java - 点在椭圆曲线上的标量乘法

Question

我正在对 NIST 指定的曲线“p192”执行椭圆曲线点算术运算。出于测试目的，我采用了 NIST Routine document for the curve p192 中所示的示例点.我得到加点和加倍点的正确答案，但对于标量乘法，我的答案不正确。由于这个原因，我无法达到是否

$ k^{-1}(kP) = P $

在哪里

$ k^{-1}.k = 1 mod p $

请帮助我了解我在哪里犯了错误。

package a;
import java.math.BigInteger;
import java.security.spec.ECPoint;
public class ScalarMultiply {
private static final BigInteger ONE = new BigInteger("1");;
static BigInteger TWO = new BigInteger("2");
    static BigInteger p = new BigInteger("6277101735386680763835789423207666416083908700390324961279");


public static ECPoint scalmult(ECPoint P, BigInteger k){
    ECPoint R =P,S = P;
    int length = k.bitLength();
    //System.out.println("length is" + length);
    byte[] binarray = new byte[length];
    for(int i=0;i<=length-1;i++){
        binarray[i] = k.mod(TWO).byteValue();
        k = k.divide(TWO);

    }
    for(int i=0;i<=length-1;i++){
        System.out.print("" + binarray[i]); 
    }

    for(int i = length - 2;i > 0;i--){
        R = doublePoint(R);
        if(binarray[i]== 1) 
            R = addPoint(R, S);
    }
return R;
}

public static ECPoint addPoint(ECPoint r, ECPoint s) {

    BigInteger slope = (r.getAffineY().subtract(s.getAffineY())).multiply(r.getAffineX().subtract(s.getAffineX()).modInverse(p)).mod(p);
    BigInteger Xout = (slope.modPow(TWO, p).subtract(r.getAffineX())).subtract(s.getAffineX()).mod(p);
    BigInteger Yout = r.getAffineY().negate().mod(p);
    Yout = Yout.add(slope.multiply(r.getAffineX().subtract(Xout))).mod(p);
    ECPoint out = new ECPoint(Xout, Yout);
    return out;
}

public static ECPoint doublePoint(ECPoint r) {
    // TODO Auto-generated method stub
    BigInteger slope = (r.getAffineX().pow(2)).multiply(new BigInteger("3"));
    slope = slope.add(new BigInteger("3"));
    slope = slope.multiply((r.getAffineY().multiply(TWO)).modInverse(p));
    BigInteger Xout = slope.pow(2).subtract(r.getAffineX().multiply(new BigInteger("2"))).mod(p);
    BigInteger Yout = (r.getAffineY().negate()).add(slope.multiply(r.getAffineX().subtract(Xout))).mod(p);
    ECPoint out = new ECPoint(Xout, Yout);
    return out;
}
}

主类是

    package a;
    import java.math.BigInteger;
    import java.security.spec.ECPoint;

    public class EccArithmetic {

    /**
     * @param args
     */
    public static void main(String[] args) {

             BigInteger xs = new BigInteger("d458e7d127ae671b0c330266d246769353a012073e97acf8", 16);
             BigInteger ys = new BigInteger
("325930500d851f336bddc050cf7fb11b5673a1645086df3b", 16);
             BigInteger xt = new BigInteger

("f22c4395213e9ebe67ddecdd87fdbd01be16fb059b9753a4", 16);
             BigInteger yt = new BigInteger

("264424096af2b3597796db48f8dfb41fa9cecc97691a9c79", 16);
             ECPoint S = new ECPoint(xs,ys);
             ECPoint T = new ECPoint(xt,yt);
             // Verifying addition 

             ECPoint Rst = ScalarMultiply.addPoint(S, T);
             BigInteger xst = new BigInteger

("48e1e4096b9b8e5ca9d0f1f077b8abf58e843894de4d0290", 16);   // Specified value of x of point R for addition  in NIST Routine example
             System.out.println("\nx-coordinate of point Rst is : " + Rst.getAffineX()); 
             System.out.println("\ny-coordinate of point Rst is : " + Rst.getAffineY());
             if(Rst.getAffineX().equals(xst))
                 System.out.println("Adding is correct");

//Verifying Doubling
BigInteger xr = new BigInteger

("30c5bc6b8c7da25354b373dc14dd8a0eba42d25a3f6e6962", 16);  // Specified value of x of point R for doubling  in NIST Routine example
             BigInteger yr = new BigInteger

("0dde14bc4249a721c407aedbf011e2ddbbcb2968c9d889cf", 16);
             ECPoint R2s = new ECPoint(xr, yr);  // Specified value of y of point R for doubling  in NIST Routine example
             System.out.println("\nx-coordinate of point R2s is : " + R2s.getAffineX()); 
             System.out.println("\ny-coordinate of point R2s is : " + R2s.getAffineY());
             System.out.println("\nx-coordinate of calculated point is : " + 

ScalarMultiply.doublePoint(S).getAffineX()); 
             System.out.println("\ny-coordinate of calculated point is : " + 

ScalarMultiply.doublePoint(S).getAffineY());
             if(R2s.getAffineX().equals(ScalarMultiply.doublePoint(S).getAffineX()))
                 System.out.println("Doubling is correct");

             xr = new BigInteger("1faee4205a4f669d2d0a8f25e3bcec9a62a6952965bf6d31", 16);  // Specified value of x of point R for scalar Multiplication  in NIST Routine example
             yr = new BigInteger("5ff2cdfa508a2581892367087c696f179e7a4d7e8260fb06", 16);   // Specified value of y of point R for scalar Multiplication  in NIST Routine example
             ECPoint Rds = new ECPoint(xr, yr);
             BigInteger d = new BigInteger

("a78a236d60baec0c5dd41b33a542463a8255391af64c74ee", 16);
             //Rs = new ECPoint(ScalarMultiply.scalmult(S, d).getAffineX(), yr);
             System.out.println("\nx-coordinate of point Rds is : " + Rds.getAffineX());
            System.out.println("\nx-coordinate of point Rds is : " + Rds.getAffineY());
            System.out.println("\nx-coordinate of calculated point is : " + ScalarMultiply.scalmult(S, 

            d).getAffineX());
            System.out.println("\nx-coordinate of calculated point is : " + ScalarMultiply.scalmult(S, 

            d).getAffineY());            
            if(Rds.getAffineX().equals(ScalarMultiply.scalmult(S, d).getAffineX()))
                 System.out.println("Scalar Multiplication is correct");

    }
}

Answer 1

addPoint 和 doublePoint 都不正确。下面编辑的JAVA代码做加倍加标量乘法，并检查加法加倍加标量乘法的结果是否正确:

ScalarMultiply.java

public class ScalarMultiply {

private static final BigInteger ONE = new BigInteger("1");;
static BigInteger TWO = new BigInteger("2");
static BigInteger p = new BigInteger("6277101735386680763835789423207666416083908700390324961279");
static BigInteger a = new BigInteger("6277101735386680763835789423207666416083908700390324961276");


public static ECPoint scalmult(ECPoint P, BigInteger kin){
    //ECPoint R=P; - incorrect
    ECPoint R = ECPoint.POINT_INFINITY,S = P;
    BigInteger k = kin.mod(p);
    int length = k.bitLength();
    //System.out.println("length is" + length);
    byte[] binarray = new byte[length];
    for(int i=0;i<=length-1;i++){
        binarray[i] = k.mod(TWO).byteValue();
        k = k.divide(TWO);
    }
    /*for(int i = length-1;i >= 0;i--){
        System.out.print("" + binarray[i]); 
    }*/

    for(int i = length-1;i >= 0;i--){
        // i should start at length-1 not -2 because the MSB of binarry may not be 1
        R = doublePoint(R);
        if(binarray[i]== 1) 
            R = addPoint(R, S);
    }
return R;
}

public static ECPoint addPoint(ECPoint r, ECPoint s) {

    if (r.equals(s))
        return doublePoint(r);
    else if (r.equals(ECPoint.POINT_INFINITY))
        return s;
    else if (s.equals(ECPoint.POINT_INFINITY))
        return r;
    BigInteger slope = (r.getAffineY().subtract(s.getAffineY())).multiply(r.getAffineX().subtract(s.getAffineX()).modInverse(p)).mod(p);
    BigInteger Xout = (slope.modPow(TWO, p).subtract(r.getAffineX())).subtract(s.getAffineX()).mod(p);
    //BigInteger Yout = r.getAffineY().negate().mod(p); - incorrect
    BigInteger Yout = s.getAffineY().negate().mod(p);
    //Yout = Yout.add(slope.multiply(r.getAffineX().subtract(Xout))).mod(p); - incorrect
    Yout = Yout.add(slope.multiply(s.getAffineX().subtract(Xout))).mod(p);
    ECPoint out = new ECPoint(Xout, Yout);
    return out;
}

public static ECPoint doublePoint(ECPoint r) {
    if (r.equals(ECPoint.POINT_INFINITY)) 
        return r;
    BigInteger slope = (r.getAffineX().pow(2)).multiply(new BigInteger("3"));
    //slope = slope.add(new BigInteger("3")); - incorrect
    slope = slope.add(a);
    slope = slope.multiply((r.getAffineY().multiply(TWO)).modInverse(p));
    BigInteger Xout = slope.pow(2).subtract(r.getAffineX().multiply(TWO)).mod(p);
    BigInteger Yout = (r.getAffineY().negate()).add(slope.multiply(r.getAffineX().subtract(Xout))).mod(p);
    ECPoint out = new ECPoint(Xout, Yout);
    return out;
}

EccArithmetic.java

public class EccArithmetic {

public static void main(String[] args) {

    BigInteger xs = new BigInteger("d458e7d127ae671b0c330266d246769353a012073e97acf8", 16);
    BigInteger ys = new BigInteger("325930500d851f336bddc050cf7fb11b5673a1645086df3b", 16);
    BigInteger xt = new BigInteger("f22c4395213e9ebe67ddecdd87fdbd01be16fb059b9753a4", 16);
    BigInteger yt = new BigInteger("264424096af2b3597796db48f8dfb41fa9cecc97691a9c79", 16);
    ECPoint S = new ECPoint(xs,ys);
    ECPoint T = new ECPoint(xt,yt);
    // Verifying addition 

    ECPoint Rst = ScalarMultiply.addPoint(S, T);
    BigInteger xst = new BigInteger("48e1e4096b9b8e5ca9d0f1f077b8abf58e843894de4d0290", 16);   // Specified value of x of point R for addition  in NIST Routine example
    System.out.println("\nx-coordinate of point Rst is : " + Rst.getAffineX()); 
    System.out.println("\ny-coordinate of point Rst is : " + Rst.getAffineY());
    if(Rst.getAffineX().equals(xst))
        System.out.println("Adding is correct");

    //Verifying Doubling
    BigInteger xr = new BigInteger("30c5bc6b8c7da25354b373dc14dd8a0eba42d25a3f6e6962", 16);  // Specified value of x of point R for doubling  in NIST Routine example
    BigInteger yr = new BigInteger("0dde14bc4249a721c407aedbf011e2ddbbcb2968c9d889cf", 16);
    ECPoint R2s = new ECPoint(xr, yr);  // Specified value of y of point R for doubling  in NIST Routine example
    System.out.println("\nx-coordinate of point R2s is : " + R2s.getAffineX()); 
    System.out.println("\ny-coordinate of point R2s is : " + R2s.getAffineY());
    System.out.println("\nx-coordinate of calculated point is : " + ScalarMultiply.doublePoint(S).getAffineX()); 
    System.out.println("\ny-coordinate of calculated point is : " +    ScalarMultiply.doublePoint(S).getAffineY());
    if(R2s.getAffineX().equals(ScalarMultiply.doublePoint(S).getAffineX()) &&
       R2s.getAffineY().equals(ScalarMultiply.doublePoint(S).getAffineY()))
        System.out.println("Doubling is correct");

    xr = new BigInteger("1faee4205a4f669d2d0a8f25e3bcec9a62a6952965bf6d31", 16);  // Specified value of x of point R for scalar Multiplication  in NIST Routine example
    yr = new BigInteger("5ff2cdfa508a2581892367087c696f179e7a4d7e8260fb06", 16);   // Specified value of y of point R for scalar Multiplication  in NIST Routine example
    ECPoint Rds = new ECPoint(xr, yr);
    BigInteger d = new BigInteger("a78a236d60baec0c5dd41b33a542463a8255391af64c74ee", 16);

    ECPoint Rs = ScalarMultiply.scalmult(S, d);

    System.out.println("\nx-coordinate of point Rds is : " + Rds.getAffineX());
    System.out.println("\ny-coordinate of point Rds is : " + Rds.getAffineY());
    System.out.println("\nx-coordinate of calculated point is : " + Rs.getAffineX());
    System.out.println("\ny-coordinate of calculated point is : " + Rs.getAffineY()); 


    if(Rds.getAffineX().equals(Rs.getAffineX()) &&
       Rds.getAffineY().equals(Rs.getAffineY()))
        System.out.println("Scalar Multiplication is correct");

}
}