Represent a finite field with characteristic and extension degree :

Specify an element of the field using polynomial coefficients:

Equivalent specification:

Specify an element of the field using an index:

Equivalent specification:

Do arithmetic:

Do polynomial algebra:

Finite field elements are atomic objects:

Extract properties of a finite field element:

Implement an error-correcting code. The Hamming code encodes a -bit message in an -bit sequence and is able to correct up to one error:

Let be a finite field with elements using the exponential element representation, let be the irreducible polynomial used to construct , and let be the generator of :

The encoded message is the coefficient list of , where the coefficient list of is the original message:

Let be the polynomial whose coefficient list is the received message:

If the received message contains no errors, then , and hence :

If the received message contains one error in position , then , and hence :

Check and correct the received message:

To decode the message, compute the coefficient list of :

The decoded message is correct when the received message has no errors or one error:

Construct orthogonal Latin squares of order for any prime power . A Latin square of order is a array such that each row and each column contains every element of a set of elements exactly once. A pair of Latin squares is said to be orthogonal if the pairs formed by juxtaposing the two arrays are all distinct:

Verify that all arrays are Latin squares:

Verify that all pairs of arrays are orthogonal:

A finite set of integers is a Sidon set if the sums for are all distinct. Construct a Sidon set of integers in , for a prime power :

Verify that is a Sidon set of length :

A de Bruijn sequence of order for an alphabet with letters is a cyclic sequence of letters of the alphabet, such that every sequence of letters appears exactly once as a subsequence of . Construct a de Bruijn sequence of order for an alphabet with letters, for a prime power :

Verify that is a de Bruijn sequence of order for an alphabet with letters:

An matrix is a Hadamard matrix if all entries of are or and H.TemplateBox[{H}, Transpose]=n I_n. Construct a Hadamard matrix of order for any prime power with TemplateBox[{{q, =, 3}, 4}, Mod]:

Implement the Rijndael S-box step used in the Advanced Encryption Standard (AES) algorithm. The first part, called a Nyberg S-box, uses multiplicative inverse in :

The second part involves an affine transformation over :

The forward S-box is the composition of the two parts:

Compute the forward S-box table in the hexadecimal notation:

Define the inverse S-box transformation:

Compute the inverse S-box table in the hexadecimal notation:

Verify that the inverse S-box is the inverse of the forward S-box:

Implement a Diffie–Hellman public key cryptosystem with a 2049-bit prime:

Find a primitive element of the field :

The first user chooses a private key :

The public key consists of , and :

The second user chooses :

To send a 2048-bit message , the second user sends and :

The first user can recover by computing :

Implement a digital signature scheme. Fix a prime and find a primitive element of :

Pick a secret integer and publish , and :

The signature for a message is a pair of positive integers less than such that . Computing the signature requires the knowledge of the secret integer :

The signature can be verified using the publicly known information:

Compute the signature for a randomly generated message:

Verify the signature:

A finite field with characteristic and extension degree has elements:

Elements of a finite field with characteristic satisfy :

Hence the mapping is a field automorphism, known as FrobeniusAutomorphism :

The field generator is a root of the field irreducible:

Use FrobeniusAutomorphism to find the remaining roots of :

Use MinimalPolynomial to find the minimal polynomial of a finite field element:

The minimal polynomial over :

The minimal polynomial over the subfield of with elements:

All elements of a finite field with elements are roots of :

In fact, :

Any irreducible polynomial of degree over has roots in a field with elements:

Use IrreduciblePolynomialQ with Modulus p to verify irreducibility over :

Use Factor with Extension ℱ to verify that f is a product of linear factors over :

Use MultiplicativeOrder to find the multiplicative order of finite field elements:

Use FiniteField [p,1] to compute over the prime field :

Compare with a result obtained using Mod :

Polynomial computation over :

Compare with a result obtained using the Modulus option:

Use FromFiniteFieldIndex to get field elements with specified indices:

FiniteFieldIndex gives indices of field elements: