算法原理参考：shamir 秘密共享

1. 大素数p的选取，这里选取的是128bit大素数

   func prime128Bit() *big.Int {
   	p := big.NewInt(2)
   	p.Exp(p, big.NewInt(127), nil)
   	p.Sub(p, big.NewInt(1))
   	return p
   }
   

2. Share和Polynomial 结构体定义，Share结构体定义了参与者拥有的秘密(x,y), Polynomial结构体定义了多项式，多项式的「知识」即该多项式的系数，知道多项式的系数，也就知道该多项式

   type Share struct {
   	x, y *big.Int
   }
   
   type Polynomial struct {
   	coefficients []*big.Int
   	modulus      *big.Int
   }
   

3. 拉格朗日插值算法实现：对定点shares，计算该多项式在x处的值，对于sss算法 ，秘密在x=0处，需要注意的是拉格朗日插值算法中有模除，需要转为乘以这个数的逆元

   func Lagrange(x, modulus *big.Int, shares ...Share) *big.Int {
   	xs := make([]*big.Int, len(shares))
   	ys := make([]*big.Int, len(shares))
   	for i := range shares {
   		xs[i] = shares[i].x
   		ys[i] = shares[i].y
   	}
   
   	numerators := make([]*big.Int, len(xs))
   	denominators := make([]*big.Int, len(ys))
   	for i := range xs {
   		numerators[i] = cal(xs, x, i)
   		denominators[i] = cal(xs, xs[i], i)
   	}
   
   	result := big.NewInt(0)
   	for i := range numerators {
   		numerators[i].Mul(numerators[i], ys[i])
   		numerators[i].Mod(numerators[i], modulus)
   		v := (&big.Int{}).Set(numerators[i].Mul(numerators[i], denominators[i].ModInverse(denominators[i],modulus)))
   		result.Add(result, v)
   	}
   	return result.Mod(result, modulus)
   }
   

4. 秘密分发：生成t-1次对多项式，若提供secret，则使用提供的秘密，否则随机生成秘密，随后计算n个参与者的秘密shares

   func SecretDistribute(secret, modulus *big.Int, t, n int64) ([]Share, *big.Int) {
   	if t > n {
   		panic("t has to be less than n")
   	}
   
   	p := NewPolynomial(t, modulus)
   	if secret != nil {
   		p.coefficients[0] = secret
   	}
   	secret = p.coefficients[0]
   
   	shares := make([]Share, n)
   	for i := int64(0); i < n; i++ {
   		x, _ := rand.Int(rand.Reader, modulus)
   		y := p.valueAt(x)
   		shares[i] = Share{x, y}
   	}
   
   	return shares, secret
   }
   

5. 验证

   func main() {
   	prime := prime128Bit()
   
   	//secret := big.NewInt(12345678987654321)
   
   	shares, secret1 := SecretDistribute(nil, prime, 5, 8)
   
   	respect := Lagrange(big.NewInt(0), prime, shares[0], shares[5], shares[4], shares[1],shares[7])
   
   	fmt.Println(secret1,respect)
   }