Question

As homework, I’m implementing Karatsuba’s algorithm and benchmarking it against a primary-school-style O(n^2) multiplication algorithm on large integers.

I guessed my only choice here was to bring the numbers to their byte array representations and then work them from there.

Well, I’m stuck here… when using the * operator, I don’t know how would I detect/correct if the number overflows a byte multiplication or adds a carry. Any ideas?

public static BigInteger simpleMultiply(BigInteger x, BigInteger y){

        //BigInteger result = x.multiply(y);

        byte [] xByteArray = x.toByteArray();
        byte [] yByteArray = y.toByteArray();

        int resultSize = xByteArray.length*yByteArray.length;

        byte [][] rowsAndColumns = new byte[resultSize][resultSize];

        for (int i =0; i<xByteArray.length;i++)
           for (int j=0; j<yByteArray.length;j++){


               rowsAndColumns[i][j] = (byte )(xByteArray[i] * yByteArray[j]); 
               // how would I detect/handle carry or overflow here?               
           }

        return null;
    }

Answer 1

The result of a byte multiplication is 2 bytes. You have to use the low order byte as the result and the high order byte as the carry (overflow).

I would also advise you to be careful of the sign of your bytes. Since bytes in Java are signed, you’ll have to either use only the low 7 bits of them or convert them to ints and correct the sign before multiplying them.

You’ll want a loop like:

        for (int i =0; i<xByteArray.length;i++)
           for (int j=0; j<yByteArray.length;j++){
               // convert bytes to ints
               int xDigit = xByteArray[i], yDigit = yByteArray[j];
               // convert signed to unsigned
               if (xDigit < 0)
                   xDigit += 256;
               if (yDigit < 0)
                   yDigit += 256;
               // compute result of multiplication
               int result = xDigit * yDigit;
               // capture low order byte
               rowsAndColumns[i][j] = (byte)(result & 0xFF);
               // get overflow (high order byte)
               int overflow = result >> 8;
               // handle overflow here
               // ...
           }