# A Classical Introduction to Cryptography Exercise BookAlgorithmic Algebra

A Classical Introduction to Cryptography Exercise Book: Algorithmic Algebra Chapter 6 Exercises Exercise 1 Captain's Age The aim of this exercise is to find the very secret age of the Captain. The only information we know is that one year ago, his age was a multiple of 3, in 2 years it will be a multiple of 5, and in 4 years it will be a multiple of 7. Deduce the Captain's age. Hint: Maybe the Captain is Chinese ... D Solution on page 142 Exercise 2 Roots in Z;, Compute the 7th root of 23 in Z& by using the Extended Euclid Algorithm and the Square-and-Multiply Algorithm. D Solution on page 142 Exercise 3 *When is Zi Cyclic? xpp be its decomposition Let n > 1 be an integer, and let n = py' x into prime numbers. We assume that for any integers i # j, we have pi # pj, that pi is prime, and that ai > 0 for 1 5 i 5 r. We consider the multiplicative group Z:. The purpose of this exercise is to find a 136 EXERCISE BOOK necessary and sufficient condition on n such that ZE is a cyclic group, i.e., such that Z; has a generator. 1 In http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png

# A Classical Introduction to Cryptography Exercise BookAlgorithmic Algebra

Springer Journals — Jan 1, 2006
23 pages      /lp/springer-journals/a-classical-introduction-to-cryptography-exercise-book-algorithmic-a8K0Al4Dzo
Publisher
Springer US
ISBN
978-0-387-27934-3
Pages
135 –158
DOI
10.1007/0-387-28835-X_6
Publisher site
See Chapter on Publisher Site

### Abstract

Chapter 6 Exercises Exercise 1 Captain's Age The aim of this exercise is to find the very secret age of the Captain. The only information we know is that one year ago, his age was a multiple of 3, in 2 years it will be a multiple of 5, and in 4 years it will be a multiple of 7. Deduce the Captain's age. Hint: Maybe the Captain is Chinese ... D Solution on page 142 Exercise 2 Roots in Z;, Compute the 7th root of 23 in Z& by using the Extended Euclid Algorithm and the Square-and-Multiply Algorithm. D Solution on page 142 Exercise 3 *When is Zi Cyclic? xpp be its decomposition Let n > 1 be an integer, and let n = py' x into prime numbers. We assume that for any integers i # j, we have pi # pj, that pi is prime, and that ai > 0 for 1 5 i 5 r. We consider the multiplicative group Z:. The purpose of this exercise is to find a 136 EXERCISE BOOK necessary and sufficient condition on n such that ZE is a cyclic group, i.e., such that Z; has a generator. 1 In

Published: Jan 1, 2006

Keywords: Elliptic Curve; Finite Field; Elliptic Curf; Advance Encryption Standard; Chinese Remainder Theorem