A Cyclic Redundancy Check (CRC) is an error-detecting code widely used in digital networks and storage devices to detect accidental changes to raw data. It involves appending a short, fixed-length binary sequence, known as the CRC or checksum, to a data block. This checksum is derived from the data using polynomial division. Upon retrieval or reception, the CRC is recalculated and compared to the transmitted checksum; if they match, the data is considered error-free; otherwise, an error is detected.
How CRC Works:
- Data Representation: The data to be transmitted is represented as a binary number.
- Polynomial Division: A predetermined divisor, represented as a binary polynomial, divides the data.
- Remainder Calculation: The remainder of this division is the CRC value.
- Transmission: The original data is transmitted along with the CRC value.
- Verification: The receiver performs the same division on the received data and compares the remainder with the transmitted CRC. If they match, the data is assumed to be error-free; otherwise, an error is detected.
Example 1:
Consider a simple example where the data to be transmitted is 1101011011, and the divisor (generator polynomial) is 1011 (which represents the polynomial x3+x+1x^3 + x + 1×3+x+1).
- Data Representation:
- Data:
1101011011 - Divisor:
1011
- Data:
- Polynomial Division:
- Append zeros to the data equal to the degree of the divisor minus one:
1101011011becomes1101011011000. - Perform binary division of
1101011011000by1011.
- Append zeros to the data equal to the degree of the divisor minus one:
- Remainder Calculation:
- The remainder after division is
101.
- The remainder after division is
- Transmission:
- Transmit the original data
1101011011along with the CRC101, resulting in1101011011101.
- Transmit the original data
- Verification:
- The receiver performs the same division on
1101011011101. - The remainder is
0, indicating no error.
- The receiver performs the same division on
Example 2:
Suppose we have the data 1011001 and the generator polynomial 1011 (which represents x3+x+1x^3 + x + 1×3+x+1).
- Data Representation:
- Data:
1011001 - Divisor:
1011
- Data:
- Polynomial Division:
- Append zeros to the data equal to the degree of the divisor minus one:
1011001becomes1011001000. - Perform binary division of
1011001000by1011.
- Append zeros to the data equal to the degree of the divisor minus one:
- Remainder Calculation:
- The remainder after division is
110.
- The remainder after division is
- Transmission:
- Transmit the original data
1011001along with the CRC110, resulting in1011001110.
- Transmit the original data
- Verification:
- The receiver performs the same division on
1011001110. - The remainder is
0, indicating no error.
- The receiver performs the same division on
Applications of CRC:
- Networking: Protocols like Ethernet and Wi-Fi use CRCs to detect errors in transmitted frames.
- Storage Devices: Hard drives and SSDs employ CRCs to ensure data integrity during read and write operations.
- File Formats: Compression formats like ZIP and image formats like PNG use CRCs to verify the integrity of stored data.
CRCs are favored for their simplicity and efficiency in detecting common transmission errors, such as single-bit errors, burst errors, and errors caused by noise in transmission channels.
Multiple-Choice Questions (MCQs):
- What is the primary purpose of a Cyclic Redundancy Check (CRC)?
- A) Data compression
- B) Error detection
- C) Data encryption
- D) Data transmission
- Answer: B) Error detection
- In CRC, what does the divisor represent?
- A) The data to be transmitted
- B) The generator polynomial
- C) The checksum
- D) The transmitted data block
- Answer: B) The generator polynomial
- Which of the following is a common application of CRC?
- A) Data encryption
- B) Error detection in network protocols
- C) Data compression
- D) Data visualization
- Answer: B) Error detection in network protocols
- What is appended to the data during CRC to form the transmitted codeword?
- A) Parity bits
- B) Checksum
- C) CRC value
- D) Data header
- Answer: C) CRC value
- Which of the following errors can CRC detect?
- A) Single-bit errors
- B) Burst errors
- C) Multiple-bit errors
- D) All of the above
- Answer: D) All of the above
- In CRC, what is the result of the division process called?
- A) Quotient
- B) Remainder
- C) Dividend
- D) Divisor
- Answer: B) Remainder
