lotus/chain/types/electionproof.go

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package types
import (
"math/big"
"github.com/filecoin-project/lotus/build"
"github.com/minio/blake2b-simd"
)
type ElectionProof struct {
WinCount int64
VRFProof []byte
}
const precision = 256
var (
expNumCoef []*big.Int
expDenoCoef []*big.Int
)
func init() {
parse := func(coefs []string) []*big.Int {
out := make([]*big.Int, len(coefs))
for i, coef := range coefs {
c, ok := new(big.Int).SetString(coef, 10)
if !ok {
panic("could not parse exp paramemter")
}
// << 256 (Q.0 to Q.256), >> 128 to transform integer params to coefficients
c = c.Lsh(c, precision-128)
out[i] = c
}
return out
}
// parameters are in integer format,
// coefficients are *2^-128 of that
num := []string{
"-648770010757830093818553637600",
"67469480939593786226847644286976",
"-3197587544499098424029388939001856",
"89244641121992890118377641805348864",
"-1579656163641440567800982336819953664",
"17685496037279256458459817590917169152",
"-115682590513835356866803355398940131328",
"340282366920938463463374607431768211456",
}
expNumCoef = parse(num)
deno := []string{
"1225524182432722209606361",
"114095592300906098243859450",
"5665570424063336070530214243",
"194450132448609991765137938448",
"5068267641632683791026134915072",
"104716890604972796896895427629056",
"1748338658439454459487681798864896",
"23704654329841312470660182937960448",
"259380097567996910282699886670381056",
"2250336698853390384720606936038375424",
"14978272436876548034486263159246028800",
"72144088983913131323343765784380833792",
"224599776407103106596571252037123047424",
"340282366920938463463374607431768211456",
}
expDenoCoef = parse(deno)
}
// expneg accepts x in Q.256 format and computes e^-x.
// It is most precise within [0, 1.725) range, where error is less than 3.4e-30.
// Over the [0, 5) range its error is less than 4.6e-15.
// Output is in Q.256 format.
func expneg(x *big.Int) *big.Int {
// exp is approximated by rational function
// polynomials of the rational function are evaluated using Horner's method
num := polyval(expNumCoef, x) // Q.256
deno := polyval(expDenoCoef, x) // Q.256
num = num.Lsh(num, precision) // Q.512
return num.Div(num, deno) // Q.512 / Q.256 => Q.256
}
// polyval evaluates a polynomial given by coefficients `p` in Q.256 format
// at point `x` in Q.256 format. Output is in Q.256.
// Coefficients should be ordered from the highest order coefficient to the lowest.
func polyval(p []*big.Int, x *big.Int) *big.Int {
// evaluation using Horner's method
res := new(big.Int).Set(p[0]) // Q.256
tmp := new(big.Int) // big.Int.Mul doesn't like when input is reused as output
for _, c := range p[1:] {
tmp = tmp.Mul(res, x) // Q.256 * Q.256 => Q.512
res = res.Rsh(tmp, precision) // Q.512 >> 256 => Q.256
res = res.Add(res, c)
}
return res
}
// computes lambda in Q.256
func lambda(power, totalPower *big.Int) *big.Int {
lam := new(big.Int).Mul(power, blocksPerEpoch.Int) // Q.0
lam = lam.Lsh(lam, precision) // Q.256
lam = lam.Div(lam /* Q.256 */, totalPower /* Q.0 */) // Q.256
return lam
}
var MaxWinCount = 3 * int64(build.BlocksPerEpoch)
type poiss struct {
lam *big.Int
pmf *big.Int
icdf *big.Int
tmp *big.Int // temporary variable for optimization
k uint64
}
// newPoiss starts poisson inverted CDF
// lambda is in Q.256 format
// returns (instance, `1-poisscdf(0, lambda)`)
// CDF value returend is reused when calling `next`
func newPoiss(lambda *big.Int) (*poiss, *big.Int) {
// pmf(k) = (lambda^k)*(e^lambda) / k!
// k = 0 here, so it simplifies to just e^-lambda
elam := expneg(lambda) // Q.256
pmf := new(big.Int).Set(elam)
// icdf(k) = 1 - ∑ᵏᵢ₌₀ pmf(i)
// icdf(0) = 1 - pmf(0)
icdf := big.NewInt(1)
icdf = icdf.Lsh(icdf, precision) // Q.256
icdf = icdf.Sub(icdf, pmf) // Q.256
k := uint64(0)
p := &poiss{
lam: lambda,
pmf: pmf,
tmp: elam,
icdf: icdf,
k: k,
}
return p, icdf
}
// next computes `k++, 1-poisscdf(k, lam)`
// return is in Q.256 format
func (p *poiss) next() *big.Int {
// incrementally compute next pmf and icdf
// pmf(k) = (lambda^k)*(e^lambda) / k!
// so pmf(k) = pmf(k-1) * lambda / k
p.k++
p.tmp.SetUint64(p.k) // Q.0
// calculate pmf for k
p.pmf = p.pmf.Div(p.pmf, p.tmp) // Q.256 / Q.0 => Q.256
// we are using `tmp` as target for multiplication as using an input as output
// for Int.Mul causes allocations
p.tmp = p.tmp.Mul(p.pmf, p.lam) // Q.256 * Q.256 => Q.512
p.pmf = p.pmf.Rsh(p.tmp, precision) // Q.512 >> 256 => Q.256
// calculate output
// icdf(k) = icdf(k-1) - pmf(k)
p.icdf = p.icdf.Sub(p.icdf, p.pmf) // Q.256
return p.icdf
}
// ComputeWinCount uses VRFProof to compute number of wins
// The algorithm is based on Algorand's Sortition with Binomial distribution
// replaced by Poisson distribution.
func (ep *ElectionProof) ComputeWinCount(power BigInt, totalPower BigInt) int64 {
h := blake2b.Sum256(ep.VRFProof)
lhs := BigFromBytes(h[:]).Int // 256bits, assume Q.256 so [0, 1)
// We are calculating upside-down CDF of Poisson distribution with
// rate λ=power*E/totalPower
// Steps:
// 1. calculate λ=power*E/totalPower
// 2. calculate elam = exp(-λ)
// 3. Check how many times we win:
// j = 0
// pmf = elam
// rhs = 1 - pmf
// for h(vrf) < rhs: j++; pmf = pmf * lam / j; rhs = rhs - pmf
lam := lambda(power.Int, totalPower.Int) // Q.256
p, rhs := newPoiss(lam)
var j int64
for lhs.Cmp(rhs) < 0 && j < MaxWinCount {
rhs = p.next()
j++
}
return j
}