# Prime Numbers in Cryptography

1. Prime numbers are often used in cryptography. Why do you think prime numbers would be more useful for creating codes than composite numbers?

2. Explain a real-world problem that you used math to solve. What mathematical expressions did you use in your problem-solving? Define your variables and explain your expression.

3. Why is it important to follow the order of operations? What are some possible outcomes when the order of operations is ignored? If you invented a new notation where the order of operations was made clear, what would you do to make it clear?

4. Explain the main differences among integers, rational numbers, real numbers, and irrational numbers. How are these used in everyday life? How would you explain the use of each to someone who did not know about the differences?

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#### Solution Preview

1. Prime numbers are often used in cryptography. Why do you think prime numbers would be more useful for creating codes than composite numbers?

Several public-key cryptography algorithms are based on large prime numbers. The term "public key" means that a known or "public" key is used to encode a message and only a recipient who knows the "private" key can decode that message. RSA encryption operates on the belief that it is "computationally infeasible" to factor a compound number that is the product of two large primes (a semiprime). The problem with this belief is that the size of the numbers that are computationally feasible to factor increases significantly as each new generation of computing device is created.

There are no efficient algorithms for finding the prime factors of numbers, so when presented with a very large number that is the product of two large primes, the factorization process takes a very long time - as the number of digits of the primes being factored increases linearly, the number of operations required to perform the factorization on any computer increases exponentially.

Read more:

1. Number Theory - Current Applications - Prime, Message, Composite, and Factors - JRank Articles ...

#### Solution Summary

The prime numbers in cryptography is examined.

Number Theory

Exercise 3 . (3 marks) Decode the following message: "79311601" knowing that the public key is

n = 8191 x 65537 = 536813567

and

α = 7582663

(I used the correspondence A<-> 01, B <-> 02, ... , Z <-> 26, 0 <-> 30, 9 <-> 39 and worked in base 41 to encode this message.)

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