Up to the 1970s, Cryptography was understood as the art of building encryption schemes, that is, the art of constructing schemes allowing secret data exchange over insecure channels. Since the 1970s, other tasks (e.g., signature schemes) have been recognized as falling within the domain of Cryptography (and even being at least as central to Cryptography). Yet the construction of encryption schemes remains, and is likely to remain, a central enterprise of Cryptography.

In this chapter we review the well–known notions of private–key and public–key encryption schemes. More importantly, we define what is meant by saying that such schemes are secure. This definitional treatment is a cornerstone of the entire area, and much of this chapter is devoted to various aspects of it. We also present several constructions of secure (private–key and public–key) encryption schemes. It turns out that using randomness during the encryption process (i.e., not only at the key–generation phase) is essential to security.

Organization. Our main treatment (i.e., Sections 5.1–5.3) refers to security under “passive” (eavesdropping) attacks. In contrast, in Section 5.4, we discuss notions of security under active attacks, culminating in robustness against chosen ciphertext attacks. Additional issues are discussed in Section 5.5.

Teaching Tip. We suggest to focus on the basic definitional treatment (i.e., Sections 5.1 and 5.2.1 – 5.2.4) and on the the feasibility of satisfying these definitions (as demonstarted by the simplest constructions provided in Sections 5.3.3 and 5.3.4.1). The overview to security under active attacks (i.e., Section 5.4.1) is also recommended.