#CJam, 14 13 bytes

q~2bf*{\2*^}*

How it works:

We first get the long multiplication results and then work our way up starting from the bottom two pairs.

q~                e# Eval the input. This puts the two numbers on stack
  2b              e# Convert the second number to binary
    f*            e# Multiply each bit of second number with the first number
                  e# This leaves an array with the candidates to be added in the long
                  e# multiplication step
      {    }*     e# Reduce on these candidates. Starting from the bottom
       \2*        e# Bit shift the lower candidate
          ^       e# XOR each other and continue

Try it online here

CJam, 14 13 bytes

q~2bf*{\2*^}*

How it works:

We first get the long multiplication results and then work our way up starting from the bottom two pairs.

q~                e# Eval the input. This puts the two numbers on stack
  2b              e# Convert the second number to binary
    f*            e# Multiply each bit of second number with the first number
                  e# This leaves an array with the candidates to be added in the long
                  e# multiplication step
      {    }*     e# Reduce on these candidates. Starting from the bottom
       \2*        e# Bit shift the lower candidate
          ^       e# XOR each other and continue

Try it online here

#CJam, 14 13 bytes

q~2bf*{\2*^}*

Try it online here

#CJam, 14 13 bytes

q~2bf*{\2*^}*

How it works:

We first get the long multiplication results and then work our way up starting from the bottom two pairs.

q~                e# Eval the input. This puts the two numbers on stack
  2b              e# Convert the second number to binary
    f*            e# Multiply each bit of second number with the first number
                  e# This leaves an array with the candidates to be added in the long
                  e# multiplication step
      {    }*     e# Reduce on these candidates. Starting from the bottom
       \2*        e# Bit shift the lower candidate
          ^       e# XOR each other and continue

Try it online here

#CJam, 14 13 bytes

I know Martin has 13 byte solution, but here goes a 14 one:

q~2bf*{\2*^}*

Try it online here

#CJam, 14 13 bytes

q~2bf*{\2*^}*

Try it online here