- Generator polynomial:
- Generator polynomial = the divisor polynomial in the polynomial division operation
- Obvious fact:
- The sender and the receiver must use the same generator polynomial to encode/decode the messages.
- How the
sender
transmits a message:
Explanation:
- The sender
divides the
message by
the generator polynomial
- The sender includes the remainder of the division in the transmitted message
- The sender
divides the
message by
the generator polynomial
- How the
receiver
checks a received message
for correctness:
Explanation:
- The receiver
divides the
message + CRC code (remainder)
by the
same generator polynomial
- The receiver will assume there is no bit error(s) if the remainder of the division is equal to 0
- The receiver
divides the
message + CRC code (remainder)
by the
same generator polynomial
- The CRC encoding procedure:
- Let:
N = length of the generator polynomial = number of bits in the generator polynomial
- Append
N−1
ZERO (0) bits
to the end
of the input message.
- Divide
the messages + (N-1) ZEROs
by the generator polynomial.
Note:
- The division operation will use the XOR operation as the subtract operation
-
Append
the remainder
of the "division"
to the input message
- The resulting message is the CRC protected message
- Let:
- Example: computing the CRC code
(= check sum) for
a message
Generator polynomial: x3 + x2 + 1 Genenrator polnomial in bit string: 1101 (1101 represents the polynomial: 1×x3 + 1×x2 + 0×x1 + 1×x0)
N = 4 (because Generator polynomial has 4 terms, or bits in "1001")
Input message: x4 + 1 Input message in bit string: 10001
Computing the CRC (check sum) for the message 10001 using Gen Polyn 1101: 1. Add N-1 (= 3) ZERO's bits to message: Message + 3 ZEROs = 10001000 2. Compute CRC by dividing "message + 3 zeros" by generator polynomial: 00011 ---------------- 1101 / 10001000 1101 <----- **** see Note below !! **** ---- (XOR) 1011 1101 ---- (XOR) 1100 1101 ---- (XOR) 000100 <---- remainder = 100CRC = 100
- Note:
how to
perform the
"division"
- Whenever the
leading bit of the
divident =
1, you get a
1 in the
quotient.
Example:
1 ------------ 1101 / 10001000 // Even when 1000 < 1101 !!! 1101 ----- XOR 0101
- The division uses the
XOR operation as
subtraction
- Whenever the
leading bit of the
divident =
1, you get a
1 in the
quotient.
- Example:
CRC generator polynomial: 11011 (N = 5) Message: 11100110
1. Add 4 0-bits to the message: 11100110 --> 111001100000 2. Divide: 10101110 --------------- 11011 / 111001100000 11011||||||| ------v|||||| 1111|||||| 0000|||||| ------v||||| 11111||||| 11011||||| ------v|||| 1000|||| 0000|||| -----v||| 10000||| 11011||| ------v|| 10110|| 11011|| ------v| 11010| 11011| ------v 00010 00000 ----- 0010 3. CRC protected message = 111001100010
- Recall:
how is the
CRC code
computed
------------------------------ CRC-polynomial / Original-Message + 000..000 .... ----------------------------- CRC-code
- A CRC protected message consists of:
Original-Message + CRC-code
- Property of
every
CRC protect message:
- The remainder of
a CRC protected message
divided by
the CRC polynomial
(used in the encoding) is:
- Always equal to ZERO !!!
In formula:
(Original-Message + CRC-code) % CRC-polynomial == 0 !!!
Reason:
Since: ------------------------------ CRC-polynomial / Original-Message + 000..000 .... ----------------------------- CRC-code We will have: ------------------------------ CRC-polynomial / Original-Message + CRC-code .... |||||||| VVVVVVVV ----------------------------- CRC-code XOR CRC-code ----------------- 000..000(The dvision will now bring down the value "CRC-code" which will be subtracted from the remaining value "CRC-code" --- therefore producting the final remainder 0)
- The remainder of
a CRC protected message
divided by
the CRC polynomial
(used in the encoding) is:
- Example:
- Previously, we have
computed the
CRC code for
a message
CRC generator polynomial: 11011 (N = 5) Message: 11100110 CRC code = 0010 CRC protected message = 111001100010
- CRC check:
Divide: Received message 111001100010 by CRC polynomial 11011
(don't care about the quotient) ---------------- 11011 / 111001100010 11011 ------ 11111 11011 ---------- 10000 11011 --------- 10110 11011 ------- 11011 11011 ------ 000000 <--- Remainder = 0 !!
- Previously, we have
computed the
CRC code for
a message