Question

UDP and TCP use 1s complement for their checksums. Suppose you have the following three 8-bit bytes: \(01010011,01100110,01110100\). What is the \(1 \mathrm{~s}\) complement of the sum of these 8-bit bytes? (Note that although UDP and TCP use 16-bit words in computing the checksum, for this problem you are being asked to consider 8-bit sums.) Show all work. Why is it that UDP takes the Is complement of the sum; that is, why not just use the sum? With the 1 s complement scheme, how does the receiver detect errors? Is it possible that a 1-bit error will go undetected? How about a 2-bit error?

Short answer

The 1's complement of the sum is \(01011001\). UDP uses it to easily verify no errors occurred by checking for an all-1s result. 1-bit errors are detected; 2-bit errors could potentially go undetected if they cancel each other out.

Step-by-step solution

Convert to Binary and Add

First, convert all given 8-bit bytes into binary form. They are already in binary form: \(01010011, 01100110, 01110100\). Now add them together using binary addition.\[\begin{align*}01010011 & \+ 01100110 & \+ 01110100 & \\hline101001101 & \end{align*}\]However, since we are constrained to 8 bits, we only take the lower 8 bits of the result: \(10100101\). There's a carry from the addition, so we add that carry to the 8-bit sum in the next step.

Add the Carry

In the previous step, there was a carry-out from the addition, which means we need to add that carry (1) to the 8-bit result to maintain 8-bit consistency.\[10100101 + 1 = 10100110\]This result is the sum before taking the 1's complement.

Take 1's Complement

To find the 1's complement, invert each bit of the 8-bit result from step 2.\(10100110\) becomes \(01011001\). This is the 1's complement of the sum of the bytes.

Error Detection Mechanism

UDP and TCP use 1's complement for checksum computation to detect errors. It allows simple error detection by summing the data and its checksum. If the result is all 1s, no error is detected. If not, there is an error. Due to the nature of 1's complement arithmetic, any 1-bit error will change a bit and affect the sum, hence will not produce all 1s upon verify. Therefore, TCP/UDP will detect a 1-bit error.

1-bit vs 2-bit Errors

A single-bit error will indeed be detected because it will disturb the balance needed for an all-1s check. For two-bit errors, if both bits were flipped compensating each other (unlikely but possible), this error might go undetected. However, such a scenario is very rare. The purpose of using 1's complement and such a process provides a high level of reliability against single-bit errors and some multi-bit errors.

Key concepts

UDP Checksums

When data is transmitted over a network using the User Datagram Protocol (UDP), it's important to verify that the data received is exactly what was sent. UDP includes a checksum to help with this. A checksum is a small-sized datum derived from a block of digital data for the purpose of detecting errors. The checksum is calculated by considering the data as a series of binary numbers. This involves sum calculations through binary addition and logical operations like 1's complement. The primary role of the checksum in UDP is error detection. If any bit changes during data transmission, the checksum will not match, indicating an error. In UDP, if a checksum calculation results in a series of all 1s, it suggests that no error occurred during transmission. This mechanism helps ensure the integrity of packets being transferred across a network, though it's worth noting that UDP is still an unreliable protocol, as it does not offer error recovery.

TCP Protocol

The Transmission Control Protocol (TCP) is another key protocol in networking that also employs checksums for error detection. Unlike UDP, TCP establishes a connection before data is sent, ensuring that the receiving end acknowledges receipt of the transmitted data. This makes TCP a reliable protocol. TCP checksums are calculated similarly to UDP checksums, but in the context of a connection-oriented transmission. TCP checksums verify data integrity by calculating the 1's complement of the binary sum of the packet's data. When a packet is received, TCP calculates the checksum and compares it to the expected value. If any mismatch is detected, TCP can request retransmission, providing a higher level of error detection and correction capability. Moreover, TCP's reliable data transfer mechanism ensures ordered delivery of packets and manages data flow effectively, making it suitable for applications where reliable transmission is crucial.

1's Complement

The 1's complement is a method used to represent negative numbers in binary and to assist in error detection within networking protocols. To compute the 1's complement of a binary number, you simply flip all the bits; change 0s to 1s and 1s to 0s. In the context of checksum calculations, the 1's complement adds an extra layer of simplicity and error detection capability. The process involves:

• Summing all the data units using binary addition.
• Taking the 1's complement of the result.

The advantage of using 1's complement arithmetic is that it simplifies error checking. If every bit in the checksum and the data results in an all 1s bit pattern (after the summation), the data is likely error-free. This characteristic is pivotal in protocols like TCP and UDP as it provides a mechanism to verify that data has been during transmission.

Binary Addition

At the heart of checksumming techniques in networking is binary addition. This is a method of adding numbers using only binary digits: 0 and 1. In binary addition, numbers are added bit by bit from the rightmost bit to the leftmost, similar to how decimal addition works. When performing binary addition, a sum of 0 and 0 results in 0, 1 and 0 results in 1, 1 and 1 results in 0 with a carry of 1, and so forth. Carry bits are very important in binary addition, especially in the context of checksums. When the binary sum exceeds the max value of an 8 or 16-bit field, the carry is wrapped around and added to the least significant bit (rightmost). This wrapping, also known as end-around carry, maintains the bit-length restrictions and is essential in checksum calculations where error detection is required across fixed-size fields.

Checksum Calculation

Checksum calculation is a technique essential in error detection, forming a crucial part of both TCP and UDP protocols. The calculation begins by dividing the data into fixed-size segments, typically 16 bits for the sake of these protocols.

• Add the segments using binary addition.
• If a carry-over occurs, add this to the least significant bit to maintain the fixed bit-length.
• Compute the 1's complement by flipping all bits.

Once the checksum value is calculated, it is appended to the packet and sent over the network. Upon reception, the data is again summed together with this checksum. If the resulting value is all 1s, the transmission is assumed error-free. This process provides a sufficient error detection mechanism, particularly for single-bit errors, although more rare multi-bit errors might not be detected. Such calculations are employed to ensure data integrity over unreliable or less dependable networking conditions.