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// Copyright (C) 2019-2024 Algorand, Inc.
// This file is part of go-algorand
//
// go-algorand is free software: you can redistribute it and/or modify
// it under the terms of the GNU Affero General Public License as
// published by the Free Software Foundation, either version 3 of the
// License, or (at your option) any later version.
//
// go-algorand is distributed in the hope that it will be useful,
// but WITHOUT ANY WARRANTY; without even the implied warranty of
// MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the
// GNU Affero General Public License for more details.
//
// You should have received a copy of the GNU Affero General Public License
// along with go-algorand. If not, see <https://www.gnu.org/licenses/>.
package stateproof
import (
"errors"
"math"
"math/big"
"math/bits"
)
// errors for the weights verification
var (
ErrSignedWeightLessThanProvenWeight = errors.New("signed weight is less than or equal to proven weight")
ErrTooManyReveals = errors.New("too many reveals in state proof")
ErrZeroSignedWeight = errors.New("signed weight cannot be zero")
ErrIllegalInputForLnApprox = errors.New("cannot calculate a ln integer value for 0")
ErrInsufficientSignedWeight = errors.New("the number of reveals is not large enough to prove that the desired weight signed, with the desired security level")
ErrNegativeNumOfRevealsEquation = errors.New("state proof creation failed: weights will not be able to satisfy the verification equation")
)
func bigInt(num uint64) *big.Int {
return (&big.Int{}).SetUint64(num)
}
// LnIntApproximation returns a uint64 approximation
func LnIntApproximation(x uint64) (uint64, error) {
if x == 0 {
return 0, ErrIllegalInputForLnApprox
}
result := math.Log(float64(x))
precision := uint64(1 << precisionBits)
expandWithPrecision := result * float64(precision)
return uint64(math.Ceil(expandWithPrecision)), nil
}
// verifyWeights makes sure that the number of reveals in the state proof is correct with respect
// to the signedWeight and a provenWeight upper bound.
// This function checks that the following inequality is satisfied
//
// numReveals * (3 * 2^b * (signedWeight^2 - 2^2d) + d * (T-1) * Y) >= ((strengthTarget) * T + numReveals * P) * Y
//
// where signedWeight/(2^d) >=1 for some integer d>=0, p = P/(2^b) >= ln(provenWeight), t = T/(2^b) >= ln(2) >= (T-1)/(2^b)
// for some integers P,T >= 0 and b=16.
//
// T and b are defined in the code as the constants ln2IntApproximation and precisionBits respectively.
// P is set to lnProvenWeight argument
// more details can be found on the Algorand's spec
func verifyWeights(signedWeight uint64, lnProvenWeight uint64, numOfReveals uint64, strengthTarget uint64) error {
if numOfReveals > MaxReveals {
return ErrTooManyReveals
}
if signedWeight == 0 {
return ErrZeroSignedWeight
}
// in order to make the code more readable and reusable we will define the following expressions:
// y = signedWeight^2 + 2^(d + 2) * signedWeight + 2^2d
// x = 3 * 2^b * (signedWeight^2 - 2^2d)
// w = d * (T - 1)
//
// numReveals * (3 * 2^b * (signedWeight^2 - 2^2d) + d * (T-1) * Y) >= ((strengthTarget) * T + numReveals * P) * Y
// /\
// ||
// \/
// numReveals * (x + w * y) >= ((strengthTarget) * T + numReveals * P) * y
y, x, w := getSubExpressions(signedWeight)
lhs := &big.Int{}
lhs.Set(w).
Mul(lhs, y).
Add(x, lhs).
Mul(bigInt(numOfReveals), lhs)
revealsTimesP := &big.Int{}
revealsTimesP.Set(bigInt(numOfReveals)).Mul(revealsTimesP, bigInt(lnProvenWeight))
rhs := &big.Int{}
rhs.Set(bigInt(strengthTarget))
rhs.Mul(rhs, bigInt(ln2IntApproximation)).
Add(rhs, revealsTimesP).
Mul(rhs, y)
if lhs.Cmp(rhs) < 0 {
return ErrInsufficientSignedWeight
}
return nil
}
// numReveals computes the number of reveals necessary to achieve the desired
// security target. We search for small integer that will satisfy the verification
// inequality checked by the verifyWeights function.
// In order to make sure the number will satisfy the verifier we will use the following inequality
//
// numReveals >= ((strengthTarget) * T * Y / (3 * 2^b * (signedWeight^2 - 2^2d) + (d * (T - 1) - P) * Y))
// where signedWeight/(2^d) >=1 for some integer d>=0, p = P/(2^b) >= ln(provenWeight), t = T/(2^b) >= ln(2) >= (T-1)/(2^b)
// for some integers P,T >= 0 and b=16.
//
// T and b are defined in the code as the constants ln2IntApproximation and precisionBits respectively,
// and P is set to lnProvenWeight argument.
//
// more details can be found on the Algorand's spec
func numReveals(signedWeight uint64, lnProvenWeight uint64, strengthTarget uint64) (uint64, error) {
// in order to make the code more readable and reusable we will define the following expressions:
// y = signedWeight^2 + 2^(d + 2) * signedWeight + 2^2d
// x = 3 * 2^b * (signedWeight^2 - 2^2d)
// w = d * (T - 1)
//
// numReveals >= ((strengthTarget) * T * Y / (3 * 2^b * (signedWeight^2 - 2^2d) + (d * (T - 1) - P) * Y))
// /\
// ||
// \/
// numReveals >= ((strengthTarget) * T * y / (x + (w - P) * y))
y, x, w := getSubExpressions(signedWeight)
// numerator = strengthTarget * ln2IntApproximation * y
numerator := bigInt(strengthTarget)
numerator.Mul(numerator, bigInt(ln2IntApproximation)).
Mul(numerator, y)
// denom = x + (w - lnProvenWeight) * y
denom := w
denom.Sub(denom, bigInt(lnProvenWeight)).
Mul(denom, y).
Add(x, denom)
if denom.Sign() <= 0 {
return 0, ErrNegativeNumOfRevealsEquation
}
// numberReveals = (numerator / denom) + 1
// by adding 1 we guarantee that the return value satisfy the inequality and therefore
// will satisfy the verifier.
// + 1 to account for the decimal point value loss due to integer division
res := numerator.Div(numerator, denom).Uint64() + 1
if res > MaxReveals {
return 0, ErrTooManyReveals
}
return res, nil
}
// getSubExpressions calculate the following expression to make the code more readable and reusable
// y = signedWeight^2 + 2^(d + 2) * signedWeight + 2^2d
// x = 3 * 2^b * (signedWeight^2 - 2^2d)
// w = d * (T - 1)
func getSubExpressions(signedWeight uint64) (y *big.Int, x *big.Int, w *big.Int) {
// find d s.t 2^(d+1) >= signedWeight >= 2^(d)
d := uint(bits.Len64(signedWeight)) - 1
signedWtPower2 := bigInt(signedWeight)
signedWtPower2.Mul(signedWtPower2, signedWtPower2)
//tmp = 2^(d+2)*signedWt
tmp := bigInt(1)
tmp.Lsh(tmp, d+2).
Mul(tmp, bigInt(signedWeight))
// Y = signedWeight^2 + 2^(d+2)*signedWeight +2^2d == signedWeight^2 + tmp +2^2d
y = bigInt(1)
y.Lsh(y, 2*d).
Add(y, tmp).
Add(y, signedWtPower2)
// x = 3*2^b*(signedWeight^2-2^2d)
x = bigInt(1)
x.Lsh(x, 2*d).
Sub(signedWtPower2, x).
Mul(x, bigInt(3)).
Mul(x, bigInt(1<<precisionBits))
// w = d*(T-1)
w = bigInt(uint64(d))
w.Mul(w, bigInt(ln2IntApproximation-1))
return
}
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