1// This package implements efficient Golidlocks arithmetic operations within Gnark. We do not use
  2// the emulated field arithmetic API, because it is too slow for our purposes. Instead, we use
  3// an efficient reduction method that leverages the fact that the modulus is a simple
  4// linear combination of powers of two.
  5package goldilocks
  6
  7// In general, methods whose name do not contain `NoReduce` can be used without any extra mental
  8// overhead. These methods act exactly as you would expect a normal field would operate.
  9//
 10// However, if you want to aggressively optimize the number of constraints in your circuit, it can
 11// be very beneficial to use the no reduction methods and keep track of the maximum number of bits
 12// your computation uses.
 13
 14// This implementation is based on the following plonky2 implementation of Goldilocks
 15// Available here: https://github.com/0xPolygonZero/plonky2/blob/main/field/src/goldilocks_field.rs#L70
 16
 17import (
 18	"fmt"
 19	"math"
 20	"math/big"
 21
 22	"github.com/consensys/gnark-crypto/field/goldilocks"
 23	"github.com/consensys/gnark/constraint/solver"
 24	"github.com/consensys/gnark/frontend"
 25	"github.com/consensys/gnark/std/math/emulated"
 26	"github.com/consensys/gnark/std/rangecheck"
 27)
 28
 29// The multiplicative group generator of the field.
 30var MULTIPLICATIVE_GROUP_GENERATOR goldilocks.Element = goldilocks.NewElement(7)
 31
 32// The two adicity of the field.
 33var TWO_ADICITY uint64 = 32
 34
 35// The power of two generator of the field.
 36var POWER_OF_TWO_GENERATOR goldilocks.Element = goldilocks.NewElement(1753635133440165772)
 37
 38// The modulus of the field.
 39var MODULUS *big.Int = emulated.Goldilocks{}.Modulus()
 40
 41// The number of bits to use for range checks on inner products of field elements.
 42var RANGE_CHECK_NB_BITS int = 140
 43
 44// Registers the hint functions with the solver.
 45func init() {
 46	solver.RegisterHint(MulAddHint)
 47	solver.RegisterHint(ReduceHint)
 48	solver.RegisterHint(InverseHint)
 49	solver.RegisterHint(SplitLimbsHint)
 50}
 51
 52// A type alias used to represent Goldilocks field elements.
 53type Variable struct {
 54	Limb frontend.Variable
 55}
 56
 57// Creates a new Goldilocks field element from an existing variable. Assumes that the element is
 58// already reduced.
 59func NewVariable(x frontend.Variable) Variable {
 60	return Variable{Limb: x}
 61}
 62
 63// The zero element in the Golidlocks field.
 64func Zero() Variable {
 65	return NewVariable(0)
 66}
 67
 68// The one element in the Goldilocks field.
 69func One() Variable {
 70	return NewVariable(1)
 71}
 72
 73// The negative one element in the Goldilocks field.
 74func NegOne() Variable {
 75	return NewVariable(MODULUS.Uint64() - 1)
 76}
 77
 78// The chip used for Goldilocks field operations.
 79type Chip struct {
 80	api          frontend.API
 81	rangeChecker frontend.Rangechecker
 82}
 83
 84// Creates a new Goldilocks Chip.
 85func New(api frontend.API) *Chip {
 86	rangeChecker := rangecheck.New(api)
 87	return &Chip{api: api, rangeChecker: rangeChecker}
 88}
 89
 90// Adds two goldilocks field elements and returns a value within the goldilocks field.
 91func (p *Chip) Add(a Variable, b Variable) Variable {
 92	return p.MulAdd(a, NewVariable(1), b)
 93}
 94
 95// Adds two goldilocks field elements and returns a value that may not be within the goldilocks field
 96// (e.g. the sum is not reduced).
 97func (p *Chip) AddNoReduce(a Variable, b Variable) Variable {
 98	return NewVariable(p.api.Add(a.Limb, b.Limb))
 99}
100
101// Subracts two goldilocks field elements and returns a value within the goldilocks field.
102func (p *Chip) Sub(a Variable, b Variable) Variable {
103	return p.MulAdd(b, NegOne(), a)
104}
105
106// Subracts two goldilocks field elements and returns a value that may not be within the goldilocks field
107// (e.g. the difference is not reduced).
108func (p *Chip) SubNoReduce(a Variable, b Variable) Variable {
109	return NewVariable(p.api.Add(a.Limb, p.api.Mul(b.Limb, NegOne().Limb)))
110}
111
112// Multiplies two goldilocks field elements and returns a value within the goldilocks field.
113func (p *Chip) Mul(a Variable, b Variable) Variable {
114	return p.MulAdd(a, b, Zero())
115}
116
117// Multiplies two goldilocks field elements and returns a value that may not be within the goldilocks field
118// (e.g. the product is not reduced).
119func (p *Chip) MulNoReduce(a Variable, b Variable) Variable {
120	return NewVariable(p.api.Mul(a.Limb, b.Limb))
121}
122
123// Multiplies two field elements and adds a field element (e.g. computes a * b + c).  The returned value
124// will be within the goldilocks field.
125func (p *Chip) MulAdd(a Variable, b Variable, c Variable) Variable {
126	result, err := p.api.Compiler().NewHint(MulAddHint, 2, a.Limb, b.Limb, c.Limb)
127	if err != nil {
128		panic(err)
129	}
130
131	quotient := NewVariable(result[0])
132	remainder := NewVariable(result[1])
133
134	cLimbCopy := p.api.Mul(c.Limb, 1)
135	lhs := p.api.MulAcc(cLimbCopy, a.Limb, b.Limb)
136	rhs := p.api.MulAcc(remainder.Limb, MODULUS, quotient.Limb)
137	p.api.AssertIsEqual(lhs, rhs)
138
139	p.RangeCheck(quotient)
140	p.RangeCheck(remainder)
141	return remainder
142}
143
144// Multiplies two field elements and adds a field element (e.g. computes a * b + c).  The returned value
145// may no be within the goldilocks field (e.g. the result is not reduced).
146func (p *Chip) MulAddNoReduce(a Variable, b Variable, c Variable) Variable {
147	cLimbCopy := p.api.Mul(c.Limb, 1)
148	return NewVariable(p.api.MulAcc(cLimbCopy, a.Limb, b.Limb))
149}
150
151// The hint used to compute MulAdd.
152func MulAddHint(_ *big.Int, inputs []*big.Int, results []*big.Int) error {
153	if len(inputs) != 3 {
154		panic("MulAddHint expects 3 input operands")
155	}
156
157	for _, operand := range inputs {
158		if operand.Cmp(MODULUS) >= 0 {
159			panic(fmt.Sprintf("%s is not in the field", operand.String()))
160		}
161	}
162
163	product := new(big.Int).Mul(inputs[0], inputs[1])
164	sum := new(big.Int).Add(product, inputs[2])
165	quotient := new(big.Int).Div(sum, MODULUS)
166	remainder := new(big.Int).Rem(sum, MODULUS)
167
168	results[0] = quotient
169	results[1] = remainder
170
171	return nil
172}
173
174// Reduces a field element x such that x % MODULUS = y.
175func (p *Chip) Reduce(x Variable) Variable {
176	// Witness a `quotient` and `remainder` such that:
177	//
178	// 		MODULUS * quotient + remainder = x
179	//
180	// Must check that offset \in [0, MODULUS) and carry \in [0, 2^RANGE_CHECK_NB_BITS) to ensure
181	// that this computation does not overflow. We use 2^RANGE_CHECK_NB_BITS to reduce the cost of the range check
182	//
183	// In other words, we assume that we at most compute a a dot product with dimension at most RANGE_CHECK_NB_BITS - 128.
184	return p.ReduceWithMaxBits(x, uint64(RANGE_CHECK_NB_BITS))
185}
186
187// Reduces a field element x such that x % MODULUS = y.
188func (p *Chip) ReduceWithMaxBits(x Variable, maxNbBits uint64) Variable {
189	// Witness a `quotient` and `remainder` such that:
190	//
191	// 		MODULUS * quotient + remainder = x
192	//
193	// Must check that remainder \in [0, MODULUS) and quotient \in [0, 2^maxNbBits) to ensure that this
194	// computation does not overflow.
195
196	result, err := p.api.Compiler().NewHint(ReduceHint, 2, x.Limb)
197	if err != nil {
198		panic(err)
199	}
200
201	quotient := result[0]
202	p.rangeChecker.Check(quotient, int(maxNbBits))
203
204	remainder := NewVariable(result[1])
205	p.RangeCheck(remainder)
206
207	p.api.AssertIsEqual(x.Limb, p.api.Add(p.api.Mul(quotient, MODULUS), remainder.Limb))
208
209	return remainder
210}
211
212// The hint used to compute Reduce.
213func ReduceHint(_ *big.Int, inputs []*big.Int, results []*big.Int) error {
214	if len(inputs) != 1 {
215		panic("ReduceHint expects 1 input operand")
216	}
217	input := inputs[0]
218	quotient := new(big.Int).Div(input, MODULUS)
219	remainder := new(big.Int).Rem(input, MODULUS)
220	results[0] = quotient
221	results[1] = remainder
222	return nil
223}
224
225// Computes the inverse of a field element x such that x * x^-1 = 1.
226func (p *Chip) Inverse(x Variable) (Variable, frontend.Variable) {
227	result, err := p.api.Compiler().NewHint(InverseHint, 1, x.Limb)
228	if err != nil {
229		panic(err)
230	}
231
232	inverse := NewVariable(result[0])
233	isZero := p.api.IsZero(x.Limb)
234	hasInv := p.api.Sub(1, isZero)
235	p.RangeCheck(inverse)
236
237	product := p.Mul(inverse, x)
238	productToCheck := p.api.Select(hasInv, product.Limb, frontend.Variable(1))
239	p.api.AssertIsEqual(productToCheck, frontend.Variable(1))
240
241	return inverse, hasInv
242}
243
244// The hint used to compute Inverse.
245func InverseHint(_ *big.Int, inputs []*big.Int, results []*big.Int) error {
246	if len(inputs) != 1 {
247		panic("InverseHint expects 1 input operand")
248	}
249
250	input := inputs[0]
251	if input.Cmp(MODULUS) == 0 || input.Cmp(MODULUS) == 1 {
252		panic("Input is not in the field")
253	}
254
255	inputGl := goldilocks.NewElement(input.Uint64())
256	resultGl := goldilocks.NewElement(0)
257
258	// Will set resultGL if inputGL == 0
259	resultGl.Inverse(&inputGl)
260
261	result := big.NewInt(0)
262	results[0] = resultGl.BigInt(result)
263
264	return nil
265}
266
267// The hint used to split a GoldilocksVariable into 2 32 bit limbs.
268func SplitLimbsHint(_ *big.Int, inputs []*big.Int, results []*big.Int) error {
269	if len(inputs) != 1 {
270		panic("SplitLimbsHint expects 1 input operand")
271	}
272
273	// The Goldilocks field element
274	input := inputs[0]
275
276	if input.Cmp(MODULUS) == 0 || input.Cmp(MODULUS) == 1 {
277		return fmt.Errorf("input is not in the field")
278	}
279
280	two_32 := big.NewInt(int64(math.Pow(2, 32)))
281
282	// The most significant bits
283	results[0] = new(big.Int).Quo(input, two_32)
284	// The least significant bits
285	results[1] = new(big.Int).Rem(input, two_32)
286
287	return nil
288}
289
290// Range checks a field element x to be less than the Golidlocks modulus 2 ^ 64 - 2 ^ 32 + 1.
291func (p *Chip) RangeCheck(x Variable) {
292	// The Goldilocks' modulus is 2^64 - 2^32 + 1, which is:
293	//
294	// 		1111111111111111111111111111111100000000000000000000000000000001
295	//
296	// in big endian binary. This function will first verify that x is at most 64 bits wide. Then it
297	// checks that if the bits[0:31] (in big-endian) are all 1, then bits[32:64] are all zero.
298
299	result, err := p.api.Compiler().NewHint(SplitLimbsHint, 2, x.Limb)
300	if err != nil {
301		panic(err)
302	}
303
304	// We check that this is a valid decomposition of the Goldilock's element and range-check each limb.
305	mostSigLimb := result[0]
306	leastSigLimb := result[1]
307	p.api.AssertIsEqual(
308		p.api.Add(
309			p.api.Mul(mostSigLimb, uint64(math.Pow(2, 32))),
310			leastSigLimb,
311		),
312		x.Limb,
313	)
314	p.rangeChecker.Check(mostSigLimb, 32)
315	p.rangeChecker.Check(leastSigLimb, 32)
316
317	// If the most significant bits are all 1, then we need to check that the least significant bits are all zero
318	// in order for element to be less than the Goldilock's modulus.
319	// Otherwise, we don't need to do any checks, since we already know that the element is less than the Goldilocks modulus.
320	shouldCheck := p.api.IsZero(p.api.Sub(mostSigLimb, uint64(math.Pow(2, 32))-1))
321	p.api.AssertIsEqual(
322		p.api.Select(
323			shouldCheck,
324			leastSigLimb,
325			frontend.Variable(0),
326		),
327		frontend.Variable(0),
328	)
329}
330
331func (p *Chip) AssertIsEqual(x, y Variable) {
332	p.api.AssertIsEqual(x.Limb, y.Limb)
333}
334
335// Computes the n'th primitive root of unity for the Goldilocks field.
336func PrimitiveRootOfUnity(nLog uint64) goldilocks.Element {
337	if nLog > TWO_ADICITY {
338		panic("nLog is greater than TWO_ADICITY")
339	}
340	res := goldilocks.NewElement(POWER_OF_TWO_GENERATOR.Uint64())
341	for i := 0; i < int(TWO_ADICITY-nLog); i++ {
342		res.Square(&res)
343	}
344	return res
345}
346
347func TwoAdicSubgroup(nLog uint64) []goldilocks.Element {
348	if nLog > TWO_ADICITY {
349		panic("nLog is greater than GOLDILOCKS_TWO_ADICITY")
350	}
351
352	var res []goldilocks.Element
353	rootOfUnity := PrimitiveRootOfUnity(nLog)
354	res = append(res, goldilocks.NewElement(1))
355
356	for i := 0; i < (1<<nLog)-1; i++ {
357		lastElement := res[len(res)-1]
358		res = append(res, *lastElement.Mul(&lastElement, &rootOfUnity))
359	}
360
361	return res
362}