12 Public-Key CryptosystemsRSA and Finite Field Cryptography-based Schemes

12.1 Introduction to Public-Key Cryptosystems

In symmetric cryptosystems, the message sender and recipient must share a common secret (i.e. a ciphering key) before encrypting and decrypting messages. The big question is: how to agree on the secret key in first place, particularly if sender and receiver never met? It looks like the “The chicken or egg” dilemma. Trust is required before exchanging the secret key in a secure way. Therefore, symmetric cryptosystems are limited to use in cases where both parties know and trust each other.

The turning point in modern cryptography occurred in 1976‒1977, when Diffie and Hellman [1] on one side and Rivest, Shamir, and Adleman [2], on the other, proposed original schemes to secure systems without requiring a unique cipher shared by both parties. The proposed schemes were and are still used to design public-key cryptosystems. The latter provide support to secure communications worldwide between people who do not a priori know each other. The first and still most widely used public-key cryptosystem is with no doubt is the RSA.

Modern cryptography is founded on the idea that the key used to encrypt messages can be made public, while the key used to decrypt messages must be kept private. As such, these systems are known as public-key cryptographic systems (also called asymmetric cryptosystems) and are based on operations easy to process in one direction, but difficult ...

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