Foundations of Asymmetric Cryptography
This chapter presents asymmetric key cryptography, also known as public-key cryptography. It introduces the first asymmetric algorithm, invented by Diffie and Hellman, as well as the ElGamal algorithm. Moreover, we describe in detail a very well-known exa
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Foundations of Asymmetric Cryptography
This chapter presents asymmetric key cryptography, also known as public-key cryptography. It introduces the first asymmetric algorithm, invented by Diffie and Hellman, as well as the ElGamal algorithm. Moreover, we describe in detail a very well-known example of an asymmetric cryptosystem called RSA.
4.1 Idea of Asymmetric Cryptography Asymmetric algorithms differ from the symmetric ones significantly. In the case of symmetric cryptography, in general the same key is used for encryption and decryption, or the decryption key can be determined from the encryption key. Hence, it can be said that only one key is used to encrypt and decrypt messages. In the case of asymmetric algorithms, a pair of two different keys is used, with computationally infeasible mutual reconstruction. Customarily, the key used for encryption is called public and the one for decryption is called private. The essence of the idea of this kind of cryptography is that a ciphertext encrypted with a given public key can be decrypted only with the corresponding private key. A public key, as its name suggests, can be publicized and widely available. Everyone may know this key and everyone can encrypt data with it. The point, as already mentioned, is that the same key cannot be applied for data decryption. This can be done only with the private key corresponding to the one used for encryption. The private key is known only to its owner and therefore only the owner is able to decrypt the data. It is also possible to encrypt data in asymmetric algorithms (e.g., RSA) using a private key. This is an example of a digital signature. In such a case a ciphertext can be decrypted only with the corresponding public key, as in the procedure of signature verification. For details see the next chapter. Distribution of the public keys does not need any secure channel, contrary to the case of symmetric keys. However, public-key cryptography requires a lot of infrastructure taking care of matching the appropriate private and public keys. Also, public C. Ko´scielny et al., Modern Cryptography Primer, DOI 10.1007/978-3-642-41386-5_4, © Springer-Verlag Berlin Heidelberg 2013
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Foundations of Asymmetric Cryptography
key computations are substantially more time consuming than those in symmetric cryptography.
4.2 The Diffie-Hellman Algorithm This algorithm was published in 1976 by Whitfield Diffie and Martin Hellman [31]. Its cryptographic power is based on the problem of computing the discrete logarithm in Zn (discussed in Chap. 2).1 The original version of the algorithm can be used only for distribution or exchange of (usually symmetric) session keys and cannot be used for encryption. However, we present it here due to its historical value and also as an introduction to the ElGamal algorithm. The latter is a well-known cipher. DiffieHellman key exchange is also widely used in Web security with SSL (Socket Layer Security) and TLS (Transport Layer Security) protocols, and in the IPSec protocol key exchange framework. L
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