How does cryptography use math

Cryptology

Mathematical theory of the design (cryptography) and the investigation (cryptanalysis) of methods of covert or protected communication over open channels.

Originally, cryptology or cryptography only meant the development of secret scripts. Caesar is said to have used a simple substitution cipher for the secret transmission of messages A.D., B.E., C.F., ... have used. However, an encryption method is not characterized by the use of an unknown assignment of characters to characters of a possibly different alphabet, but rather the use of an unknown key with which such an assignment can be defined.

A cryptographic system is understood to mean a lot M. of plaintext, a lot C. of ciphers, a lot K of keys and as well as encryption and decryption functions E. : M. × KC. and D. : C. × KM. so that

for all news mM. is. The keys k' and k″ Are also called keys and counter keys. Are the keys k' and k″ Identical or easy to derive from one another, one speaks of a symmetrical encryption method. Is on the other hand from knowing the key k′ The counter key k″ Not or only very difficult to calculate (exponential effort), so it is more precisely difficult without the opposite key k″ To decrypt an encrypted message, one speaks of an asymmetric procedure. With the symmetrical Caesar cipher is k′? 3, k″ & Equals; −3 and on the 20 letters x of the Latin alphabet applies

The security of a cryptosystem is based on the selection of suitable functions E. and D.that should be easy to calculate and the choice of a sufficiently large key k′. Encryption method, the security of which is mainly based on the fact that the functions E. and D. are not known are generally not safe for long. So the Enigma was broken soon after its structure was known. If the keys are chosen poorly or too short for encryption, it is often sufficient to try out all possible keys (brute force) to decrypt an unknown text.

An encryption method that is absolutely secure, but impractical because the key is much too long, is obtained by encrypting a message with a flow cipher and an unpredictable random sequence of the same length (for example by bit-wise XOR). Then the unsolicited decryptor can at best get an estimate of the length of the message, but no information about the message itself. In practice, combinations of symmetric and asymmetric methods are often used to guarantee sufficient security and the speed of the encryption. Even with software implementations of the AES candidates on a personal computer (500 MHz), encryption rates of 50 Mbit / sec are achieved.