What is group ring and field in cryptography?

A RING is a set equipped with two operations, called addition and multiplication. A RING is a GROUP under addition and satisfies some of the properties of a group for multiplication. A FIELD is a GROUP under both addition and multiplication.

Why do we care about groups in cryptography?

Hence we are using groups here (or rather some practical proofs from group theory) because they are the correct tool, they make it easy to reason about the algorithm and prove the correctness, and very easy to implement.

What is ring cryptography?

Cryptographic systems are derived using units in group rings. Combinations of types of units in group rings give units not of any particular type. This includes cases of taking powers of units and products of such powers and adds the complexity of the {\em discrete logarithm} problem to the system.

What is the fundamental difference between groups and rings?

The main difference between groups and rings is that rings have two binary operations (usually called addition and multiplication) instead of just one binary operation. If you forget about multiplication, then a ring becomes a group with respect to addition (the identity is 0 and inverses are negatives).

What is field in cryptography and network security?

Key Points. A field is a set of elements on which two arithmetic operations (addition and multiplication) have been defined and which has the properties of ordinary arithmetic, such as closure, associativity, commutativity, distributivity, and having both additive and multiplicative inverses.

What is field in cryptography?

Finite Fields, also known as Galois Fields, are cornerstones for understanding any cryptography. A field can be defined as a set of numbers that we can add, subtract, multiply and divide together and only ever end up with a result that exists in our set of numbers.

What is the purpose of ring theory?

Ring theory studies the structure of rings, their representations, or, in different language, modules, special classes of rings (group rings, division rings, universal enveloping algebras), as well as an array of properties that proved to be of interest both within the theory itself and for its applications, such as …

Why do we study ring theory?

So, why study ring and group theory? You might study them because you’ve got a research question that somehow involves symmetry — constraint problems in computer science can be solved more efficiently when a little is known about the solution space.

What is a ring in group theory?

Definition. A ring is a set R equipped with two binary operations + (addition) and ⋅ (multiplication) satisfying the following three sets of axioms, called the ring axioms. R is an abelian group under addition, meaning that: (a + b) + c = a + (b + c) for all a, b, c in R (that is, + is associative).

What is field in rings?

A field is a ring where the multiplication is commutative and every nonzero element has a multiplicative inverse. There are rings that are not fields. For example, the ring of integers Z is not a field since for example 2 has no multiplicative inverse in Z.

Why finite fields are used in cryptography?

A Finite Field denoted by Fp, where p is a prime number, works well with cryptographic algorithms like AES, RSA , etc. because of the following reasons: We need to decrypt the encrypted message, this is only possible when a unique (bijective) inverse of a function is available.