forked from bnb-chain/tss-lib
/
safe_prime.go
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/
safe_prime.go
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// Copyright © 2019 Binance
//
// This file is part of Binance. The full Binance copyright notice, including
// terms governing use, modification, and redistribution, is contained in the
// file LICENSE at the root of the source code distribution tree.
package common
import (
"context"
"crypto/rand"
"errors"
"fmt"
"io"
"math/big"
"sync"
"sync/atomic"
)
const (
primeTestN = 30
)
type (
GermainSafePrime struct {
q,
p *big.Int // p = 2q + 1
}
)
func (sgp *GermainSafePrime) Prime() *big.Int {
return sgp.q
}
func (sgp *GermainSafePrime) SafePrime() *big.Int {
return sgp.p
}
func (sgp *GermainSafePrime) Validate() bool {
return probablyPrime(sgp.q) &&
getSafePrime(sgp.q).Cmp(sgp.p) == 0 &&
probablyPrime(sgp.p)
}
// ----- //
func getSafePrime(p *big.Int) *big.Int {
i := new(big.Int)
i.Mul(p, two)
i.Add(i, one)
return i
}
func probablyPrime(prime *big.Int) bool {
return prime != nil && prime.ProbablyPrime(primeTestN)
}
// ----- //
// The following code is a modified copy of: https://github.com/didiercrunch/paillier/blob/753322e473bf8ee20267c7824e68ae47360cc69b/safe_prime_generator.go
// It is an implementation of the algorithm described in "Safe Prime Generation with a Combined Sieve" https://eprint.iacr.org/2003/186.pdf
// The code is the original Go implementation of rand.Prime optimized for
// generating safe (Sophie Germain) primes.
// A safe prime is a prime number of the form 2p + 1, where p is also a prime.
// Note from Author (https://github.com/pdyraga):
// I've adapted a Go code for generating random numbers by inserting some
// optimisations that will allow us to generate safe primes faster than
// with the previous, naive approach.
//
// First of all, having q which can be prime, we first check whether q%3=1.
// If that's true, there is no chance p=2q+1 is prime. It lets us to reject
// candidate numbers quicker without running an expensive primality tests.
//
// Also, before we run a primality test for q, we may check p=2q+1 against
// the primes between 3-53 (We are limited by Go's uint64 range).
//
// If all those conditions are met and we know p is prime, it's enough to
// check Pocklington criterion for q instead of running an expensive
// primality test for it.
// smallPrimes is a list of small, prime numbers that allows us to rapidly
// exclude some fraction of composite candidates when searching for a random
// prime. This list is truncated at the point where smallPrimesProduct exceeds
// a uint64. It does not include two because we ensure that the candidates are
// odd by construction.
var smallPrimes = []uint8{
3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53,
}
// smallPrimesProduct is the product of the values in smallPrimes and allows us
// to reduce a candidate prime by this number and then determine whether it's
// coprime to all the elements of smallPrimes without further big.Int
// operations.
var smallPrimesProduct = new(big.Int).SetUint64(16294579238595022365)
// ErrGeneratorCancelled is an error returned from GetRandomSafePrimesConcurrent
// when the work of the generator has been cancelled as a result of the context
// being done (cancellation or timeout).
var ErrGeneratorCancelled = fmt.Errorf("generator work cancelled")
// GetRandomSafePrimesConcurrent tries to find safe primes concurrently.
// The returned results are safe primes `p` and prime `q` such that `p=2q+1`.
// Concurrency level can be controlled with the `concurrencyLevel` parameter.
// If a safe prime could not be found before the context is done, the error
// is returned. Also, if at least one search process failed, error is returned
// as well.
//
// How fast we generate a prime number is mostly a matter of luck and it depends
// on how lucky we are with drawing the first bytes.
// With today's multi-core processors, we can execute the process on multiple
// cores concurrently, accept the first valid result and cancel the rest of
// work. This way, with the same finding algorithm, we can get the result
// faster.
//
// Concurrency level should be set depending on what `bitLen` of prime is
// expected. For example, as of today, on a typical workstation, for 512-bit
// safe prime, `concurrencyLevel` should be set to `1` as generating the prime
// of this length is a matter of milliseconds for a single core.
// For 1024-bit safe prime, `concurrencyLevel` should be usually set to at least
// `2` and for 2048-bit safe prime, `concurrencyLevel` must be set to at least
// `4` to get the result in a reasonable time.
//
// This function generates safe primes of at least 6 `bitLen`. For every
// generated safe prime, the two most significant bits are always set to `1`
// - we don't want the generated number to be too small.
func GetRandomSafePrimesConcurrent(ctx context.Context, bitLen, numPrimes int, concurrency int) ([]*GermainSafePrime, error) {
if bitLen < 6 {
return nil, errors.New("safe prime size must be at least 6 bits")
}
if numPrimes < 1 {
return nil, errors.New("numPrimes should be > 0")
}
primeCh := make(chan *GermainSafePrime, concurrency*numPrimes)
errCh := make(chan error, concurrency*numPrimes)
primes := make([]*GermainSafePrime, 0, numPrimes)
waitGroup := &sync.WaitGroup{}
defer close(primeCh)
defer close(errCh)
defer waitGroup.Wait()
generatorCtx, cancelGeneratorCtx := context.WithCancel(ctx)
defer cancelGeneratorCtx()
for i := 0; i < concurrency; i++ {
waitGroup.Add(1)
runGenPrimeRoutine(
generatorCtx, primeCh, errCh, waitGroup, rand.Reader, bitLen,
)
}
needed := int32(numPrimes)
for {
select {
case result := <-primeCh:
primes = append(primes, result)
if atomic.AddInt32(&needed, -1) <= 0 {
return primes[:numPrimes], nil
}
case err := <-errCh:
return nil, err
case <-ctx.Done():
return nil, ErrGeneratorCancelled
}
}
}
// Starts a Goroutine searching for a safe prime of the specified `pBitLen`.
// If succeeds, writes prime `p` and prime `q` such that `p = 2q+1` to the
// `primeCh`. Prime `p` has a bit length equal to `pBitLen` and prime `q` has
// a bit length equal to `pBitLen-1`.
//
// The algorithm is as follows:
// 1. Generate a random odd number `q` of length `pBitLen-1` with two the most
// significant bits set to `1`.
// 2. Execute preliminary primality test on `q` checking whether it is coprime
// to all the elements of `smallPrimes`. It allows to eliminate trivial
// cases quickly, when `q` is obviously no prime, without running an
// expensive final primality tests.
// If `q` is coprime to all of the `smallPrimes`, then go to the point 3.
// If not, add `2` and try again. Do it at most 10 times.
// 3. Check the potentially prime `q`, whether `q = 1 (mod 3)`. This will
// happen for 50% of cases.
// If it is, then `p = 2q+1` will be a multiple of 3, so it will be obviously
// not a prime number. In this case, add `2` and try again. Do it at most 10
// times. If `q != 1 (mod 3)`, go to the point 4.
// 4. Now we know `q` is potentially prime and `p = 2q+1` is not a multiple of
// 3. We execute a preliminary primality test on `p`, checking whether
// it is coprime to all the elements of `smallPrimes` just like we did for
// `q` in point 2. If `p` is not coprime to at least one element of the
// `smallPrimes`, then go back to point 1.
// If `p` is coprime to all the elements of `smallPrimes`, go to point 5.
// 5. At this point, we know `q` is potentially prime, and `p=q+1` is also
// potentially prime. We need to execute a final primality test for `q`.
// We apply Miller-Rabin and Baillie-PSW tests. If they succeed, it means
// that `q` is prime with a very high probability. Knowing `q` is prime,
// we use Pocklington's criterion to prove the primality of `p=2q+1`, that
// is, we execute Fermat primality test to base 2 checking whether
// `2^{p-1} = 1 (mod p)`. It's significantly faster than running full
// Miller-Rabin and Baillie-PSW for `p`.
// If `q` and `p` are found to be prime, return them as a result. If not, go
// back to the point 1.
func runGenPrimeRoutine(
ctx context.Context,
primeCh chan<- *GermainSafePrime,
errCh chan<- error,
waitGroup *sync.WaitGroup,
rand io.Reader,
pBitLen int,
) {
qBitLen := pBitLen - 1
b := uint(qBitLen % 8)
if b == 0 {
b = 8
}
bytes := make([]byte, (qBitLen+7)/8)
p := new(big.Int)
q := new(big.Int)
bigMod := new(big.Int)
go func() {
defer waitGroup.Done()
for {
select {
case <-ctx.Done():
return
default:
_, err := io.ReadFull(rand, bytes)
if err != nil {
errCh <- err
return
}
// Clear bits in the first byte to make sure the candidate has
// a size <= bits.
bytes[0] &= uint8(int(1<<b) - 1)
// Don't let the value be too small, i.e, set the most
// significant two bits.
// Setting the top two bits, rather than just the top bit,
// means that when two of these values are multiplied together,
// the result isn't ever one bit short.
if b >= 2 {
bytes[0] |= 3 << (b - 2)
} else {
// Here b==1, because b cannot be zero.
bytes[0] |= 1
if len(bytes) > 1 {
bytes[1] |= 0x80
}
}
// Make the value odd since an even number this large certainly
// isn't prime.
bytes[len(bytes)-1] |= 1
q.SetBytes(bytes)
// Calculate the value mod the product of smallPrimes. If it's
// a multiple of any of these primes we add two until it isn't.
// The probability of overflowing is minimal and can be ignored
// because we still perform Miller-Rabin tests on the result.
bigMod.Mod(q, smallPrimesProduct)
mod := bigMod.Uint64()
NextDelta:
for delta := uint64(0); delta < 1<<20; delta += 2 {
m := mod + delta
for _, prime := range smallPrimes {
if m%uint64(prime) == 0 && (qBitLen > 6 || m != uint64(prime)) {
continue NextDelta
}
}
if delta > 0 {
bigMod.SetUint64(delta)
q.Add(q, bigMod)
}
// If `q = 1 (mod 3)`, then `p` is a multiple of `3` so it's
// obviously no prime and such `q` should be rejected.
// This will happen in 50% of cases and we should detect
// and eliminate them early.
//
// Explanation:
// If q = 1 (mod 3) then there exists a q' such that:
// q = 3q' + 1
//
// Since p = 2q + 1:
// p = 2q + 1 = 2(3q' + 1) + 1 = 6q' + 2 + 1 = 6q' + 3 =
// = 3(2q' + 1)
// So `p` is a multiple of `3`.
qMod3 := new(big.Int).Mod(q, big.NewInt(3))
if qMod3.Cmp(big.NewInt(1)) == 0 {
continue NextDelta
}
// p = 2q+1
p.Mul(q, big.NewInt(2))
p.Add(p, big.NewInt(1))
if !isPrimeCandidate(p) {
continue NextDelta
}
break
}
// There is a tiny possibility that, by adding delta, we caused
// the number to be one bit too long. Thus we check BitLen
// here.
if q.ProbablyPrime(20) &&
isPocklingtonCriterionSatisfied(p) &&
q.BitLen() == qBitLen {
if sgp := (&GermainSafePrime{p: p, q: q}); sgp.Validate() {
primeCh <- &GermainSafePrime{p: p, q: q}
}
p, q = new(big.Int), new(big.Int)
}
}
}
}()
}
// Pocklington's criterion can be used to prove the primality of `p = 2q + 1`
// once one has proven the primality of `q`.
// With `q` prime, `p = 2q + 1`, and `p` passing Fermat's primality test to base
// `2` that `2^{p-1} = 1 (mod p)` then `p` is prime as well.
func isPocklingtonCriterionSatisfied(p *big.Int) bool {
return new(big.Int).Exp(
big.NewInt(2),
new(big.Int).Sub(p, big.NewInt(1)),
p,
).Cmp(big.NewInt(1)) == 0
}
func isPrimeCandidate(number *big.Int) bool {
m := new(big.Int).Mod(number, smallPrimesProduct).Uint64()
for _, prime := range smallPrimes {
if m%uint64(prime) == 0 && m != uint64(prime) {
return false
}
}
return true
}