Pairing-Based Cryptography

Abstract

Following the patterns of the modern world, it is justifiable to say that Data is one of the most valuable assets today. This change in perspective has resulted in a usefulness and popularity boost to previously neglected fields like Information security and cryptography. Cryptography, i.e. the protection of Data and messages by converting them into a senseless/unreadable format, is an age-old concept. From the Roman times where it was used for conveying covert battle plans between generals in the army, to a much later time, when it was used for sending secret messages in wars between nations, to now, when it is used to protect every strand of data in a variety of uses from social messaging and networking sites to bank accounts for the privacy of users and national secrets. Over the years, cryptography has been modified countless times and yet, each form it has taken has had the sole purpose of being nearly impossible to crack, i.e. decrypt without knowing the secret keys.

Out of the many methods/algorithms used for Encryption, each one has unique implementations, strengths and weaknesses. Pairing-based cryptography is one of the best methods known to us. It takes advantage of the Diffie–Hellman approach to make cracking the code difficult, and at the same time, it keeps computation fast. It is based on the pairing of elements from two cryptographic groups (a set based on/enveloping a binary operation which connects every two elements of the group to a third). The Diffie–Hellman Key Exchange works on the assumption that there are no secure channels, i.e. third parties (Hackers for instance) have access to every encrypted message being communicated. There are many procedures used for making groups and rings involved in the generation of our cryptographic groups like the (modified) Weil pairing, the Tate-Lichtenbaum Pairing, Eta pairing and Ate pairing. The directions provided by the method implemented result in different sub-problems and advantages which result in different security levels of our encryption technique. The combination of these pros, cons and uniqueness acts as different methodologies for the implementation of pairing-based cryptography. Although modifications to algorithms and inventions to new approaches keep being explored every day, the backbone of a vast majority of these implementations, however, has the same concept.

This book chapter gives an introduction to pairing-based cryptography, the associated mathematical concepts, definitions and procedures and associated algorithms used for implementation. Since the main motive behind cryptography is to aid in the field of Information Security, the fulcrum of issues faced/areas of judgement for all encryption techniques to be implemented is the un-crackability/strength of the algorithm used; the reverse-engineering methods for these algorithms will also be discussed. Furthermore, there are many implementation techniques being discovered everyday which when combined with existing algorithms have scope for improvement in the future. Some of which are also mentioned.