APPLICATIONS OF GROUP THEORY TO CRYPTOGRAPHY
Algebraic structures is made up of a set or a group of sets called carriers, underlying sets, or sorts that suits some algebraic axioms. Some axioms are associativity, invertibility, closure, and identity. The number systems of the structures of mathematics follow these axioms. In abstract algebra, the group axioms permit flexibility in the handling of extremely wide-ranging mathematical origins even as they maintain their vital structural aspects.
Group theory of abstract algebra and mathematics examines the algebraic structures called groups. In abstract algebra, the idea of a group Is fundamental in its study. In mathematics, groups always persist. It has effectively affected various elements of algebra.
[i]Cryptography is the science of encoding information so that only certain specified people can decode it. A central problem to cryptography is the problem regarding key exchange. Functions of methods for key exchange are usually one-directional. Processes on key exchange are frequently based on processes that are not hard to compute while their opposites are not easy to decode. To wit, finding a polynomial complexity through a one way function is simple to find but proving that a similar complexity has no inverse function is the hard part as the most practical inverse function might not be in existence yet. The origin for creating the public key cryptosystem was the intricacy of deciphering equations over algebraic structures.
Group theory also influences cryptography in several ways.
[ii]The ElGamal encryption system (in cryptography) is an asymmetric key encryption algorithm for public-key cryptography which is based on the Diffie-Hellman key exchange. This type of encryption system is utilized in later forms by PGP, the software GNU Privacy Guard, and other various cryptosystems.
Any cyclic group G identifies ElGamal encryption. The complexity of a problem in G correlated to calculating discrete logarithms and the padding plan utilized on communication and messages are the securities of ElGamal.
ElGamal encryption is categorically compliant and malleable thus it is not protected against the chosen cipher text attack.
It is probabilistic. Various probable cipher texts may be encrypted using a plaintext with the result that the encryption generates a growth of 2.1 in range to cipher text.
It needs two exponentiations but these are separate from the message and can be calculated in advance if required.
The encryption algorithm, decryption algorithm, and the key generator are the three components of ElGamal encryption.
[iii]Elliptic curve cryptography is (also) an approach to public key cryptography based on the algebraic structure of elliptic curves over finite fields. These are also applied to a number of integer factorization algorithms that have relevance in cryptography.
Public key cryptography is worked on by sets of prime order formed in elliptic-curve cryptography. This type of cryptographical methodology profits from the adaptability of geometrical objects; thus, group structures along with the complex structure of groups, makes calculating the discrete logarithm very difficult. The elliptic curve’s range or size dictates the impediment of the problem. Using a significantly smaller curve group, the security an RSA-based system can provide is attainable. Making use of a smaller curve group lessens the need for requirements on transmission and storage.
Of late, cryptographic primitives using a variety of elliptical curve groups were launched. The cryptographic primitives’ systems and designs provide identity-based encryption along with signcryption, proxy re-encryption, key agreement, and pairing-based signatures.
[i] Web.williams.edu
[ii] En.wikipedia.org
[iii] E.n.wikipedia.org
Credit:ivythesis.typepad.com
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