forked from coinbase/kryptology
/
fp.go
694 lines (617 loc) · 18.8 KB
/
fp.go
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package bls12381
import (
"encoding/binary"
"fmt"
"io"
"math/big"
"github.com/berry-block/kryptology/internal"
"github.com/berry-block/kryptology/pkg/core/curves/native"
)
// fp field element mod p
type fp [Limbs]uint64
var (
modulus = fp{
0xb9feffffffffaaab,
0x1eabfffeb153ffff,
0x6730d2a0f6b0f624,
0x64774b84f38512bf,
0x4b1ba7b6434bacd7,
0x1a0111ea397fe69a,
}
halfModulus = fp{
0xdcff_7fff_ffff_d556,
0x0f55_ffff_58a9_ffff,
0xb398_6950_7b58_7b12,
0xb23b_a5c2_79c2_895f,
0x258d_d3db_21a5_d66b,
0x0d00_88f5_1cbf_f34d,
}
// 2^256 mod p
r = fp{
0x760900000002fffd,
0xebf4000bc40c0002,
0x5f48985753c758ba,
0x77ce585370525745,
0x5c071a97a256ec6d,
0x15f65ec3fa80e493,
}
// 2^512 mod p
r2 = fp{
0xf4df1f341c341746,
0x0a76e6a609d104f1,
0x8de5476c4c95b6d5,
0x67eb88a9939d83c0,
0x9a793e85b519952d,
0x11988fe592cae3aa,
}
// 2^768 mod p
r3 = fp{
0xed48ac6bd94ca1e0,
0x315f831e03a7adf8,
0x9a53352a615e29dd,
0x34c04e5e921e1761,
0x2512d43565724728,
0x0aa6346091755d4d,
}
biModulus = new(big.Int).SetBytes([]byte{
0x1a, 0x01, 0x11, 0xea, 0x39, 0x7f, 0xe6, 0x9a, 0x4b, 0x1b, 0xa7, 0xb6, 0x43, 0x4b, 0xac, 0xd7, 0x64, 0x77, 0x4b, 0x84, 0xf3, 0x85, 0x12, 0xbf, 0x67, 0x30, 0xd2, 0xa0, 0xf6, 0xb0, 0xf6, 0x24, 0x1e, 0xab, 0xff, 0xfe, 0xb1, 0x53, 0xff, 0xff, 0xb9, 0xfe, 0xff, 0xff, 0xff, 0xff, 0xaa, 0xab},
)
)
// inv = -(p^{-1} mod 2^64) mod 2^64
const inv = 0x89f3_fffc_fffc_fffd
const hashBytes = 64
// IsZero returns 1 if fp == 0, 0 otherwise
func (f *fp) IsZero() int {
t := f[0]
t |= f[1]
t |= f[2]
t |= f[3]
t |= f[4]
t |= f[5]
return int(((int64(t) | int64(-t)) >> 63) + 1)
}
// IsNonZero returns 1 if fp != 0, 0 otherwise
func (f *fp) IsNonZero() int {
t := f[0]
t |= f[1]
t |= f[2]
t |= f[3]
t |= f[4]
t |= f[5]
return int(-((int64(t) | int64(-t)) >> 63))
}
// IsOne returns 1 if fp == 1, 0 otherwise
func (f *fp) IsOne() int {
return f.Equal(&r)
}
// Cmp returns -1 if f < rhs
// 0 if f == rhs
// 1 if f > rhs
func (f *fp) Cmp(rhs *fp) int {
gt := uint64(0)
lt := uint64(0)
for i := 5; i >= 0; i-- {
// convert to two 64-bit numbers where
// the leading bits are zeros and hold no meaning
// so rhs - f actually means gt
// and f - rhs actually means lt.
rhsH := rhs[i] >> 32
rhsL := rhs[i] & 0xffffffff
lhsH := f[i] >> 32
lhsL := f[i] & 0xffffffff
// Check the leading bit
// if negative then f > rhs
// if positive then f < rhs
gt |= (rhsH - lhsH) >> 32 & 1 &^ lt
lt |= (lhsH - rhsH) >> 32 & 1 &^ gt
gt |= (rhsL - lhsL) >> 32 & 1 &^ lt
lt |= (lhsL - rhsL) >> 32 & 1 &^ gt
}
// Make the result -1 for <, 0 for =, 1 for >
return int(gt) - int(lt)
}
// Equal returns 1 if fp == rhs, 0 otherwise
func (f *fp) Equal(rhs *fp) int {
t := f[0] ^ rhs[0]
t |= f[1] ^ rhs[1]
t |= f[2] ^ rhs[2]
t |= f[3] ^ rhs[3]
t |= f[4] ^ rhs[4]
t |= f[5] ^ rhs[5]
return int(((int64(t) | int64(-t)) >> 63) + 1)
}
// LexicographicallyLargest returns 1 if
// this element is strictly lexicographically larger than its negation
// 0 otherwise
func (f *fp) LexicographicallyLargest() int {
var ff fp
ff.fromMontgomery(f)
_, borrow := sbb(ff[0], halfModulus[0], 0)
_, borrow = sbb(ff[1], halfModulus[1], borrow)
_, borrow = sbb(ff[2], halfModulus[2], borrow)
_, borrow = sbb(ff[3], halfModulus[3], borrow)
_, borrow = sbb(ff[4], halfModulus[4], borrow)
_, borrow = sbb(ff[5], halfModulus[5], borrow)
return (int(borrow) - 1) & 1
}
// Sgn0 returns the lowest bit value
func (f *fp) Sgn0() int {
t := new(fp).fromMontgomery(f)
return int(t[0] & 1)
}
// SetOne fp = r
func (f *fp) SetOne() *fp {
f[0] = r[0]
f[1] = r[1]
f[2] = r[2]
f[3] = r[3]
f[4] = r[4]
f[5] = r[5]
return f
}
// SetZero fp = 0
func (f *fp) SetZero() *fp {
f[0] = 0
f[1] = 0
f[2] = 0
f[3] = 0
f[4] = 0
f[5] = 0
return f
}
// SetUint64 fp = rhs
func (f *fp) SetUint64(rhs uint64) *fp {
f[0] = rhs
f[1] = 0
f[2] = 0
f[3] = 0
f[4] = 0
f[5] = 0
return f.toMontgomery(f)
}
// Random generates a random field element
func (f *fp) Random(reader io.Reader) (*fp, error) {
var t [WideFieldBytes]byte
n, err := reader.Read(t[:])
if err != nil {
return nil, err
}
if n != WideFieldBytes {
return nil, fmt.Errorf("can only read %d when %d are needed", n, WideFieldBytes)
}
return f.Hash(t[:]), nil
}
// Hash converts the byte sequence into a field element
func (f *fp) Hash(input []byte) *fp {
dst := []byte("BLS12381_XMD:SHA-256_SSWU_RO_")
xmd := native.ExpandMsgXmd(native.EllipticPointHasherSha256(), input, dst, hashBytes)
var t [WideFieldBytes]byte
copy(t[:hashBytes], internal.ReverseScalarBytes(xmd))
return f.SetBytesWide(&t)
}
// toMontgomery converts this field to montgomery form
func (f *fp) toMontgomery(a *fp) *fp {
// arg.R^0 * R^2 / R = arg.R
return f.Mul(a, &r2)
}
// fromMontgomery converts this field from montgomery form
func (f *fp) fromMontgomery(a *fp) *fp {
// Mul by 1 is division by 2^256 mod q
//out.Mul(arg, &[native.FieldLimbs]uint64{1, 0, 0, 0})
return f.montReduce(&[Limbs * 2]uint64{a[0], a[1], a[2], a[3], a[4], a[5], 0, 0, 0, 0, 0, 0})
}
// Neg performs modular negation
func (f *fp) Neg(a *fp) *fp {
// Subtract `arg` from `modulus`. Ignore final borrow
// since it can't underflow.
var t [Limbs]uint64
var borrow uint64
t[0], borrow = sbb(modulus[0], a[0], 0)
t[1], borrow = sbb(modulus[1], a[1], borrow)
t[2], borrow = sbb(modulus[2], a[2], borrow)
t[3], borrow = sbb(modulus[3], a[3], borrow)
t[4], borrow = sbb(modulus[4], a[4], borrow)
t[5], _ = sbb(modulus[5], a[5], borrow)
// t could be `modulus` if `arg`=0. Set mask=0 if self=0
// and 0xff..ff if `arg`!=0
mask := a[0] | a[1] | a[2] | a[3] | a[4] | a[5]
mask = -((mask | -mask) >> 63)
f[0] = t[0] & mask
f[1] = t[1] & mask
f[2] = t[2] & mask
f[3] = t[3] & mask
f[4] = t[4] & mask
f[5] = t[5] & mask
return f
}
// Square performs modular square
func (f *fp) Square(a *fp) *fp {
var r [2 * Limbs]uint64
var carry uint64
r[1], carry = mac(0, a[0], a[1], 0)
r[2], carry = mac(0, a[0], a[2], carry)
r[3], carry = mac(0, a[0], a[3], carry)
r[4], carry = mac(0, a[0], a[4], carry)
r[5], r[6] = mac(0, a[0], a[5], carry)
r[3], carry = mac(r[3], a[1], a[2], 0)
r[4], carry = mac(r[4], a[1], a[3], carry)
r[5], carry = mac(r[5], a[1], a[4], carry)
r[6], r[7] = mac(r[6], a[1], a[5], carry)
r[5], carry = mac(r[5], a[2], a[3], 0)
r[6], carry = mac(r[6], a[2], a[4], carry)
r[7], r[8] = mac(r[7], a[2], a[5], carry)
r[7], carry = mac(r[7], a[3], a[4], 0)
r[8], r[9] = mac(r[8], a[3], a[5], carry)
r[9], r[10] = mac(r[9], a[4], a[5], 0)
r[11] = r[10] >> 63
r[10] = (r[10] << 1) | r[9]>>63
r[9] = (r[9] << 1) | r[8]>>63
r[8] = (r[8] << 1) | r[7]>>63
r[7] = (r[7] << 1) | r[6]>>63
r[6] = (r[6] << 1) | r[5]>>63
r[5] = (r[5] << 1) | r[4]>>63
r[4] = (r[4] << 1) | r[3]>>63
r[3] = (r[3] << 1) | r[2]>>63
r[2] = (r[2] << 1) | r[1]>>63
r[1] = r[1] << 1
r[0], carry = mac(0, a[0], a[0], 0)
r[1], carry = adc(0, r[1], carry)
r[2], carry = mac(r[2], a[1], a[1], carry)
r[3], carry = adc(0, r[3], carry)
r[4], carry = mac(r[4], a[2], a[2], carry)
r[5], carry = adc(0, r[5], carry)
r[6], carry = mac(r[6], a[3], a[3], carry)
r[7], carry = adc(0, r[7], carry)
r[8], carry = mac(r[8], a[4], a[4], carry)
r[9], carry = adc(0, r[9], carry)
r[10], carry = mac(r[10], a[5], a[5], carry)
r[11], _ = adc(0, r[11], carry)
return f.montReduce(&r)
}
// Double this element
func (f *fp) Double(a *fp) *fp {
return f.Add(a, a)
}
// Mul performs modular multiplication
func (f *fp) Mul(arg1, arg2 *fp) *fp {
// Schoolbook multiplication
var r [2 * Limbs]uint64
var carry uint64
r[0], carry = mac(0, arg1[0], arg2[0], 0)
r[1], carry = mac(0, arg1[0], arg2[1], carry)
r[2], carry = mac(0, arg1[0], arg2[2], carry)
r[3], carry = mac(0, arg1[0], arg2[3], carry)
r[4], carry = mac(0, arg1[0], arg2[4], carry)
r[5], r[6] = mac(0, arg1[0], arg2[5], carry)
r[1], carry = mac(r[1], arg1[1], arg2[0], 0)
r[2], carry = mac(r[2], arg1[1], arg2[1], carry)
r[3], carry = mac(r[3], arg1[1], arg2[2], carry)
r[4], carry = mac(r[4], arg1[1], arg2[3], carry)
r[5], carry = mac(r[5], arg1[1], arg2[4], carry)
r[6], r[7] = mac(r[6], arg1[1], arg2[5], carry)
r[2], carry = mac(r[2], arg1[2], arg2[0], 0)
r[3], carry = mac(r[3], arg1[2], arg2[1], carry)
r[4], carry = mac(r[4], arg1[2], arg2[2], carry)
r[5], carry = mac(r[5], arg1[2], arg2[3], carry)
r[6], carry = mac(r[6], arg1[2], arg2[4], carry)
r[7], r[8] = mac(r[7], arg1[2], arg2[5], carry)
r[3], carry = mac(r[3], arg1[3], arg2[0], 0)
r[4], carry = mac(r[4], arg1[3], arg2[1], carry)
r[5], carry = mac(r[5], arg1[3], arg2[2], carry)
r[6], carry = mac(r[6], arg1[3], arg2[3], carry)
r[7], carry = mac(r[7], arg1[3], arg2[4], carry)
r[8], r[9] = mac(r[8], arg1[3], arg2[5], carry)
r[4], carry = mac(r[4], arg1[4], arg2[0], 0)
r[5], carry = mac(r[5], arg1[4], arg2[1], carry)
r[6], carry = mac(r[6], arg1[4], arg2[2], carry)
r[7], carry = mac(r[7], arg1[4], arg2[3], carry)
r[8], carry = mac(r[8], arg1[4], arg2[4], carry)
r[9], r[10] = mac(r[9], arg1[4], arg2[5], carry)
r[5], carry = mac(r[5], arg1[5], arg2[0], 0)
r[6], carry = mac(r[6], arg1[5], arg2[1], carry)
r[7], carry = mac(r[7], arg1[5], arg2[2], carry)
r[8], carry = mac(r[8], arg1[5], arg2[3], carry)
r[9], carry = mac(r[9], arg1[5], arg2[4], carry)
r[10], r[11] = mac(r[10], arg1[5], arg2[5], carry)
return f.montReduce(&r)
}
// MulBy3b returns arg * 12 or 3 * b
func (f *fp) MulBy3b(arg *fp) *fp {
var a, t fp
a.Double(arg) // 2
t.Double(&a) // 4
a.Double(&t) // 8
a.Add(&a, &t) // 12
return f.Set(&a)
}
// Add performs modular addition
func (f *fp) Add(arg1, arg2 *fp) *fp {
var t fp
var carry uint64
t[0], carry = adc(arg1[0], arg2[0], 0)
t[1], carry = adc(arg1[1], arg2[1], carry)
t[2], carry = adc(arg1[2], arg2[2], carry)
t[3], carry = adc(arg1[3], arg2[3], carry)
t[4], carry = adc(arg1[4], arg2[4], carry)
t[5], _ = adc(arg1[5], arg2[5], carry)
// Subtract the modulus to ensure the value
// is smaller.
return f.Sub(&t, &modulus)
}
// Sub performs modular subtraction
func (f *fp) Sub(arg1, arg2 *fp) *fp {
d0, borrow := sbb(arg1[0], arg2[0], 0)
d1, borrow := sbb(arg1[1], arg2[1], borrow)
d2, borrow := sbb(arg1[2], arg2[2], borrow)
d3, borrow := sbb(arg1[3], arg2[3], borrow)
d4, borrow := sbb(arg1[4], arg2[4], borrow)
d5, borrow := sbb(arg1[5], arg2[5], borrow)
// If underflow occurred on the final limb, borrow 0xff...ff, otherwise
// borrow = 0x00...00. Conditionally mask to add the modulus
borrow = -borrow
d0, carry := adc(d0, modulus[0]&borrow, 0)
d1, carry = adc(d1, modulus[1]&borrow, carry)
d2, carry = adc(d2, modulus[2]&borrow, carry)
d3, carry = adc(d3, modulus[3]&borrow, carry)
d4, carry = adc(d4, modulus[4]&borrow, carry)
d5, _ = adc(d5, modulus[5]&borrow, carry)
f[0] = d0
f[1] = d1
f[2] = d2
f[3] = d3
f[4] = d4
f[5] = d5
return f
}
// Sqrt performs modular square root
func (f *fp) Sqrt(a *fp) (*fp, int) {
// Shank's method, as p = 3 (mod 4). This means
// exponentiate by (p+1)/4. This only works for elements
// that are actually quadratic residue,
// so check the result at the end.
var c, z fp
z.pow(a, &fp{
0xee7fbfffffffeaab,
0x07aaffffac54ffff,
0xd9cc34a83dac3d89,
0xd91dd2e13ce144af,
0x92c6e9ed90d2eb35,
0x0680447a8e5ff9a6,
})
c.Square(&z)
wasSquare := c.Equal(a)
f.CMove(f, &z, wasSquare)
return f, wasSquare
}
// Invert performs modular inverse
func (f *fp) Invert(a *fp) (*fp, int) {
// Exponentiate by p - 2
t := &fp{}
t.pow(a, &fp{
0xb9feffffffffaaa9,
0x1eabfffeb153ffff,
0x6730d2a0f6b0f624,
0x64774b84f38512bf,
0x4b1ba7b6434bacd7,
0x1a0111ea397fe69a,
})
wasInverted := a.IsNonZero()
f.CMove(a, t, wasInverted)
return f, wasInverted
}
// SetBytes converts a little endian byte array into a field element
// return 0 if the bytes are not in the field, 1 if they are
func (f *fp) SetBytes(arg *[FieldBytes]byte) (*fp, int) {
var borrow uint64
t := &fp{}
t[0] = binary.LittleEndian.Uint64(arg[:8])
t[1] = binary.LittleEndian.Uint64(arg[8:16])
t[2] = binary.LittleEndian.Uint64(arg[16:24])
t[3] = binary.LittleEndian.Uint64(arg[24:32])
t[4] = binary.LittleEndian.Uint64(arg[32:40])
t[5] = binary.LittleEndian.Uint64(arg[40:])
// Try to subtract the modulus
_, borrow = sbb(t[0], modulus[0], 0)
_, borrow = sbb(t[1], modulus[1], borrow)
_, borrow = sbb(t[2], modulus[2], borrow)
_, borrow = sbb(t[3], modulus[3], borrow)
_, borrow = sbb(t[4], modulus[4], borrow)
_, borrow = sbb(t[5], modulus[5], borrow)
// If the element is smaller than modulus then the
// subtraction will underflow, producing a borrow value
// of 1. Otherwise, it'll be zero.
mask := int(borrow)
return f.CMove(f, t.toMontgomery(t), mask), mask
}
// SetBytesWide takes 96 bytes as input and treats them as a 512-bit number.
// Attributed to https://github.com/zcash/pasta_curves/blob/main/src/fields/Fp.rs#L255
// We reduce an arbitrary 768-bit number by decomposing it into two 384-bit digits
// with the higher bits multiplied by 2^384. Thus, we perform two reductions
//
// 1. the lower bits are multiplied by r^2, as normal
// 2. the upper bits are multiplied by r^2 * 2^384 = r^3
//
// and computing their sum in the field. It remains to see that arbitrary 384-bit
// numbers can be placed into Montgomery form safely using the reduction. The
// reduction works so long as the product is less than r=2^384 multiplied by
// the modulus. This holds because for any `c` smaller than the modulus, we have
// that (2^384 - 1)*c is an acceptable product for the reduction. Therefore, the
// reduction always works so long as `c` is in the field; in this case it is either the
// constant `r2` or `r3`.
func (f *fp) SetBytesWide(a *[WideFieldBytes]byte) *fp {
d0 := &fp{
binary.LittleEndian.Uint64(a[:8]),
binary.LittleEndian.Uint64(a[8:16]),
binary.LittleEndian.Uint64(a[16:24]),
binary.LittleEndian.Uint64(a[24:32]),
binary.LittleEndian.Uint64(a[32:40]),
binary.LittleEndian.Uint64(a[40:48]),
}
d1 := &fp{
binary.LittleEndian.Uint64(a[48:56]),
binary.LittleEndian.Uint64(a[56:64]),
binary.LittleEndian.Uint64(a[64:72]),
binary.LittleEndian.Uint64(a[72:80]),
binary.LittleEndian.Uint64(a[80:88]),
binary.LittleEndian.Uint64(a[88:96]),
}
// d0*r2 + d1*r3
d0.Mul(d0, &r2)
d1.Mul(d1, &r3)
return f.Add(d0, d1)
}
// SetBigInt initializes an element from big.Int
// The value is reduced by the modulus
func (f *fp) SetBigInt(bi *big.Int) *fp {
var buffer [FieldBytes]byte
t := new(big.Int).Set(bi)
t.Mod(t, biModulus)
t.FillBytes(buffer[:])
copy(buffer[:], internal.ReverseScalarBytes(buffer[:]))
_, _ = f.SetBytes(&buffer)
return f
}
// Set copies a into fp
func (f *fp) Set(a *fp) *fp {
f[0] = a[0]
f[1] = a[1]
f[2] = a[2]
f[3] = a[3]
f[4] = a[4]
f[5] = a[5]
return f
}
// SetLimbs converts an array into a field element
// by converting to montgomery form
func (f *fp) SetLimbs(a *[Limbs]uint64) *fp {
return f.toMontgomery((*fp)(a))
}
// SetRaw converts a raw array into a field element
// Assumes input is already in montgomery form
func (f *fp) SetRaw(a *[Limbs]uint64) *fp {
f[0] = a[0]
f[1] = a[1]
f[2] = a[2]
f[3] = a[3]
f[4] = a[4]
f[5] = a[5]
return f
}
// Bytes converts a field element to a little endian byte array
func (f *fp) Bytes() [FieldBytes]byte {
var out [FieldBytes]byte
t := new(fp).fromMontgomery(f)
binary.LittleEndian.PutUint64(out[:8], t[0])
binary.LittleEndian.PutUint64(out[8:16], t[1])
binary.LittleEndian.PutUint64(out[16:24], t[2])
binary.LittleEndian.PutUint64(out[24:32], t[3])
binary.LittleEndian.PutUint64(out[32:40], t[4])
binary.LittleEndian.PutUint64(out[40:], t[5])
return out
}
// BigInt converts this element into the big.Int struct
func (f *fp) BigInt() *big.Int {
buffer := f.Bytes()
return new(big.Int).SetBytes(internal.ReverseScalarBytes(buffer[:]))
}
// Raw converts this element into the a [FieldLimbs]uint64
func (f *fp) Raw() [Limbs]uint64 {
t := new(fp).fromMontgomery(f)
return *t
}
// CMove performs conditional select.
// selects arg1 if choice == 0 and arg2 if choice == 1
func (f *fp) CMove(arg1, arg2 *fp, choice int) *fp {
mask := uint64(-choice)
f[0] = arg1[0] ^ ((arg1[0] ^ arg2[0]) & mask)
f[1] = arg1[1] ^ ((arg1[1] ^ arg2[1]) & mask)
f[2] = arg1[2] ^ ((arg1[2] ^ arg2[2]) & mask)
f[3] = arg1[3] ^ ((arg1[3] ^ arg2[3]) & mask)
f[4] = arg1[4] ^ ((arg1[4] ^ arg2[4]) & mask)
f[5] = arg1[5] ^ ((arg1[5] ^ arg2[5]) & mask)
return f
}
// CNeg conditionally negates a if choice == 1
func (f *fp) CNeg(a *fp, choice int) *fp {
var t fp
t.Neg(a)
return f.CMove(f, &t, choice)
}
// Exp raises base^exp.
func (f *fp) Exp(base, exp *fp) *fp {
e := (&fp{}).fromMontgomery(exp)
return f.pow(base, e)
}
func (f *fp) pow(base, e *fp) *fp {
var tmp, res fp
res.SetOne()
for i := len(e) - 1; i >= 0; i-- {
for j := 63; j >= 0; j-- {
res.Square(&res)
tmp.Mul(&res, base)
res.CMove(&res, &tmp, int(e[i]>>j)&1)
}
}
f[0] = res[0]
f[1] = res[1]
f[2] = res[2]
f[3] = res[3]
f[4] = res[4]
f[5] = res[5]
return f
}
// montReduce performs the montgomery reduction
func (f *fp) montReduce(r *[2 * Limbs]uint64) *fp {
// Taken from Algorithm 14.32 in Handbook of Applied Cryptography
var r1, r2, r3, r4, r5, r6, r7, r8, r9, r10, r11, carry, k uint64
var rr fp
k = r[0] * inv
_, carry = mac(r[0], k, modulus[0], 0)
r1, carry = mac(r[1], k, modulus[1], carry)
r2, carry = mac(r[2], k, modulus[2], carry)
r3, carry = mac(r[3], k, modulus[3], carry)
r4, carry = mac(r[4], k, modulus[4], carry)
r5, carry = mac(r[5], k, modulus[5], carry)
r6, r7 = adc(r[6], 0, carry)
k = r1 * inv
_, carry = mac(r1, k, modulus[0], 0)
r2, carry = mac(r2, k, modulus[1], carry)
r3, carry = mac(r3, k, modulus[2], carry)
r4, carry = mac(r4, k, modulus[3], carry)
r5, carry = mac(r5, k, modulus[4], carry)
r6, carry = mac(r6, k, modulus[5], carry)
r7, r8 = adc(r7, r[7], carry)
k = r2 * inv
_, carry = mac(r2, k, modulus[0], 0)
r3, carry = mac(r3, k, modulus[1], carry)
r4, carry = mac(r4, k, modulus[2], carry)
r5, carry = mac(r5, k, modulus[3], carry)
r6, carry = mac(r6, k, modulus[4], carry)
r7, carry = mac(r7, k, modulus[5], carry)
r8, r9 = adc(r8, r[8], carry)
k = r3 * inv
_, carry = mac(r3, k, modulus[0], 0)
r4, carry = mac(r4, k, modulus[1], carry)
r5, carry = mac(r5, k, modulus[2], carry)
r6, carry = mac(r6, k, modulus[3], carry)
r7, carry = mac(r7, k, modulus[4], carry)
r8, carry = mac(r8, k, modulus[5], carry)
r9, r10 = adc(r9, r[9], carry)
k = r4 * inv
_, carry = mac(r4, k, modulus[0], 0)
r5, carry = mac(r5, k, modulus[1], carry)
r6, carry = mac(r6, k, modulus[2], carry)
r7, carry = mac(r7, k, modulus[3], carry)
r8, carry = mac(r8, k, modulus[4], carry)
r9, carry = mac(r9, k, modulus[5], carry)
r10, r11 = adc(r10, r[10], carry)
k = r5 * inv
_, carry = mac(r5, k, modulus[0], 0)
rr[0], carry = mac(r6, k, modulus[1], carry)
rr[1], carry = mac(r7, k, modulus[2], carry)
rr[2], carry = mac(r8, k, modulus[3], carry)
rr[3], carry = mac(r9, k, modulus[4], carry)
rr[4], carry = mac(r10, k, modulus[5], carry)
rr[5], _ = adc(r11, r[11], carry)
return f.Sub(&rr, &modulus)
}