Exam 2: Introduction to Number Theory

the first assertion of the CRt, concerning arithmetic operations, follows from the rules for modular arithmetic.

the scheme where you can find the greatest common divisor of two integers by repetitive application of the division algorithm is known as the Brady algorithm.

Basic concepts from number theory that are needed for understanding finite fields include divisibility, the Euclidian algorithm, and modular arithmetic.

the rules for ordinary arithmetic involving addition, subtraction, and multiplication carry over into modular arithmetic. 9.two theorems that play important roles in public-key cryptography are Fermat's theorem and Euler's theorem.

For many cryptographic algorithms, it is necessary to select one or more very large prime numbers.

One of the useful features of the Chinese remainder theorem is that it provides a way to manipulate potentially very large numbers mod M in terms of tuples of smaller numbers.

the Chinese Remainder theorem is believed to have been discovered by the Chinese mathematician Agrawal in 100 A.D.

If b|a we say that b is a divisor of A.

All integers have primitive roots.

Unlike ordinary addition, there is not an additive inverse to each integer in modular arithmetic.

the algorithm credited to Euclid for easily finding the greatest common divisor of two integers has broad significance in cryptography.

two integers a and b are said to be congruent modulo n, if (a mod n) = (b mod n).

the primitive roots for the prime number 19 are 2, 3, 10, 13, 14 and 15.