Euler's criterion

https://brilliant.org/wiki/eulers-criterion

 

https://proofwiki.org/wiki/Euler%27s_Criterion/Quadratic_Residue

 

https://mathweb.ucsd.edu/~apollack/8_305S_course_notes_quadratic_reciprocity.pdf

 

 

that an integer 'a' has a square root (aka is a quadratic residue) modulo a prime if a^((p-1)/2) ≡ 1 (mod p)

prime is ≡ 3 (mod 4)

r = ±a^((p+1)/4) (mod p)

prime is ≡ 5 (mod 8)

//  alpha ==  (2*a)^((p-5)/8)    mod p
//  beta  ==  2*a*alpha^2        mod p  is a square root of -1
//  b     ==  a*alpha*(beta-1)   mod p  is a square root of a




https://www.rfc-editor.org/rfc/rfc9380.html#section-6.6.2-4.2.2.2

https://link.springer.com/content/pdf/10.1007/978-3-642-14623-7_13.pdf

 

 

 

package aa

import (
    "crypto/elliptic"
    "fmt"
    "math/big"
    "testing"
)

func g(x, b, p *big.Int) *big.Int {
    x3 := new(big.Int).Mul(x, x)
    x3.Mul(x3, x)

    threeX := new(big.Int).Lsh(x, 1)
    threeX.Add(threeX, x)

    x3.Sub(x3, threeX)
    x3.Add(x3, b)
    x3.Mod(x3, p)

    return x3
}

func findZ(i int64, c elliptic.Curve) int64 {
    p := c.Params().P
    b := c.Params().B
    pm1 := new(big.Int).Sub(p, big.NewInt(1))
    pm1.Rsh(pm1, 1)
    for ; ; i-- {
       t := big.NewInt(i)
       t.Exp(t, pm1, p)
       if t.Cmp(big.NewInt(1)) == 0 {
          continue
       }

       t = big.NewInt(i * -3)
       t.ModInverse(t, p)
       t.Mul(t, b)
       y := g(t, b, p)
       y.Exp(y, pm1, p)
       if y.Cmp(big.NewInt(1)) != 0 {
          continue
       }
       return i
    }
}

func hex(s string, base int) *big.Int {
    r, _ := new(big.Int).SetString(s, base)
    return r
}

var sm2 = &elliptic.CurveParams{
    P:       hex("FFFFFFFEFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFFF00000000FFFFFFFFFFFFFFFF", 16),
    N:       hex("FFFFFFFEFFFFFFFFFFFFFFFFFFFFFFFF7203DF6B21C6052B53BBF40939D54123", 16),
    B:       hex("28E9FA9E9D9F5E344D5A9E4BCF6509A7F39789F515AB8F92DDBCBD414D940E93", 16),
    Gx:      hex("32C4AE2C1F1981195F9904466A39C9948FE30BBFF2660BE1715A4589334C74C7", 16),
    Gy:      hex("BC3736A2F4F6779C59BDCEE36B692153D0A9877CC62A474002DF32E52139F0A0", 16),
    BitSize: 256,
    Name:    "sm2",
}

func TestZ(t *testing.T) {
    c := sm2
    r := int64(-1)
    for i := 0; i < 9; i++ {
       r--
       r = findZ(r, c)
       fmt.Println(r)
    }
}