Little/Big Endian

Little Endian and Big Endian are two byte-ordering conventions used to represent multi-byte integers in memory or during data transmission.

Consider an integer encoded on $ n $ bytes. In Big Endian, the most significant byte (MSB) is stored at the lowest memory address. In Little Endian, the least significant byte (LSB) is stored at the lowest memory address.

It is important to note that the order of bits within a byte does not change, only the position of the bytes is modified.

To encode an integer in Little Endian:

Write the integer in base $ 256 $, i.e. as a sum of bytes: $$ N = \sum_{i=0}^{n-1} b_i \times 256^i $$

Each $ b_i $ corresponds to a byte with $ 0 \leq b_i < 256 $.

Write the bytes in increasing order of indices $ i $ (from least significant to most significant).

To encode an integer in Big Endian:

Write the integer in base $ 256 $, i.e. as a sum of bytes: $ N = \sum_{i=0}^{n-1} b_i \times 256^i $

Each $ b_i $ corresponds to a byte with $ 0 \leq b_i < 256 $.

Write the bytes in decreasing order of indices $ i $ (from most significant to least significant).

To decode a sequence of bytes in Little Endian:

Note the bytes $ b_0, b_1, \dots, b_{n-1} $ in reading order (from least significant to most significant).

Compute: $ N = \sum_{i=0}^{n-1} b_i \times 256^i $

To decode a sequence of bytes in Big Endian:

Note the bytes $ b_0, b_1, \dots, b_{n-1} $ in reading order (from most significant to least significant).

Compute: $ N = \sum_{i=0}^{n-1} b_i \times 256^{n-1-i} $

The existence of these two conventions comes from historical and technical architectural choices.

Some hardware architectures use Little Endian (such as x86), while others support Big Endian or both (some ARM architectures are bi-endian).

Little Endian simplifies certain arithmetic operations on variable-size integers, because the least significant byte is directly accessible at the base address, which facilitates carry propagation.

Big Endian more closely matches the usual positional representation of numbers (most significant digits first), which can make human inspection and some lexicographic comparisons easier.

In networking, a single convention called 'network byte order' (Big Endian) is used to ensure interoperability between heterogeneous systems.