Cryptographic algorithms have been used not only to create robust ciphertexts but also to generate cryptograms that, contrary to the classic goal of cryptography, are meant to be broken. These cryptograms, generally called puzzles, require the use of a certain amount of resources to be solved, hence introducing a cost that is often regarded as a time delay---though it could involve other metrics as well, such as bandwidth. These powerful features have made puzzles the core of many security protocols, acquiring increasing importance in the IT security landscape. The concept of a puzzle has subsequently been extended to other types of schemes that do not use cryptographic functions, such as CAPTCHAs, which are used to discriminate humans from machines. Overall, puzzles have experienced a renewed interest with the advent of Bitcoin, which uses a CPU-intensive puzzle as proof of work. In this paper, we provide a comprehensive study of the most important puzzle construction schemes available in the literature, categorizing them according to several attributes, such as resource type, verification type, and applications. We have redefined the term puzzle by collecting and integrating the scattered notions used in different works, to cover all the existing applications. Moreover, we provide an overview of the possible applications, identifying key requirements and different design approaches. Finally, we highlight the features and limitations of each approach, providing a useful guide for the future development of new puzzle schemes.
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