"Let’s start with some context. The story begins 1,800 years ago with a mathematician named Diophantus, who lived in Alexandria and whose work gave rise to many things we learn in high school algebra."

"The ideas he developed to solve one particular problem described something that is heavily studied in mathematics and called an elliptic curve."

"Cryptography is a beautiful area involving elegant mathematics."

"The same time, the results are practical and used in real-world systems."

"If a browser wants to talk to a bank, it needs to exchange keys to encrypt the communication."

"Whitfield Diffie and Martin Hellman, working at Stanford, invented an ingenious way to do this."

"The problem is that to make their method secure, you have to make the numbers involved quite large, and that makes it slow."

"I head the applied cryptography group and co-direct the computer security lab at Stanford University."

"My group’s research focuses on applications of cryptography to computer security."

"Our work includes cryptosystems with novel properties, cryptography for blockchains, Web security, and cryptanalysis."

"We propose a fully functional identity-based encryption scheme (IBE)."

"The scheme has chosen ciphertext security in the random oracle model assuming an elliptic curve variant of the computational Diffie-Hellman problem."

"Our system is based on the Weil pairing."

"We give precise definitions for secure identity based encryption schemes and give several applications for such systems."

"We introduce a short signature scheme based on the Computational Diffie-Hellman assumption on certain elliptic and hyper-elliptic curves."

"The signature length is half the size of a DSA signature for a similar level of security."

"Our short signature scheme is designed for systems where signatures are typed in by a human or signatures are sent over a low-bandwidth channel."

"We construct two efficient Identity Based Encryption (IBE) systems that are selective identity secure without the random oracle model."

"Selective identity secure IBE is a slightly weaker security model than the standard security model for IBE."

"In this model the adversary must commit ahead of time to the identity that it intends to attack, whereas in the standard model the adversary is allowed to choose this identity adaptively."

"Our first secure IBE system extends to give a selective identity Hierarchical IBE secure without random oracles."