The Cyclic Redundancy Check article at Wikipedia says the polynomial is x^4 + x + 1. There is also a pretty good description of how the checksum is computed.
Here is an algorithm for CRC16. I know it's not what you asked for, but it should be relatively straightforward to adapt it for 4 bits.
public ushort calculate(byte[] bytes)
{
int crc = 0xFFFF; // initial value
// loop, calculating CRC for each byte of the string
for (int byteIndex = 0; byteIndex < bytes.Length; byteIndex++)
{
ushort bit = 0x80; // initialize bit currently being tested
for (int bitIndex = 0; bitIndex < 8; bitIndex++)
{
bool xorFlag = ((crc & 0x8000) == 0x8000);
crc <<= 1;
if (((bytes[byteIndex] & bit) ^ (ushort)0xff) != (ushort)0xff)
{
crc = crc + 1;
}
if (xorFlag)
{
crc = crc ^ 0x1021;
}
bit >>= 1;
}
}
return (ushort)crc;
}
http://www.experts-exchange.com/Programming/Languages/C_Sharp/Q_24775723.html
Also, there is this guide to computing checksums:
http://www.ross.net/crc/download/crc_v3.txt
"Everything you wanted to know about CRC algorithms, but were afraid
to ask for fear that errors in your understanding might be detected."