Achyut BharadwajSeptember 2021
cryptography,
RSA,
encryption,
decryption,
one-way function,
factorization,
prime,
modular arithmetic,
modular inverse,
kid-RSA,
Euclid's division,
gcd,
Bezout,
Euler,
totient,
Fermat,
primality test,
Miller-Rabin A presentation on the mathematics of cryptography, with an emphasis on RSA. A link to the slides is here.
Basic Cryptography
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Sending Messages
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Say, there are 3 people, Aditi, Bhaskar and Diti.
Aditi wants to send the message “HELLO” to Bhaskar, using her computer.
Computers, however, can only store numbers. How will she send the
message “HELLO” to Bhaskar?
To store characters, there is a code called the ASCII code. In ASCII,
each character is given a value in binary.
“HELLO” is coded as: 1001000 1000101 1001100 1001100 1001111
So, Aditi has to send the above code to Bhaskar.
...Achyut Bharadwaj, Saee Patil, Valentio Iverson, Danny LiAugust 2021
Acknowledgements
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This work was carried out as part of a research project at PROMYS-2021. I am a grateful recipient of the Mehta Fellowship to the PROMYS programs in 2021 and 2022.
Definition and Preliminaries Notation
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Let us define a polynomial $f(x)$ to be
$$f(x) = \sum_{i = 0}^n a_i x^i$$ for some $a_i \in \mathbb{C}$, for all
$0 \le i \le n$ and $n \in \mathbb{N}_0$. We define the degree of a
polynomial to be the largest non-negative integer $n$ such that
$a_n \not= 0$ and let degree of the polynomial $f(x) = 0$ be $0$. Degree
of a polynomial $f$ will be denoted as $\text{deg}(f)$.
...Achyut Bharadwaj, Saee Patil, Valentio Iverson, Danny LiAugust 2021
Acknowledgements
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This work was carried out as part of a research project at PROMYS-2021. I am a grateful recipient of the Mehta Fellowship to the PROMYS programs in 2021 and 2022.
Preliminary Definitions
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Finding Patterns
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(1) $$1,4,7,10,\cdots\;\;?$$
Here’s a harder one:
(2) $$3,4,6,9,13,\cdots\;\;?$$
Notations
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(3) $$\Delta\left(f\right)(x)=f(x+1)-f(x)$$
(4) $$\Delta^{n+1}(f)=\Delta\left(\Delta^n(f)\right)$$
What is $\Delta\left(f\right)$ when $f$ is $1,4,7,10,\cdots$?
What is $\Delta\left(f\right)$ when $f$ is $3,4,6,9,13\cdots$?
What is $\Delta^2(f)$ when $f$ is $3,4,6,9,13\cdots$?
...Achyut BharadwajJanuary 2020
In this article, we will be going through some of the basic concepts
required to understand Kepler’s laws.
Angular Displacement and Velocity
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Angular Displacement
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Just as displacement is defined as the change in position of an object,
angular displacement is defined as the angle swept by an object
travelling in a circular path.
Let’s look at an example. Say, an object travels from point $A$ to point
$B$. In this process, say it covers an angle of $\theta$ (see Fig. 1)
...