This monograph illustrates important notions in security reductions and essential techniques in security reductions for group-based cryptosystems. Using digital signatures and encryption as examples, the authors explain how to program correct security reductions for those cryptographic primitives. Various schemes are selected and re-proven in this book to demonstrate and exemplify correct security reductions.

This book is suitable for researchers and graduate students engaged with public-key cryptography.

Notions, Definitions and Models.- Identity-Based Encryption.- Foundations of Group-Based Cryptography.- Foundations of Security Reduction.- Digital Signatures With Random Oracles.- Digital Signatures Without Random Oracles.- Public Key Encryption With Random Oracles.- Public Key Encryption Without Random Oracles.- Identity-Based Encryption With Random Oracles.- Identity-Based Encryption Without Random Oracles.