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.MulAcc(c.Limb, a.Limb, b.Limb)
135	rhs := p.api.MulAcc(remainder.Limb, MODULUS, quotient.Limb)
136	p.api.AssertIsEqual(lhs, rhs)
137
138	p.RangeCheck(quotient)
139	p.RangeCheck(remainder)
140	return remainder
141}
142
143// Multiplies two field elements and adds a field element such that x * y + z = c within the
144// Golidlocks field without reducing.
145func (p *Chip) MulAddNoReduce(a Variable, b Variable, c Variable) Variable {
146	return NewVariable(p.api.MulAcc(c.Limb, a.Limb, b.Limb))
147}
148
149// The hint used to compute MulAdd.
150func MulAddHint(_ *big.Int, inputs []*big.Int, results []*big.Int) error {
151	if len(inputs) != 3 {
152		panic("MulAddHint expects 3 input operands")
153	}
154
155	for _, operand := range inputs {
156		if operand.Cmp(MODULUS) >= 0 {
157			panic(fmt.Sprintf("%s is not in the field", operand.String()))
158		}
159	}
160
161	product := new(big.Int).Mul(inputs[0], inputs[1])
162	sum := new(big.Int).Add(product, inputs[2])
163	quotient := new(big.Int).Div(sum, MODULUS)
164	remainder := new(big.Int).Rem(sum, MODULUS)
165
166	results[0] = quotient
167	results[1] = remainder
168
169	return nil
170}
171
172// Reduces a field element x such that x % MODULUS = y.
173func (p *Chip) Reduce(x Variable) Variable {
174	// Witness a `quotient` and `remainder` such that:
175	//
176	// 		MODULUS * quotient + remainder = x
177	//
178	// Must check that offset \in [0, MODULUS) and carry \in [0, 2^RANGE_CHECK_NB_BITS) to ensure
179	// that this computation does not overflow. We use 2^RANGE_CHECK_NB_BITS to reduce the cost of the range check
180	//
181	// In other words, we assume that we at most compute a a dot product with dimension at most RANGE_CHECK_NB_BITS - 128.
182
183	result, err := p.api.Compiler().NewHint(ReduceHint, 2, x.Limb)
184	if err != nil {
185		panic(err)
186	}
187
188	quotient := result[0]
189	p.rangeChecker.Check(quotient, RANGE_CHECK_NB_BITS)
190
191	remainder := NewVariable(result[1])
192	p.RangeCheck(remainder)
193	return remainder
194}
195
196// Reduces a field element x such that x % MODULUS = y.
197func (p *Chip) ReduceWithMaxBits(x Variable, maxNbBits uint64) Variable {
198	// Witness a `quotient` and `remainder` such that:
199	//
200	// 		MODULUS * quotient + remainder = x
201	//
202	// Must check that remainder \in [0, MODULUS) and quotient \in [0, 2^maxNbBits) to ensure that this
203	// computation does not overflow.
204
205	result, err := p.api.Compiler().NewHint(ReduceHint, 2, x.Limb)
206	if err != nil {
207		panic(err)
208	}
209
210	quotient := result[0]
211	p.rangeChecker.Check(quotient, int(maxNbBits))
212
213	remainder := NewVariable(result[1])
214	p.RangeCheck(remainder)
215	return remainder
216}
217
218// The hint used to compute Reduce.
219func ReduceHint(_ *big.Int, inputs []*big.Int, results []*big.Int) error {
220	if len(inputs) != 1 {
221		panic("ReduceHint expects 1 input operand")
222	}
223	input := inputs[0]
224	quotient := new(big.Int).Div(input, MODULUS)
225	remainder := new(big.Int).Rem(input, MODULUS)
226	results[0] = quotient
227	results[1] = remainder
228	return nil
229}
230
231// Computes the inverse of a field element x such that x * x^-1 = 1.
232func (p *Chip) Inverse(x Variable) Variable {
233	result, err := p.api.Compiler().NewHint(InverseHint, 1, x.Limb)
234	if err != nil {
235		panic(err)
236	}
237
238	inverse := NewVariable(result[0])
239	p.RangeCheck(inverse)
240
241	product := p.Mul(inverse, x)
242	p.api.AssertIsEqual(product.Limb, frontend.Variable(1))
243	return inverse
244}
245
246// The hint used to compute Inverse.
247func InverseHint(_ *big.Int, inputs []*big.Int, results []*big.Int) error {
248	if len(inputs) != 1 {
249		panic("InverseHint expects 1 input operand")
250	}
251
252	input := inputs[0]
253	if input.Cmp(MODULUS) == 0 || input.Cmp(MODULUS) == 1 {
254		panic("Input is not in the field")
255	}
256
257	inputGl := goldilocks.NewElement(input.Uint64())
258	resultGl := goldilocks.NewElement(0)
259	resultGl.Inverse(&inputGl)
260
261	result := big.NewInt(0)
262	results[0] = resultGl.BigInt(result)
263
264	return nil
265}
266
267// Computes a field element raised to some power.
268func (p *Chip) Exp(x Variable, k *big.Int) Variable {
269	if k.IsUint64() && k.Uint64() == 0 {
270		return One()
271	}
272
273	e := k
274	if k.Sign() == -1 {
275		panic("Unsupported negative exponent. Need to implement inversion.")
276	}
277
278	z := x
279	for i := e.BitLen() - 2; i >= 0; i-- {
280		z = p.Mul(z, z)
281		if e.Bit(i) == 1 {
282			z = p.Mul(z, x)
283		}
284	}
285
286	return z
287}
288
289// The hint used to split a GoldilocksVariable into 2 32 bit limbs.
290func SplitLimbsHint(_ *big.Int, inputs []*big.Int, results []*big.Int) error {
291	if len(inputs) != 1 {
292		panic("SplitLimbsHint expects 1 input operand")
293	}
294
295	// The Goldilocks field element
296	input := inputs[0]
297
298	if input.Cmp(MODULUS) == 0 || input.Cmp(MODULUS) == 1 {
299		return fmt.Errorf("input is not in the field")
300	}
301
302	two_32 := big.NewInt(int64(math.Pow(2, 32)))
303
304	// The most significant bits
305	results[0] = new(big.Int).Quo(input, two_32)
306	// The least significant bits
307	results[1] = new(big.Int).Rem(input, two_32)
308
309	return nil
310}
311
312// Range checks a field element x to be less than the Golidlocks modulus 2 ^ 64 - 2 ^ 32 + 1.
313func (p *Chip) RangeCheck(x Variable) {
314	// The Goldilocks' modulus is 2^64 - 2^32 + 1, which is:
315	//
316	// 		1111111111111111111111111111111100000000000000000000000000000001
317	//
318	// in big endian binary. This function will first verify that x is at most 64 bits wide. Then it
319	// checks that if the bits[0:31] (in big-endian) are all 1, then bits[32:64] are all zero.
320
321	result, err := p.api.Compiler().NewHint(SplitLimbsHint, 2, x.Limb)
322	if err != nil {
323		panic(err)
324	}
325
326	// We check that this is a valid decomposition of the Goldilock's element and range-check each limb.
327	mostSigLimb := result[0]
328	leastSigLimb := result[1]
329	p.api.AssertIsEqual(
330		p.api.Add(
331			p.api.Mul(mostSigLimb, uint64(math.Pow(2, 32))),
332			leastSigLimb,
333		),
334		x.Limb,
335	)
336	p.rangeChecker.Check(mostSigLimb, 32)
337	p.rangeChecker.Check(leastSigLimb, 32)
338
339	// If the most significant bits are all 1, then we need to check that the least significant bits are all zero
340	// in order for element to be less than the Goldilock's modulus.
341	// Otherwise, we don't need to do any checks, since we already know that the element is less than the Goldilocks modulus.
342	shouldCheck := p.api.IsZero(p.api.Sub(mostSigLimb, uint64(math.Pow(2, 32))-1))
343	p.api.AssertIsEqual(
344		p.api.Select(
345			shouldCheck,
346			leastSigLimb,
347			frontend.Variable(0),
348		),
349		frontend.Variable(0),
350	)
351}
352
353func (p *Chip) AssertIsEqual(x, y Variable) {
354	p.api.AssertIsEqual(x.Limb, y.Limb)
355}
356
357// Computes the n'th primitive root of unity for the Goldilocks field.
358func PrimitiveRootOfUnity(nLog uint64) goldilocks.Element {
359	if nLog > TWO_ADICITY {
360		panic("nLog is greater than TWO_ADICITY")
361	}
362	res := goldilocks.NewElement(POWER_OF_TWO_GENERATOR.Uint64())
363	for i := 0; i < int(TWO_ADICITY-nLog); i++ {
364		res.Square(&res)
365	}
366	return res
367}
368
369func TwoAdicSubgroup(nLog uint64) []goldilocks.Element {
370	if nLog > TWO_ADICITY {
371		panic("nLog is greater than GOLDILOCKS_TWO_ADICITY")
372	}
373
374	var res []goldilocks.Element
375	rootOfUnity := PrimitiveRootOfUnity(nLog)
376	res = append(res, goldilocks.NewElement(1))
377
378	for i := 0; i < (1 << nLog); i++ {
379		lastElement := res[len(res)-1]
380		res = append(res, *lastElement.Mul(&lastElement, &rootOfUnity))
381	}
382
383	return res
384}