field arithmetic is implemented in terms of the arithmetic of integers modulo p.
- A binary field is the field GF(2m) which contains 2m elements for some m (called the degree of the field). The elements of this field are the bit strings of length m, and the field arithmetic is implemented in terms of operations on the bits.
The following table gives the sizes of the various underlying fields. By ||p|| is meant the length of the binary expansion of the integer p.
| Symmetric | Example | ||
| CV Length | Algorithm | Prime Field | Binary Field |
| 80 | SKIPJACK | ||p|| = 192 | m = 163 |
| 112 | Triple-DES | ||p|| = 224 | m = 233 |
| 128 | AES Small | ||p|| = 256 | m = 283 |
| 192 | AES Medium | ||p|| = 384 | m = 409 |
| 256 | AES Large | ||p|| = 521 | m = 571 |
1.3 Choice of Basis
To describe the arithmetic of a binary field, it is first necessary to specify how a bit string is to be interpreted. This is referred to as choosing a basis for the field. There are two common types of bases: a polynomial basis and a normal basis.
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