'Can you tell an equation has an integer solution by collecting data from

modular arithmetic' Zhang investigates whether and how an equation can be solved by restoring this local data to a global piece of information - like finding a Pythagorean triple.

The main bottleneck of public key algorithms is that they are slower compared to symmetric-key simply because of their foundation in

modular arithmetic. Hence, how to make a faster and more efficient implementation of public key algorithms such as RSA is a challenge to researchers in cryptography field.

* Charlie Du, a CCHS freshman, who was admitted to AwesomeMath at the University of Texas-Dallas campus, where he will focus on coursework in computational geometry and

modular arithmetic.

Several applications in cryptography such as ciphering and deciphering of asymmetric algorithms, the creation and verification of digital signatures, and secure key exchange mechanisms require excessive use of the basic finite field

modular arithmetic operations addition, multiplication, and the calculation of the multiplicative inverse.

RSA is heavily based on the

modular arithmetic of large integers and the whole RSA includes three parts: key generation, encryption and decryption process.

His topics are integers,

modular arithmetic, quadratic reciprocity and primitive roots, secrets, arithmetic functions, algebraic numbers, rational and irrational numbers, diophantine equations, elliptic curves, dynamical systems, and polynomials.

"Cracking Codes with Python" also shows how to: Combine loops, variables, and flow control statements into real working programs; Use dictionary files to instantly detect whether decrypted messages are valid English or gibberish; Create test programs to make sure that your code encrypts and decrypts correctly; Code (and hack!) a working example of the affine cipher, which uses

modular arithmetic to encrypt a message; Break ciphers with techniques such as brute-force and frequency analysis.

Key words:

Modular Arithmetic, Congruence, Divisibility, b-adic expension.

An exam on

modular arithmetic was elaborated in both paper and electronic formats (with the same questions) to measure the effects of the treatments on students' outcomes.

(1) Algorithms often are based on

modular arithmetic, and a simple example illustrates the idea.

[4,5] proposed integer on the homomorphism scheme based on

modular arithmetic, but the execution efficiency is low; Smart et al.

My favourite sessions were

Modular Arithmetic and Continued Fractions.