#### Publication Year

2011

#### Keywords

Cocks IBE Scheme, Clifford Cocks, quadratic reciprocity, problem solving, cryptography

#### Disciplines

Applied Mathematics | Mathematics | Statistics and Probability

#### Abstract

The Cocks IBE Scheme was first proposed by Clifford Cocks in 2001 at the 8th IMA International Conference on Cryptography and Coding. It is a cryptosystem that allows identity-based encryption (IBE) and is based on the difficulty of the composite quadratic residue problem, which has to do with when a number is a square modulo a positive integer *n*. [3] An identity-based encryption scheme is a type of public-key cryptosystem where the public key is some unique information about the identity of the user (e.g. a user's email address or important dates in his/her life). [5]

This cryptosystem depends on the sender and receiver knowing solutions to public quadratic equations modulo a large composite number *M*, where *M* is the product of two very large primes, *P* and *Q*. Both *P* and *Q* are usually 600-700 digits, making *M* extremely large, and factorization of *M* nearly impossible. This, in part, makes Cocks IBE Scheme a very secure system, and it can be shown, as is proved later in this paper, that being able to crack the scheme would be equivalent to solving the composite quadratic residue problem, which is currently unsolved.

#### Department 1 Awarding Honors Status

Mathematics

#### Recommended Citation

Prager, S.
(2011).
*The Cocks IBE Scheme: The Legendre Symbol and Quadratic Reciprocity* (Undergraduate honors thesis, University of Redlands).
Retrieved from https://inspire.redlands.edu/cas_honors/502

#### Creative Commons License

This work is licensed under a Creative Commons Attribution-Noncommercial 4.0 License