我必须将两个场元素（f1和f2）与使用Weierstrass方程定义的椭圆曲线上的基点P相乘，即y2 = x3 + ax + b。我知道在标量乘法中，为了得到结果，这个点被添加了很多次。我有一个问题，如果标量乘法（（f1 + f2）* P）等于（f1 * P + f2 * P）？我已经在Jpair库中编写了代码，但我得到了不同的观点。而且，当我使用BigInt数字时，我得到同样的观点。有人能告诉我为什么在进行标量乘法时我会得到两个不同点？
import java.util.Random;
import uk.ac.ic.doc.jpair.api.FieldElement;
import uk.ac.ic.doc.jpair.pairing.BigInt;
import uk.ac.ic.doc.jpair.pairing.EllipticCurve;
import uk.ac.ic.doc.jpair.pairing.Fp;
import uk.ac.ic.doc.jpair.pairing.Point;
public class testCode {
public static void main(String [] args)
{
Random rnd = new Random();
//initialise the finite field Fp given a prime number p
BigInt p = new BigInt("8780710799663312522437781984754049815806883199414208211028653399266475630880222957078625179422662221423155858769582317459277713367317481324925129998224791",10);
Fp field = new Fp(p);
//System.out.println("order" +field.getOrder());
//initialise the elliptic curve. The curve can be specified given the field and the coefficient a and b
//of the Weierstrass equation:
//y2 = x3 +ax +b
//For example, in the case of the supersingular curve //y2 = x3 + x, a = 1 b = 0 ,
EllipticCurve ec = new EllipticCurve (field,BigInt.ONE,BigInt.ZERO);
//You also need to know the group order r and the cofactor
BigInt r = new BigInt("730750818665451621361119245571504901405976559617",10);
BigInt cof = new BigInt("12016012264891146079388821366740534204802954401251311822919615131047207289359704531102844802183906537786776",10);
/*------------------------------------------------BEGIN SETUP ---------------------------------------------------------*/
Point ecBasePoint = ec.getBasePoint(rnd, r, cof);
// scALAR MULITPLICATION
Point p1 = ec.multiply(ecBasePoint, BigInt.valueOf(2));
Point p2 = ec.multiply(ecBasePoint, BigInt.valueOf(3));
Point p3 = ec.add(p1, p2);
Point p4 = ec.multiply(ecBasePoint, BigInt.valueOf(5));
if(p4.equals(p3))
{
System.out.println("P3 equals P4");
}
else{System.out.println("P3 NOT equals P4");}
// pick two random elements in Zp
FieldElement f1 = field.randomElement(rnd);
FieldElement f2 = field.randomElement(rnd);
Point p5 = ec.multiply(ecBasePoint, (BigInt) f1);
Point p6 = ec.multiply(ecBasePoint, (BigInt) f2);
Point p7 = ec.add(p5, p6);
Point p8 = ec.multiply(ecBasePoint, (BigInt) field.add(f1, f2));
if(p8.equals(p7))
{
System.out.println("P7 equals P8");
}
else{System.out.println("P7 NOT equals P8");}
}
}
--------------新代码-----
import java.util.Random;
import uk.ac.ic.doc.jpair.api.FieldElement;
import uk.ac.ic.doc.jpair.pairing.BigInt;
import uk.ac.ic.doc.jpair.pairing.EllipticCurve;
import uk.ac.ic.doc.jpair.pairing.Fp;
import uk.ac.ic.doc.jpair.pairing.Point;
import uk.ac.ic.doc.jpair.ibeNew.Util;
/*
* ECC CPABE Scheme coding
*
* */
public class eccThird {
static final String charSet ="UTF-8";
public static void main(String [] args)
{
Random rnd = new Random();
//initialise the finite field Fp given a prime number p
BigInt p = new BigInt("8780710799663312522437781984754049815806883199414208211028653399266475630880222957078625179422662221423155858769582317459277713367317481324925129998224791",10);
Fp field = new Fp(p);
System.out.println("order" +field.getOrder());
//initialise the elliptic curve. The curve can be specified given the field and the coefficient a and b
//of the Weierstrass equation:
//y2 = x3 +ax +b
//For example, in the case of the supersingular curve //y2 = x3 + x, a = 1 b = 0 ,
EllipticCurve ec = new EllipticCurve (field,BigInt.ONE,BigInt.ZERO);
//You also need to know the group order r and the cofactor
BigInt r = new BigInt("730750818665451621361119245571504901405976559617",10);
BigInt cof = new BigInt("12016012264891146079388821366740534204802954401251311822919615131047207289359704531102844802183906537786776",10);
// attribute universe = U = {"attribute1","attribute2","attribute3","attribute4"}
// Obtaining hashes of attributes
String attribute1 ="Attribute1";
String attribute2 ="Attribute2";
String attribute3 ="Attribute3";
String attribute4 ="Attribute4";
byte[] bAttribute1 = null;
byte[] bAttribute2 = null;
byte[] bAttribute3 = null;
byte[] bAttribute4 = null;
try {
bAttribute1 = attribute1.getBytes(charSet);
bAttribute2 = attribute2.getBytes(charSet);
bAttribute3 = attribute3.getBytes(charSet);
bAttribute4 = attribute4.getBytes(charSet);
} catch (UnsupportedEncodingException e1) {
// TODO Auto-generated catch block
e1.printStackTrace();
}
System.out.println("Byte Attribute 1" +bAttribute1);
// Hashes attribute bytes to field elements
FieldElement attribute1H4 =Util.hashToField(bAttribute1, field);
FieldElement attribute2H4 =Util.hashToField(bAttribute2, field);
FieldElement attribute3H4 =Util.hashToField(bAttribute3, field);
FieldElement attribute4H4 =Util.hashToField(bAttribute4, field);
int numberAttributes = 5;
/*------------------------------------------------BEGIN SETUP ---------------------------------------------------------*/
Point ecBasePoint = ec.getBasePoint(rnd, r, cof);
// pick three random private keys alpa, k1, and k2 in Zp
FieldElement alphaSK = field.randomElement(rnd);
FieldElement k1SK = field.randomElement(rnd);
FieldElement k2SK = field.randomElement(rnd);
// Generate public key components P_i, U_i, and V_i for all attributes
// generate alpha^i components
FieldElement alphaSKZero = field.pow(alphaSK, BigInt.ZERO);
FieldElement alphaSKOne = field.pow(alphaSK, BigInt.ONE);
FieldElement alphaSKTwo = field.pow(alphaSK, BigInt.valueOf(2));
FieldElement alphaSKThree = field.pow(alphaSK, BigInt.valueOf(3));
FieldElement alphaSKFour = field.pow(alphaSK, BigInt.valueOf(4));
FieldElement alphaSKFive = field.pow(alphaSK, BigInt.valueOf(5));
// Generate P_i components
Point PiZero = ec.multiply(ecBasePoint, (BigInt) alphaSKZero);
Point PiOne = ec.multiply(ecBasePoint, (BigInt) alphaSKOne);
Point PiTwo = ec.multiply(ecBasePoint, (BigInt) alphaSKTwo);
Point PiThree = ec.multiply(ecBasePoint, (BigInt) alphaSKThree);
Point PiFour = ec.multiply(ecBasePoint, (BigInt) alphaSKFour);
Point PiFive = ec.multiply(ecBasePoint, alphaSKFive);
// Generate U_i components
Point UiZero = ec.multiply(PiZero, (BigInt) k1SK);
Point UiOne = ec.multiply(PiOne, (BigInt) k1SK);
Point UiTwo = ec.multiply(PiTwo, (BigInt) k1SK);
Point UiThree = ec.multiply(PiThree, (BigInt) k1SK);
Point UiFour = ec.multiply(PiFour, (BigInt) k1SK);
Point UiFive = ec.multiply(PiFive, (BigInt)k1SK);
// Generate V_i components
Point ViZero = ec.multiply(PiZero, (BigInt) k2SK);
Point ViOne = ec.multiply(PiOne, (BigInt) k2SK);
Point ViTwo = ec.multiply(PiTwo, (BigInt) k2SK);
Point ViThree = ec.multiply(PiThree, (BigInt) k2SK);
Point ViFour = ec.multiply(PiFour, (BigInt) k2SK);
Point ViFive = ec.multiply(PiFive, (BigInt) k2SK);
//System.out.println("Alpha alphaSKTwo = "+ PiZero);
/*------------------------------------------------ END SETUP ---------------------------------------------------------*/
/*------------------------------------------------ BEGIN KEYGEN ---------------------------------------------------------*/
// generating KeyGen Polynomial
FieldElement keyGenPoly = field.add(alphaSK, attribute3H4);
// two random numbers in Zp field
FieldElement ru = field.randomElement(rnd);
FieldElement tu = field.randomElement(rnd);
if(ru.equals(tu))
{
System.out.println("ru and tu are same");
}
else { System.out.println("ru and tu are not same"); }
//compute su
FieldElement suTemp1 = field.multiply(k2SK, ru);
FieldElement suTemp2 = field.subtract(field.inverse(keyGenPoly),suTemp1);
FieldElement su = field.multiply(field.inverse(k1SK),suTemp2);
//computing u1 and u2
FieldElement u1Temp = field.multiply(k1SK, tu);
FieldElement u1 = field.add(ru, u1Temp);
FieldElement u2Temp = field.multiply(k2SK, tu);
FieldElement u2 = field.subtract(su, u2Temp);
//System.out.println("KeyGen Polynomial "+ u1);
// checking if field.inverse(keyGenPoly) = k1su + k2ru
FieldElement test1 = field.add(field.multiply(k1SK, su), field.multiply(k2SK, ru));
if(test1.equals(field.inverse(keyGenPoly)))
{
/*System.out.println("Poly Inverse = " + field.inverse(keyGenPoly));
System.out.println("Condition = " + test1); */
System.out.println("KeyGen Condition Satisfied ");
}
/*------------------------------------------------ END KEYGEN ---------------------------------------------------------*/
}
}