt(a *Element[T]) *Element[T] {
+	return f.reduceAndOp(f.sqrt, f.sqrtPreCond, a, nil)
+}
+
+func (f *Field[T]) sqrtPreCond(a, _ *Element[T]) (nextOverflow uint, err error) {
+	mulOf, err := f.mulPreCond(a, a)
+	if err != nil {
+		return mulOf, err
+	}
+	return f.subPreCond(a, &Element[T]{overflow: mulOf})
+}
+
+func (f *Field[T]) sqrt(a, _ *Element[T], _ uint) *Element[T] {
 	// omit width assertion as is done in Mul below
 	if !f.fParams.IsPrime() {
 		panic("modulus not a prime")

std/math/emulated/regression_test.go

@@ -0,0 +1,96 @@
+package emulated
+
+import (
+	"testing"
+
+	"github.com/consensys/gnark-crypto/ecc"
+	"github.com/consensys/gnark/frontend"
+	"github.com/consensys/gnark/std/math/emulated/emparams"
+	"github.com/consensys/gnark/test"
+)
+
+type testIssue867Circuit struct {
+	A, B Element[emparams.BW6761Fp]
+	// Res  Element[emparams.BW6761Fp]
+}
+
+func (c *testIssue867Circuit) Define(api frontend.API) error {
+	f, err := NewField[emparams.BW6761Fp](api)
+	if err != nil {
+		return err
+	}
+	f.Div(&c.A, &c.B)
+	return nil
+}
+
+func TestIssue867Division(t *testing.T) {
+	ac := Element[emparams.BW6761Fp]{
+		Limbs:    make([]frontend.Variable, 23),
+		overflow: 72,
+		internal: true,
+	}
+	bc := Element[emparams.BW6761Fp]{
+		Limbs:    make([]frontend.Variable, 23),
+		overflow: 118,
+		internal: true,
+	}
+	a := Element[emparams.BW6761Fp]{
+		Limbs: []frontend.Variable{
+			"42152055455558239410918856365347934987",
+			"284096672345318884331906744025751994436",
+			"614038234500395071520584452045854556036",
+			"650791570547517983681540124631549305906",
+			"1015426967328780486983265586758233548314",
+			"1547576981782048968201287533960518281592",
+			"2241388874175174864705300518494392715473",
+			"2272776063523064079117175149406828917348",
+			"2108034997939354830629994253650658395093",
+			"2496638612325105562443710658519436256202",
+			"2516535160396097216063035076559950097396",
+			"2851622448053604742119299279390439519900",
+			"1745307545739213351161058618528459959073",
+			"1611333894983919176759905895818287768762",
+			"1466950300593143231048571873514528370337",
+			"1413148907163753703359561383721137311278",
+			"1106632639185048629298472160281839189795",
+			"223485414995874678448933378505248108260",
+			"498894012096407228249522197291620119508",
+			"66456820497382616360969181916422224388",
+			"418687747092845373939573067221695941200",
+			"929739253206827109560966422852970160",
+			"516235675893857489524743064330668",
+		},
+		overflow: 72,
+		internal: true,
+	}
+	b := Element[emparams.BW6761Fp]{
+		Limbs: []frontend.Variable{
+			"1225996432692727976601767167989361746138121889728902418",
+			"1225996432692726898457503589717765703850589432813740752",
+			"1225996432692727601202002035504082609034686265426224534",
+			"1225996432692727863800627065585502400025688697839624688",
+			"1225996432692726895840280725330019244557638565515208816",
+			"1225996432692728388244995609757388577021081951481986580",
+			"1225996432692726531118348372019113404206525154000451880",
+			"1225996432692726447731005019972770765499945195971883848",
+			"1225996432692727744357066134208375801833920991141279930",
+			"1225996432692726890371778721001659042419189436317645960",
+			"1225996432692724521956555855212298329189894636085023472",
+			"1225996432692727669349649012761793837315979800575415838",
+			"1225996432692728342310921852357706757313945745570174496",
+			"1225996432692727633381606696459772297983693097833527678",
+			"1225996432692727495940982271275341035852610611137245606",
+			"1225996432692728288709439232109402555258221126516325320",
+			"1225996432692726848720378195875416806833757401249877456",
+			"1225996432692727083894468774663851695073095308036985598",
+			"1225996432692728230861854324679838743877597377148752472",
+			"1225996432692726866241709451789414581782249958783970800",
+			"1225996432692728208615724450642876467256539219504282232",
+			"1225996432692728509987163825647123386984338675020339462",
+			"1225996432692728509159175754185978068065410668081643970",
+		},
+		overflow: 118,
+		internal: true,
+	}
+	test.IsSolved(&testIssue867Circuit{A: ac, B: bc}, &testIssue867Circuit{A: a, B: b}, ecc.BLS12_381.ScalarField())
+}