Question

Suppose that a message 1001110010100011 is transmitted using the Internet Checksum (4-bit word). What is the value of the checksum?

Short answer

The 4-bit checksum is 1101.

Step-by-step solution

Divide the Message into 4-bit Words

The given message is 1001110010100011. We need to divide this 16-bit message into four parts, each containing 4 bits: 1. 1001 2. 1100 3. 1010 4. 0011

Convert Binary Words to Decimal

Now convert each 4-bit binary word into decimal form: 1. 1001 in binary is 9 in decimal. 2. 1100 in binary is 12 in decimal. 3. 1010 in binary is 10 in decimal. 4. 0011 in binary is 3 in decimal.

Sum the Decimal Values

Add all the decimal values obtained from Step 2: 9 + 12 + 10 + 3 = 34.

Apply Modulo Operation with 16

Use modulo 16 to wrap the sum within the 4-bit range, since we're working with 4-bit words: 34 mod 16 = 2. Therefore, the carry value is 2 in this context.

Calculate the Checksum

Subtract the carry value from 15 (since a 4-bit word can hold up to a maximum value of 15): 15 - 2 = 13. Now, convert 13 back into binary format for a 4-bit checksum, which gives us 1101.

Key concepts

Binary to Decimal Conversion

Binary to decimal conversion is a fundamental process in many computing applications. It involves translating a number from its binary format (base 2) to its decimal equivalent (base 10). This is crucial when interpreting digital data, as computers inherently operate using binary data.
To convert a binary number to decimal, each digit is multiplied by two raised to the power of its position, starting from the right (0th position). For example, the binary number 1001 is converted as follows:

• The rightmost digit (1) is in the 0th position: \(1 \times 2^0 = 1\).
• The next digit (0) in the 1st position: \(0 \times 2^1 = 0\).
• Then, the digit (0) in the 2nd position: \(0 \times 2^2 = 0\).
• Finally, the leftmost digit (1) in the 3rd position: \(1 \times 2^3 = 8\).

Adding these values together, \(8 + 0 + 0 + 1 = 9\), we find that 1001 in binary is equivalent to 9 in decimal. This step is essential for further processing and analyses in tasks like calculating internet checksums.

Modulo Operation

The modulo operation is an arithmetic operation that finds the remainder after division of one number by another. Symbolically, the modulo operation is represented by the mod symbol and is crucial in many computing contexts, including when working with data representations like checksums.
In the context of the Internet Checksum, the modulo operation ensures that the sum of binary words wraps within the bounds of a specified range. For a 4-bit word, this range is 0 to 15. For example, if the sum of decimal values from binary conversions exceeds this range, the modulo operation helps wrap this sum to fit within the 4-bit limit.
Consider the sum 34 from the exercise, calculated from a series of binary words. To normalize this sum to fit within the 4-bit word range, we use the modulo operation:

• 34 mod 16 yields a remainder of 2.

This number, 2, represents the overflow value and play a role in determining the final checksum by showing how much the original sum surpasses the maximum value that can be held by a 4-bit word.

4-bit Words

4-bit words are a compact data representation, commonly used in systems that require minimal data granularity to ensure efficiency and conserve resources. Each word can represent a decimal value ranging from 0 to 15, accommodating up to 16 different states.
In the context of the Internet Checksum, messages are divided into 4-bit words for processing. For example, the 16-bit message 1001110010100011 from the exercise was subdivided into four distinct 4-bit words:

• 1001
• 1100
• 1010
• 0011

Dividing data into 4-bit segments simplifies operations like checksums and error checking, as you can manage overflow efficiently and minimize data handling complexity. Processing is simpler and faster with smaller units, making such a division a logical choice in building reliable and resource-conservative digital communication systems.