Cryptology - I: Appendix A - Review of Cryptosystems

Cryptology - I: Appendix C - Review of Cryptosystems

Instructors: R.E. Newman-Wolfe and M.S. Schmalz

In this class, we often express cryptologic operations in terms of a formal notation. This section contains an overview of notation associated with cryptographic transforms (Section C-1) and cryptosystems that incorporate such trnsforms (Section C-2). In Section C-3, we discuss information-theoretic concepts of data security, and summarize high-level formalisms associated with cryptanalysis in Section C-4.

C-1. Cryptographic Transforms.

An encryption or decryption transform accepts a message and a key and yields another message. In the encryption step, the input message is plaintext and the output is ciphertext. For decryption, the input (ciphertext) is transformed into plaintext at the transform's output.

Definition. An encryption transform T_E : F^X × K -> G^Y, where K denotes the keyspace and G = F or X = Y is usual. Here, the input of T_E is plaintext and the output is ciphertext.
Observation. If |Y| > |X|, then we call the output of T_E a homophonic cipher.
Definition. A decryption transform T_D : G^Y × K -> F^X, where K denotes the keyspace and G = F or X = Y is usual. Here, the input of T_E is ciphertext and the output is plaintext.
Observation. Let k K and plaintext aF^X. It is customary that a = T_D(T_E(a)).

C-2. Cryptosystems.

The fundamental purpose of cryptography is to enable two parties to communicate over an insecure channel such that an adversary cannot understand their communication. This concept can be expressed formally in terms of the following notation.

Definition. A cryptosystem is a five-tuple (P,C,K,E,D), where
1. P is a finite set of possible plaintexts;
2. C is a finite set of possible ciphertexts;
3. K, the keyspace, is a finite set of possible keys; and
4. E and D are sets of encryption and decryption transforms;
such that a = T_D(T_E(a)), for all aP.
Observation. Let us call the two parties conversing over a secure channel Alice and Bob, and the adversary will be named Oscar. The paradigm for secure communication with cryptographic protocols is illustrated in Figure 1.

Figure 1. Alice sends a message a to Bob over an insecure channel as ciphertext c, with a key sent to Bob over a secure channel. Meanwhile, Oscar listens to the insecure channel, which he can intercept. The dotted line indicates that Alice controls the keysource.
-- TO BE CONTINUED --

C-3. Concepts of Security.

C-4. Cryptanalytic Oracles.

This concludes our discussion of cryptosystems for this class. Additional statistical concepts will be defined as they are introduced in theory development.

References

[Mau91] Maurer, U.M. "A universal statistical test for random bit generators", In A.J. Menezes and S. A. Vanstone, Eds., Proceedings CRYPTO 90, Springer Lecture Notes in Computer Science 537:409-420 (1991).