使用RSA进行模块乘法的加密会导致Java卡出错

Question

你好，我正在做一个关于Java卡的项目，这意味着很多模乘。我用RSA密码系统在这个平台上实现了模乘，但它似乎适用于某些数字

public byte[] modMultiply(byte[] x, short xOffset, short xLength, byte[] y,
        short yOffset, short yLength, short tempOutoffset) {

    //copy x value to temporary rambuffer
    Util.arrayCopy(x, xOffset, tempBuffer, tempOutoffset, xLength);


    // copy the y value to match th size of rsa_object
    Util.arrayFillNonAtomic(eempromTempBuffer, (short)0, (byte) (Configuration.LENGTH_RSAOBJECT_MODULUS-1),(byte)0x00);
    Util.arrayCopy(y,yOffset,eempromTempBuffer,(short)(Configuration.LENGTH_RSAOBJECT_MODULUS - yLength),yLength);

    // x+y
    if (JBigInteger.add(x,xOffset,xLength, eempromTempBuffer,
            (short)0,Configuration.LENGTH_MODULUS)) ;
    if(this.isGreater(x, xOffset, xLength, tempBuffer,Configuration.TEMP_OFFSET_MODULUS, Configuration.LENGTH_MODULUS)>0)
    {
        JBigInteger.subtract(x,xOffset,xLength, tempBuffer,
                Configuration.TEMP_OFFSET_MODULUS, Configuration.LENGTH_MODULUS);
    }

    //(x+y)2
    mRsaCipherForSquaring.init(mRsaPublicKekForSquare, Cipher.MODE_ENCRYPT);

    mRsaCipherForSquaring.doFinal(x, xOffset, Configuration.LENGTH_RSAOBJECT_MODULUS, x,
            xOffset); // OK

    mRsaCipherForSquaring.doFinal(tempBuffer, tempOutoffset, Configuration.LENGTH_RSAOBJECT_MODULUS, tempBuffer, tempOutoffset); // OK


    if (JBigInteger.subtract(x, xOffset, Configuration.LENGTH_MODULUS, tempBuffer, tempOutoffset,
            Configuration.LENGTH_MODULUS)) {
        JBigInteger.add(x, xOffset, Configuration.LENGTH_MODULUS, tempBuffer,
                Configuration.TEMP_OFFSET_MODULUS, Configuration.LENGTH_MODULUS);
    } 

    mRsaCipherForSquaring.doFinal(eempromTempBuffer, yOffset, Configuration.LENGTH_RSAOBJECT_MODULUS, eempromTempBuffer, yOffset); //OK 


    if (JBigInteger.subtract(x, xOffset, Configuration.LENGTH_MODULUS, eempromTempBuffer, yOffset,
            Configuration.LENGTH_MODULUS)) {

        JBigInteger.add(x, xOffset, Configuration.LENGTH_MODULUS, tempBuffer,
                Configuration.TEMP_OFFSET_MODULUS, Configuration.LENGTH_MODULUS);

    }
    // ((x+y)^2 - x^2 -y^2)/2
    JBigInteger.modular_division_by_2(x, xOffset,Configuration. LENGTH_MODULUS, tempBuffer, Configuration.TEMP_OFFSET_MODULUS, Configuration.LENGTH_MODULUS);
    return x;
}


public static boolean add(byte[] x, short xOffset, short xLength, byte[] y,
        short yOffset, short yLength) {
    short digit_mask = 0xff;
    short digit_len = 0x08;
    short result = 0;
    short i = (short) (xLength + xOffset - 1);
    short j = (short) (yLength + yOffset - 1);

    for (; i >= xOffset; i--, j--) {
        result = (short) (result + (short) (x[i] & digit_mask) + (short) (y[j] & digit_mask));

        x[i] = (byte) (result & digit_mask);
        result = (short) ((result >> digit_len) & digit_mask);
    }
    while (result > 0 && i >= xOffset) {
        result = (short) (result + (short) (x[i] & digit_mask));
        x[i] = (byte) (result & digit_mask);
        result = (short) ((result >> digit_len) & digit_mask);
        i--;
    }

    return result != 0;
}
public static boolean subtract(byte[] x, short xOffset, short xLength, byte[] y,
        short yOffset, short yLength) {
    short digit_mask = 0xff;
    short i = (short) (xLength + xOffset - 1);
    short j = (short) (yLength + yOffset - 1);
    short carry = 0;
    short subtraction_result = 0;

    for (; i >= xOffset && j >= yOffset; i--, j--) {
        subtraction_result = (short) ((x[i] & digit_mask)
                - (y[j] & digit_mask) - carry);
        x[i] = (byte) (subtraction_result & digit_mask);
        carry = (short) (subtraction_result < 0 ? 1 : 0);
    }
    for (; i >= xOffset && carry > 0; i--) {
        if (x[i] != 0)
            carry = 0;
        x[i] -= 1;
    }

    return carry > 0;
}



 public short isGreater(byte[] x,short xOffset,short xLength,byte[] y ,short yOffset,short yLength)
    {
        if(xLength > yLength)
            return (short)1;
        if(xLength < yLength)
            return (short)(-1);
        short digit_mask = 0xff;
        short digit_len = 0x08;
        short result = 0;
        short i = (short) (xLength + xOffset - 1);
        short j = (short) (yLength + yOffset - 1);

        for (; i >= xOffset; i--, j--) {
            result = (short) (result + (short) (x[i] & digit_mask) - (short) (y[j] & digit_mask));
            if(result > 0)
                return (short)1;
            if(result < 0)
                return (short)-1;
        }
        return 0;
    }

该代码适用于较小的数字，但在较大的数字上失败

Answer 1

下面是一个非常简单的单元测试，其中包含（希望）代码的工作变体：

package test.java.so;

import java.math.BigInteger;
import java.util.Random;

import javacard.framework.JCSystem;
import javacard.framework.Util;
import javacard.security.KeyBuilder;
import javacard.security.RSAPublicKey;
import javacardx.crypto.Cipher;

import org.apache.commons.lang3.ArrayUtils;
import org.bouncycastle.util.Arrays;
import org.junit.Assert;
import org.junit.Test;

import sutil.test.AbstractTest;

public class So36966764_Test extends AbstractTest {

    private static final int NUM_BITS = 1024;

    // Dummy
    static class Configuration {
        public static final short LENGTH_MODULUS = NUM_BITS/8;
        public static final short LENGTH_RSAOBJECT_MODULUS = LENGTH_MODULUS;
        public static final short TEMP_OFFSET_MODULUS = 0;
        public static final short TEMP_OFFSET_RESULT = LENGTH_MODULUS;
    }

    private byte[] tempBuffer = JCSystem.makeTransientByteArray((short)(Configuration.TEMP_OFFSET_RESULT+Configuration.LENGTH_MODULUS), JCSystem.CLEAR_ON_DESELECT);
    private byte[] eempromTempBuffer = new byte[Configuration.LENGTH_MODULUS]; // Why EEPROM?
    private RSAPublicKey mRsaPublicKekForSquare = (RSAPublicKey)KeyBuilder.buildKey(KeyBuilder.TYPE_RSA_PUBLIC, (short)NUM_BITS, false);
    private Cipher mRsaCipherForSquaring = Cipher.getInstance(Cipher.ALG_RSA_NOPAD, false);

    // Assuming xLength==yLength==LENGTH_MODULUS
    public byte[] modMultiply(byte[] x, short xOffset, short xLength, byte[] y, short yOffset, short yLength, short tempOutoffset) {

        //copy x value to temporary rambuffer
        Util.arrayCopy(x, xOffset, tempBuffer, tempOutoffset, xLength);

        // copy the y value to match th size of rsa_object
        Util.arrayFillNonAtomic(eempromTempBuffer, (short)0, (short) (Configuration.LENGTH_RSAOBJECT_MODULUS-1),(byte)0x00);
        Util.arrayCopy(y,yOffset,eempromTempBuffer,(short)(Configuration.LENGTH_RSAOBJECT_MODULUS - yLength),yLength);

        // x+y
        if(add(x,xOffset,xLength, eempromTempBuffer, (short)0,Configuration.LENGTH_MODULUS)) {
            subtract(x,xOffset,xLength, tempBuffer, Configuration.TEMP_OFFSET_MODULUS, Configuration.LENGTH_MODULUS);
        }
        while(isGreater(x, xOffset, xLength, tempBuffer,Configuration.TEMP_OFFSET_MODULUS, Configuration.LENGTH_MODULUS)>0) {
            subtract(x,xOffset,xLength, tempBuffer,Configuration.TEMP_OFFSET_MODULUS, Configuration.LENGTH_MODULUS);
        }

        //(x+y)2
        mRsaCipherForSquaring.init(mRsaPublicKekForSquare, Cipher.MODE_ENCRYPT);
        mRsaCipherForSquaring.doFinal(x, xOffset, Configuration.LENGTH_RSAOBJECT_MODULUS, x, xOffset); // OK

        mRsaCipherForSquaring.doFinal(tempBuffer, tempOutoffset, Configuration.LENGTH_RSAOBJECT_MODULUS, tempBuffer, tempOutoffset); // OK

        if (subtract(x, xOffset, Configuration.LENGTH_MODULUS, tempBuffer, tempOutoffset,
                Configuration.LENGTH_MODULUS)) {
            add(x, xOffset, Configuration.LENGTH_MODULUS, tempBuffer,
                    Configuration.TEMP_OFFSET_MODULUS, Configuration.LENGTH_MODULUS);
        }

        /*WRONG OFFSET mRsaCipherForSquaring.doFinal(eempromTempBuffer, yOffset, Configuration.LENGTH_RSAOBJECT_MODULUS, eempromTempBuffer, yOffset); */
        mRsaCipherForSquaring.doFinal(eempromTempBuffer, (short)0, Configuration.LENGTH_RSAOBJECT_MODULUS, eempromTempBuffer, (short)0); //OK

        /*WRONG OFFSET if (subtract(x, xOffset, Configuration.LENGTH_MODULUS, eempromTempBuffer, yOffset,*/
        if (subtract(x, xOffset, Configuration.LENGTH_MODULUS, eempromTempBuffer, (short)0,Configuration.LENGTH_MODULUS)) {
            add(x, xOffset, Configuration.LENGTH_MODULUS, tempBuffer,
                    Configuration.TEMP_OFFSET_MODULUS, Configuration.LENGTH_MODULUS);
        }
        // ((x+y)^2 - x^2 -y^2)/2
        modular_division_by_2(x, xOffset,Configuration. LENGTH_MODULUS, tempBuffer, Configuration.TEMP_OFFSET_MODULUS, Configuration.LENGTH_MODULUS);
        return x;
    }

    public static boolean add(byte[] x, short xOffset, short xLength, byte[] y, short yOffset, short yLength) {
        short digit_mask = 0xff;
        short digit_len = 0x08;
        short result = 0;
        short i = (short) (xLength + xOffset - 1);
        short j = (short) (yLength + yOffset - 1);

        for (; i >= xOffset; i , j ) {
            result = (short) (result + (short) (x[i] & digit_mask) + (short) (y[j] & digit_mask));

            x[i] = (byte) (result & digit_mask);
            result = (short) ((result >> digit_len) & digit_mask);
        }
        while (result > 0 && i >= xOffset) {
            result = (short) (result + (short) (x[i] & digit_mask));
            x[i] = (byte) (result & digit_mask);
            result = (short) ((result >> digit_len) & digit_mask);
            i ;
        }

        return result != 0;
    }

    public static boolean subtract(byte[] x, short xOffset, short xLength, byte[] y, short yOffset, short yLength) {
        short digit_mask = 0xff;
        short i = (short) (xLength + xOffset - 1);
        short j = (short) (yLength + yOffset - 1);
        short carry = 0;
        short subtraction_result = 0;

        for (; i >= xOffset && j >= yOffset; i , j ) {
            subtraction_result = (short) ((x[i] & digit_mask)
                    - (y[j] & digit_mask) - carry);
            x[i] = (byte) (subtraction_result & digit_mask);
            carry = (short) (subtraction_result < 0 ? 1 : 0);
        }
        for (; i >= xOffset && carry > 0; i ) {
            if (x[i] != 0)
                carry = 0;
            x[i] -= 1;
        }

        return carry > 0;
    }

    public short isGreater(byte[] x,short xOffset,short xLength,byte[] y ,short yOffset,short yLength)
    {
        // Beware: this part is not tested
        while(xLength>yLength) {
            if(x[xOffset++]!=0x00) {
                return 1; // x is greater
            }
            xLength ;
        }
        while(yLength>xLength) {
            if(y[yOffset++]!=0x00) {
                return -1; // y is greater
            }
            yLength ;
        }
        // Beware: this part is not tested END
        for(short i = 0; i < xLength; i++) {
            if (x[xOffset] != y[yOffset]) {
                short srcShort = (short)(x[xOffset]&(short)0xFF);
                short dstShort = (short)(y[yOffset]&(short)0xFF);
                return ( ((srcShort > dstShort) ? (byte)1 : (byte)-1));
            }
            xOffset++;
            yOffset++;
        }
        return 0;
    }

    private void modular_division_by_2(byte[] input, short inOffset, short inLength, byte[] modulus, short modulusOffset, short modulusLength) {
        short carry = 0;
        short digit_mask = 0xff;
        short digit_first_bit_mask = 0x80;
        short lastIndex = (short) (inOffset + inLength - 1);

        short i = inOffset;
        if ((byte) (input[lastIndex] & 0x01) != 0) {
            if (add(input, inOffset, inLength, modulus, modulusOffset,
                    modulusLength)) {
                carry = digit_first_bit_mask;
            }
        }

        for (; i <= lastIndex; i++) {
            if ((input[i] & 0x01) == 0) {
                input[i] = (byte) (((input[i] & digit_mask) >> 1) | carry);
                carry = 0;
            } else {
                input[i] = (byte) (((input[i] & digit_mask) >> 1) | carry);
                carry = digit_first_bit_mask;
            }
        }
    }

    @Test
    public void testModMultiply() {
        Random r = new Random(12345L);
        for(int iiii=0;iiii<10;iiii++) {
            BigInteger modulus = BigInteger.probablePrime(NUM_BITS, r);
            System.out.println(" M = " + modulus);
            byte[] modulusBytes = normalize(modulus.toByteArray());
            Util.arrayCopyNonAtomic(modulusBytes, (short)0, tempBuffer, Configuration.TEMP_OFFSET_MODULUS, Configuration.LENGTH_MODULUS);

            mRsaPublicKekForSquare.setModulus(modulusBytes, (short)0, (short)modulusBytes.length);
            mRsaPublicKekForSquare.setExponent(new byte[] {0x02}, (short)0, (short)1);

            for(int iii=0;iii<1000;iii++) {
                BigInteger x = new BigInteger(NUM_BITS, r).mod(modulus);
                System.out.println(" x = " + x);
                BigInteger y = new BigInteger(NUM_BITS, r).mod(modulus);
                System.out.println(" y = " + y);
                BigInteger accResult;
                {
                    byte[] xBytes = normalize(x.toByteArray());
                    byte[] yBytes = normalize(y.toByteArray());
                    byte[] accResultBytes = modMultiply(xBytes, (short)0, (short)xBytes.length, yBytes, (short)0, (short)yBytes.length, Configuration.TEMP_OFFSET_RESULT);
                    accResult = new BigInteger(1, accResultBytes);
                }
                System.out.println(" Qr = " + accResult);
                BigInteger realResult = x.multiply(y).mod(modulus);
                System.out.println(" Rr = " + realResult);
                Assert.assertEquals(realResult, accResult);
            }
        }
    }

    private byte[] normalize(byte[] xBytes) {
        if(xBytes.length<Configuration.LENGTH_MODULUS) {
            xBytes = ArrayUtils.addAll(new byte[Configuration.LENGTH_MODULUS-xBytes.length], xBytes);
        }
        if(xBytes.length>Configuration.LENGTH_MODULUS) {
            Assert.assertEquals(xBytes[0], 0x00);
            xBytes=Arrays.copyOfRange(xBytes, 1, xBytes.length);
        }
        return xBytes;
    }
}

出了什么问题：

1. isGreater()方法虽然可以使用减法来比较数字，但比较从最重要的字节开始并在第一次不匹配时停止的相应字节要容易得多（更快）。（在减法的情况下，您需要完成减法并返回最终结果的符号，您的原始代码以第一个“不匹配”结尾。）

2. x+y溢出在上一次编辑中，您应该保留加法溢出情况下的模减法（即当add()返回true时）

3. 在两个地方将偏移量转换为eempromTempBuffer，您使用了yOffset并且应该使用0（用“错误偏移量”注释掉）

4. 将Configuration.LENGTH_RSAOBJECT_MODULUS-1铸造到byte对于较大的模数长度值来说不是一个好主意

一些（随机）评论：

• 测试使用已经提到的jcardsim来工作

• 代码假定x和y的长度都是LENGTH_MODULUS（以及LENGTH_RSAOBJECT_MODULUS等于LENGTH_MODULUS）

• 把eempromTempBuffer放在非易失性存储器中不是个好主意

• 您的代码与this code非常相似，这很有趣

• 关于这个主题的有趣读物是here（第4.2.3节）

祝你好运

免责声明：我不是密码专家，也不是数学家，所以请验证我的想法