The toughest part of the material in these sections was using the extended euclidean algorithm to find the inverse of a. That is, find s and t such that as + nt = 1. From this our value s is our inverse. Therefore the inverse of a is conrgruent to s (mod n). In previous classes I have seen the euclidean algorithm, but this was my first exposure to the extended euclidean algorithm. In some cases we have been able to use a "guess and check" mehtod of finding s and t for as + nt = 1. The easiest way form me to understand this concept was to see it as the linear combination of a and n to achieve the result 1. However, we can not solve directly for s and t since we have one equation and two unknowns. This is why we need the extended Euclidean algorithm.
(Reflective)
In thinking about our encryption project, I was trying to think of a way to encrypt a message using basic math functions. I started by seeing how addition or multiplication would work. This presented a problem since adding or multiplying would produce results outside of the range of numbers established before encrypting the message. This section on congruence made me realize that modular arithmetic is very well suited for encryption. It would allow me to simplify my method of encryption and still keep the numbers within a specified range. This will be extremely useful in cryptography, especially since simplifying the method often means the computer programming associated with the encryption will be easier as well.
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