問(wèn)題描述

您好，我正在研究 Java Card 上的一個(gè)項(xiàng)目，這意味著很多模乘.我設(shè)法在這個(gè)平臺(tái)上使用 RSA 密碼系統(tǒng)實(shí)現(xiàn)了模乘，但它似乎適用于某些數(shù)字.

Hello I'm working on a project on Java Card which implies a lot of modulo-multiplication. I managed to implement an modulo-multiplication on this platform using RSA cryptosystem but it seems to work for certain numbers.

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;
    }

該代碼適用于小數(shù)字，但對(duì)于較大的數(shù)字則失敗

The code works well for little number but fails on bigger one

推薦答案

我設(shè)法通過(guò)改變乘法的數(shù)學(xué)公式解決了這個(gè)問(wèn)題.我在更新的代碼下面發(fā)布了.

I managed to solve the problem by changing the mathematical formula of multiplication.I posted below the updated code.

private byte[] multiply(byte[] x, short xOffset, short xLength, byte[] y,
        short yOffset, short yLength,short tempOutoffset)
{
    normalize();
    //copy x value to temporary rambuffer
    Util.arrayFillNonAtomic(tempBuffer, tempOutoffset,(short) (Configuration.LENGTH_RSAOBJECT_MODULUS+tempOutoffset),(byte)0x00);
    Util.arrayCopy(x, xOffset, tempBuffer, (short)(Configuration.LENGTH_RSAOBJECT_MODULUS - xLength), xLength);

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

    Util.arrayFillNonAtomic(ram_y_prime, IConsts.OFFSET_START, (short) (Configuration.LENGTH_RSAOBJECT_MODULUS-1),(byte)0x00);
    Util.arrayCopy(y,yOffset,ram_y_prime,(short)(Configuration.LENGTH_RSAOBJECT_MODULUS - yLength),yLength);

    Util.arrayFillNonAtomic(ram_x, IConsts.OFFSET_START, (short) (Configuration.LENGTH_RSAOBJECT_MODULUS-1),(byte)0x00);
    Util.arrayCopy(x,xOffset,ram_x,(short)(Configuration.LENGTH_RSAOBJECT_MODULUS - xLength),xLength);

    // if x>y
    if(this.isGreater(ram_x, IConsts.OFFSET_START, Configuration.LENGTH_RSAOBJECT_MODULUS, ram_y,IConsts.OFFSET_START, Configuration.LENGTH_MODULUS)>0)
    {

        // x <- x-y
        JBigInteger.subtract(ram_x,IConsts.OFFSET_START,Configuration.LENGTH_RSAOBJECT_MODULUS, ram_y,
                IConsts.OFFSET_START, Configuration.LENGTH_RSAOBJECT_MODULUS);
    }
    else
    {

        // y <- y-x
        JBigInteger.subtract(ram_y_prime,IConsts.OFFSET_START,Configuration.LENGTH_RSAOBJECT_MODULUS, ram_x,
                IConsts.OFFSET_START, Configuration.LENGTH_MODULUS);
         // ramy stores the (y-x) values copy value to ram_x
        Util.arrayCopy(ram_y_prime, IConsts.OFFSET_START,ram_x,IConsts.OFFSET_START,Configuration.LENGTH_RSAOBJECT_MODULUS);

    }

        //|x-y|2
        mRsaCipherForSquaring.init(mRsaPublicKekForSquare, Cipher.MODE_ENCRYPT);
        mRsaCipherForSquaring.doFinal(ram_x, IConsts.OFFSET_START, Configuration.LENGTH_RSAOBJECT_MODULUS, ram_x,
                IConsts.OFFSET_START); // OK

        // x^2
        mRsaCipherForSquaring.doFinal(tempBuffer, tempOutoffset, Configuration.LENGTH_RSAOBJECT_MODULUS, tempBuffer, tempOutoffset); // OK

        // y^2
        mRsaCipherForSquaring.doFinal(ram_y,IConsts.OFFSET_START, Configuration.LENGTH_RSAOBJECT_MODULUS, ram_y,IConsts.OFFSET_START); //OK 



        if (JBigInteger.add(ram_y, IConsts.OFFSET_START, Configuration.LENGTH_MODULUS, tempBuffer, tempOutoffset,
                Configuration.LENGTH_MODULUS)) {
              // y^2 + x^2 
            JBigInteger.subtract(ram_y, IConsts.OFFSET_START, Configuration.LENGTH_MODULUS, tempBuffer,
                    Configuration.TEMP_OFFSET_MODULUS, Configuration.LENGTH_MODULUS);
        } 


        //  x^2 + y^2
        if (JBigInteger.subtract(ram_y, IConsts.OFFSET_START, Configuration.LENGTH_MODULUS, ram_x, IConsts.OFFSET_START,
                Configuration.LENGTH_MODULUS)) {

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

問(wèn)題在于，對(duì)于相同數(shù)字 a 和 b 在 1024 位上的某些數(shù)字，總和 a+b 克服了值 p 的模數(shù).在上面的代碼中，我從 a+b 中減去 p 值以使 RSA 正常工作.但這不是數(shù)學(xué)上的東西正確，因?yàn)?(a+b)^2 mod p 與 ((a+b) mod p)^2 mod p 不同.通過(guò)將公式從 ((x+y)^2 -x^2 -y^2)/2 更改為 (x^2 + y^2 - (xy)^2)/2 我確信我永遠(yuǎn)不會(huì)溢出，因?yàn)?ab 小于 p.基于 上面的鏈接 我改變了將所有操作移至 RAM 中的代碼.

The problem was that for some numbers for same numbers a and b on 1024 bits the sum a+b overcome the value p of the modulus.In above code I subtract from a+b the p value in order to make the RSA functioning.But this thing is not mathematically correct because (a+b)^2 mod p is different from ((a+b) mod p)^2 mod p . By changing the formula from ((x+y)^2 -x^2 -y^2)/2 to (x^2 + y^2 - (x-y)^2)/2 I was sure I will never have overflow because a-b is smaller than p. Based on link above I changed the code moving all the operations in RAM.

這篇關(guān)于使用 RSA 進(jìn)行模乘會(huì)導(dǎo)致 Java Card 出錯(cuò)的文章就介紹到這了，希望我們推薦的答案對(duì)大家有所幫助，也希望大家多多支持html5模板網(wǎng)！