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 threshold maximum number of bits at which we must reduce the element.
 42var REDUCE_NB_BITS_THRESHOLD uint8 = 254 - 64
 43
 44// The number of bits to use for range checks on inner products of field elements.
 45var RANGE_CHECK_NB_BITS int = 140
 46
 47// Registers the hint functions with the solver.
 48func init() {
 49	solver.RegisterHint(MulAddHint)
 50	solver.RegisterHint(ReduceHint)
 51	solver.RegisterHint(InverseHint)
 52	solver.RegisterHint(SplitLimbsHint)
 53}
 54
 55// A type alias used to represent Goldilocks field elements.
 56type Variable struct {
 57	Limb frontend.Variable
 58}
 59
 60// Creates a new Goldilocks field element from an existing variable. Assumes that the element is
 61// already reduced.
 62func NewVariable(x frontend.Variable) Variable {
 63	return Variable{Limb: x}
 64}
 65
 66// The zero element in the Golidlocks field.
 67func Zero() Variable {
 68	return NewVariable(0)
 69}
 70
 71// The one element in the Goldilocks field.
 72func One() Variable {
 73	return NewVariable(1)
 74}
 75
 76// The negative one element in the Goldilocks field.
 77func NegOne() Variable {
 78	return NewVariable(MODULUS.Uint64() - 1)
 79}
 80
 81// The chip used for Goldilocks field operations.
 82type Chip struct {
 83	api          frontend.API
 84	rangeChecker frontend.Rangechecker
 85}
 86
 87// Creates a new Goldilocks Chip.
 88func New(api frontend.API) *Chip {
 89	rangeChecker := rangecheck.New(api)
 90	return &Chip{api: api, rangeChecker: rangeChecker}
 91}
 92
 93// Adds two field elements such that x + y = z within the Golidlocks field.
 94func (p *Chip) Add(a Variable, b Variable) Variable {
 95	return p.MulAdd(a, NewVariable(1), b)
 96}
 97
 98// Adds two field elements such that x + y = z within the Golidlocks field without reducing.
 99func (p *Chip) AddNoReduce(a Variable, b Variable) Variable {
100	return NewVariable(p.api.Add(a.Limb, b.Limb))
101}
102
103// Subtracts two field elements such that x + y = z within the Golidlocks field.
104func (p *Chip) Sub(a Variable, b Variable) Variable {
105	return p.MulAdd(b, NewVariable(MODULUS.Uint64()-1), a)
106}
107
108// Subtracts two field elements such that x + y = z within the Golidlocks field without reducing.
109func (p *Chip) SubNoReduce(a Variable, b Variable) Variable {
110	return NewVariable(p.api.Add(a.Limb, p.api.Mul(b.Limb, MODULUS.Uint64()-1)))
111}
112
113// Multiplies two field elements such that x * y = z within the Golidlocks field.
114func (p *Chip) Mul(a Variable, b Variable) Variable {
115	return p.MulAdd(a, b, Zero())
116}
117
118// Multiplies two field elements such that x * y = z within the Golidlocks field without reducing.
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 such that x * y + z = c within the
124// Golidlocks 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	lhs := p.api.Mul(a.Limb, b.Limb)
135	lhs = p.api.Add(lhs, c.Limb)
136	rhs := p.api.Add(p.api.Mul(quotient.Limb, MODULUS), remainder.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 such that x * y + z = c within the
145// Golidlocks field without reducing.
146func (p *Chip) MulAddNoReduce(a Variable, b Variable, c Variable) Variable {
147	return p.AddNoReduce(p.MulNoReduce(a, b), c)
148}
149
150// The hint used to compute MulAdd.
151func MulAddHint(_ *big.Int, inputs []*big.Int, results []*big.Int) error {
152	if len(inputs) != 3 {
153		panic("MulAddHint expects 3 input operands")
154	}
155
156	for _, operand := range inputs {
157		if operand.Cmp(MODULUS) >= 0 {
158			panic(fmt.Sprintf("%s is not in the field", operand.String()))
159		}
160	}
161
162	product := new(big.Int).Mul(inputs[0], inputs[1])
163	sum := new(big.Int).Add(product, inputs[2])
164	quotient := new(big.Int).Div(sum, MODULUS)
165	remainder := new(big.Int).Rem(sum, MODULUS)
166
167	results[0] = quotient
168	results[1] = remainder
169
170	return nil
171}
172
173// Reduces a field element x such that x % MODULUS = y.
174func (p *Chip) Reduce(x Variable) Variable {
175	// Witness a `quotient` and `remainder` such that:
176	//
177	// 		MODULUS * quotient + remainder = x
178	//
179	// Must check that offset \in [0, MODULUS) and carry \in [0, 2^RANGE_CHECK_NB_BITS) to ensure
180	// that this computation does not overflow. We use 2^RANGE_CHECK_NB_BITS to reduce the cost of the range check
181	//
182	// In other words, we assume that we at most compute a a dot product with dimension at most RANGE_CHECK_NB_BITS - 128.
183
184	result, err := p.api.Compiler().NewHint(ReduceHint, 2, x.Limb)
185	if err != nil {
186		panic(err)
187	}
188
189	quotient := result[0]
190	p.rangeChecker.Check(quotient, RANGE_CHECK_NB_BITS)
191
192	remainder := NewVariable(result[1])
193	p.RangeCheck(remainder)
194	return remainder
195}
196
197// Reduces a field element x such that x % MODULUS = y.
198func (p *Chip) ReduceWithMaxBits(x Variable, maxNbBits uint64) Variable {
199	// Witness a `quotient` and `remainder` such that:
200	//
201	// 		MODULUS * quotient + remainder = x
202	//
203	// Must check that remainder \in [0, MODULUS) and quotient \in [0, 2^maxNbBits) to ensure that this
204	// computation does not overflow.
205
206	result, err := p.api.Compiler().NewHint(ReduceHint, 2, x.Limb)
207	if err != nil {
208		panic(err)
209	}
210
211	quotient := result[0]
212	p.rangeChecker.Check(quotient, int(maxNbBits))
213
214	remainder := NewVariable(result[1])
215	p.RangeCheck(remainder)
216	return remainder
217}
218
219// The hint used to compute Reduce.
220func ReduceHint(_ *big.Int, inputs []*big.Int, results []*big.Int) error {
221	if len(inputs) != 1 {
222		panic("ReduceHint expects 1 input operand")
223	}
224	input := inputs[0]
225	quotient := new(big.Int).Div(input, MODULUS)
226	remainder := new(big.Int).Rem(input, MODULUS)
227	results[0] = quotient
228	results[1] = remainder
229	return nil
230}
231
232// Computes the inverse of a field element x such that x * x^-1 = 1.
233func (p *Chip) Inverse(x Variable) Variable {
234	result, err := p.api.Compiler().NewHint(InverseHint, 1, x.Limb)
235	if err != nil {
236		panic(err)
237	}
238
239	inverse := NewVariable(result[0])
240	product := p.Mul(inverse, x)
241	p.api.AssertIsEqual(product.Limb, frontend.Variable(1))
242	return inverse
243}
244
245// The hint used to compute Inverse.
246func InverseHint(_ *big.Int, inputs []*big.Int, results []*big.Int) error {
247	if len(inputs) != 1 {
248		panic("InverseHint expects 1 input operand")
249	}
250
251	input := inputs[0]
252	if input.Cmp(MODULUS) == 0 || input.Cmp(MODULUS) == 1 {
253		panic("Input is not in the field")
254	}
255
256	inputGl := goldilocks.NewElement(input.Uint64())
257	resultGl := goldilocks.NewElement(0)
258	resultGl.Inverse(&inputGl)
259
260	result := big.NewInt(0)
261	results[0] = resultGl.BigInt(result)
262
263	return nil
264}
265
266// Computes a field element raised to some power.
267func (p *Chip) Exp(x Variable, k *big.Int) Variable {
268	if k.IsUint64() && k.Uint64() == 0 {
269		return One()
270	}
271
272	e := k
273	if k.Sign() == -1 {
274		panic("Unsupported negative exponent. Need to implement inversion.")
275	}
276
277	z := x
278	for i := e.BitLen() - 2; i >= 0; i-- {
279		z = p.Mul(z, z)
280		if e.Bit(i) == 1 {
281			z = p.Mul(z, x)
282		}
283	}
284
285	return z
286}
287
288// The hint used to split a GoldilocksVariable into 2 32 bit limbs.
289func SplitLimbsHint(_ *big.Int, inputs []*big.Int, results []*big.Int) error {
290	if len(inputs) != 1 {
291		panic("SplitLimbsHint expects 1 input operand")
292	}
293
294	// The Goldilocks field element
295	input := inputs[0]
296
297	if input.Cmp(MODULUS) == 0 || input.Cmp(MODULUS) == 1 {
298		return fmt.Errorf("input is not in the field")
299	}
300
301	two_32 := big.NewInt(int64(math.Pow(2, 32)))
302
303	// The most significant bits
304	results[0] = new(big.Int).Quo(input, two_32)
305	// The least significant bits
306	results[1] = new(big.Int).Rem(input, two_32)
307
308	return nil
309}
310
311// Range checks a field element x to be less than the Golidlocks modulus 2 ^ 64 - 2 ^ 32 + 1.
312func (p *Chip) RangeCheck(x Variable) {
313	// The Goldilocks' modulus is 2^64 - 2^32 + 1, which is:
314	//
315	// 		1111111111111111111111111111111100000000000000000000000000000001
316	//
317	// in big endian binary. This function will first verify that x is at most 64 bits wide. Then it
318	// checks that if the bits[0:31] (in big-endian) are all 1, then bits[32:64] are all zero.
319
320	result, err := p.api.Compiler().NewHint(SplitLimbsHint, 2, x.Limb)
321	if err != nil {
322		panic(err)
323	}
324
325	// We check that this is a valid decomposition of the Goldilock's element and range-check each limb.
326	mostSigLimb := result[0]
327	leastSigLimb := result[1]
328	p.api.AssertIsEqual(
329		p.api.Add(
330			p.api.Mul(mostSigLimb, uint64(math.Pow(2, 32))),
331			leastSigLimb,
332		),
333		x.Limb,
334	)
335	p.rangeChecker.Check(mostSigLimb, 32)
336	p.rangeChecker.Check(leastSigLimb, 32)
337
338	// If the most significant bits are all 1, then we need to check that the least significant bits are all zero
339	// in order for element to be less than the Goldilock's modulus.
340	// Otherwise, we don't need to do any checks, since we already know that the element is less than the Goldilocks modulus.
341	shouldCheck := p.api.IsZero(p.api.Sub(mostSigLimb, uint64(math.Pow(2, 32))-1))
342	p.api.AssertIsEqual(
343		p.api.Select(
344			shouldCheck,
345			leastSigLimb,
346			frontend.Variable(0),
347		),
348		frontend.Variable(0),
349	)
350}
351
352func (p *Chip) AssertIsEqual(x, y Variable) {
353	p.api.AssertIsEqual(x.Limb, y.Limb)
354}
355
356// Computes the n'th primitive root of unity for the Goldilocks field.
357func PrimitiveRootOfUnity(nLog uint64) goldilocks.Element {
358	if nLog > TWO_ADICITY {
359		panic("nLog is greater than TWO_ADICITY")
360	}
361	res := goldilocks.NewElement(POWER_OF_TWO_GENERATOR.Uint64())
362	for i := 0; i < int(TWO_ADICITY-nLog); i++ {
363		res.Square(&res)
364	}
365	return res
366}
367
368func TwoAdicSubgroup(nLog uint64) []goldilocks.Element {
369	if nLog > TWO_ADICITY {
370		panic("nLog is greater than GOLDILOCKS_TWO_ADICITY")
371	}
372
373	var res []goldilocks.Element
374	rootOfUnity := PrimitiveRootOfUnity(nLog)
375	res = append(res, goldilocks.NewElement(1))
376
377	for i := 0; i < (1 << nLog); i++ {
378		lastElement := res[len(res)-1]
379		res = append(res, *lastElement.Mul(&lastElement, &rootOfUnity))
380	}
381
382	return res
383}