Cryptography Reference
In-Depth Information
Chapter 3
Elliptic Curve Cryptography
In This Chapter
• Basic concepts of field theory
• Why elliptic curve cryptography
• Pairing-based cryptography
• Weil and Tate pairing
3.1 Introduction
Modern cryptography is one of the building blocks in the design of wireless sensor net-
works (WSN). The two fundamental categories are symmetric key schemes and asym-
metric key schemes. In symmetric key schemes, the communicating pair has to agree
on secret and authentic keying material before initiating encrypted communication.
Hence, a major drawback with such schemes is the requirement of an authenticated and
confidential channel for distributing keying material. The need for a constant trusted
third party weakens the applicability of symmetric key cryptography in distributed
networks. Furthermore, problems with key management (Chapter 6) and nonrepu-
diation services in symmetric key schemes exacerbate its usage in an ad-hoc network
such as WSN. In 1975, Whitfield Diffie and Martin Hellman introduced public-key
cryptographic schemes that required only an authentic key exchange between the com-
municating parties (Diffie and Hellman 1976). This was the first discrete log sys-
tem. In this scheme, each communicating entity had a private and a corresponding
public-key pair such that retrieving the private key from the corresponding public key
was computationally infeasible. This revolutionary approach laid the foundation for
modern cryptography. In 1977, the first fully conceptualized public-key cryptographic
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