Divisibility, Factors and Euclid's Algorithms

An outline of the fundamentals of number theory
Cryptography is built on the foundations of algebra and number theory, which we will hopefully cover well enough here.
Divisibility
​aaa
 is said to be divisible by bbb
 if there is another integer ccc
 such that a=bca=bca=bc
. In this case, bbb
 is said to be a factor of aaa
. This is denoted by a∣ba \mid ba∣b
 if aaa
 divides bbb
, and a∤ba \nmid ba∤b
 if it does not.
Greatest Common Divisor
Given two integers aaa
 and bbb
, the greatest common divisor gcd(a,b)gcd(a,b)gcd(a,b)
 (also known as the highest common factor) is the largest integer ppp
 where p∣ap \mid ap∣a
 and p∣bp \mid bp∣b
.
Euclidean Algorithm
Given aaa
 and bbb
, we can write an equation linking the two:
a=b⋅q+ra = b \cdot q + ra=b⋅q+r
​Where qqq
 and rrr
 are integers which fit the equation with r<br < br<b
. Basically, bbb
 divides into aaa
 a maximum of qqq
 times with a remainder of rrr
. But here's where the trick lies.
Every term added together in that equation must be divisible by gcd(a,b)gcd(a,b)gcd(a,b)
 because if we treat the gcd as ggg
 we can say that a=k1g,b=k2ga=k_1g,b=k_2ga=k1​g,b=k2​g
​:
k1g=k2g⋅q+rk_1g = k_2g \cdot q + rk1​g=k2​g⋅q+r
Meaning rrr
 has to be some integer multiple of ggg
​.
But now, if we think outside the box, we realise that both bbb
 and rrr
 are divisible by gcd(a,b)gcd(a,b)gcd(a,b)
... so we can just calculate gcd(b,r)gcd(b, r)gcd(b,r)
!
This is quite a leap forward which will require a bit of thinking, but let's break it down algebraically:
a=b⋅q0+r1b=r1⋅q1+r2r1=r2⋅q2+r3r2=r3⋅q3+r4a = b \cdot q_0 + r_1 \\
b = r_1 \cdot q_1 + r_2 \\
r_1 = r_2 \cdot q_2 + r_3 \\
r_2 = r_3 \cdot q_3 + r_4 \\a=b⋅q0​+r1​b=r1​⋅q1​+r2​r1​=r2​⋅q2​+r3​r2​=r3​⋅q3​+r4​
And so on. But when does it stop? When do we stop taking the GCD? Well we can stop taking the GCD once rn=0r_n = 0rn​=0
, in which case rn−1∣rn−2r_{n-1} \mid r_{n-2}rn−1​∣rn−2​
 and as a result we can take rn−1r_{n-1}rn−1​
 as the GCD!
I highly recommend you think about this for a bit until it makes sense to you, and make sure to use other resources if it helps!
Example
Let’s say we want to find the GCD of 8075 and 16283. First, we can write it in the form a=b⋅q+ra= b \cdot q+r a=b⋅q+r
:
16283=8075⋅2+13316283 = 8075·2 + 13316283=8075⋅2+133
And now we attempt to calculate the GCD of 8075 and 133.
8075=133⋅60+95133=95⋅1+3895=38⋅2+1938=19⋅2+08075 = 133 \cdot 60 + 95 \\
    133 = 95 \cdot 1 + 38 \\
    95 = 38 \cdot 2 + 19 \\
    38 = 19 \cdot 2 + 08075=133⋅60+95133=95⋅1+3895=38⋅2+1938=19⋅2+0
Therefore the GCD of 16283 and 8075 is 19.
Extended Euclidean Algorithm
We can take the Euclidean Algorithm a step further and calculate, in addition to the GCD, u,v∈Zu,v \in \mathbb{Z}u,v∈Z
 for a,ba,ba,b
 which sum to the GCD, i.e.
au+bv=gcd(a,b) au + bv = gcd(a,b) au+bv=gcd(a,b)
This extension of the algorithm is invaluable for calculating modular inverses of numbers and is based on using the Euclidean Algorithm to calculate the GCD then writing it in terms of other numbers, repeating the process for the smallest non-GCD number until we reach an equation with only the GCD and the two starting numbers. Let's work with the example above, writing an equation for the GCD:
19=95−38⋅219 = 95 - 38 \cdot 219=95−38⋅2
Now 38 is the smallest non-GCD number, and we can write it in terms of the larger number in the sequence of equations written in the example, then repeat for the next smallest:
19=95−38⋅2=95−(133−95⋅1)⋅2=95−(133⋅2+95⋅−2)=95⋅3+133⋅−2=(8075+133⋅−60)⋅3+133⋅−2=8075⋅3+133⋅−180+133⋅−2=8075⋅3+133⋅−182=8075⋅3+(16283+8075⋅−2)⋅−182=8075⋅3+16283⋅−182+8075⋅364=8075⋅367+16283⋅−18219 = 95 - 38 \cdot 2 \\
    = 95 - (133 - 95 \cdot 1) \cdot 2 \\
    = 95 - (133 \cdot 2 + 95 \cdot -2) \\
    = 95 \cdot 3 + 133 \cdot -2 \\
    = (8075 + 133 \cdot -60) \cdot 3 + 133 \cdot -2 \\
    = 8075 \cdot 3 + 133 \cdot -180 + 133 \cdot -2 \\
    = 8075 \cdot 3 + 133 \cdot -182 \\
    = 8075 \cdot 3 + (16283 + 8075 \cdot -2) \cdot -182 \\
    = 8075 \cdot 3 + 16283 \cdot -182 + 8075 \cdot 364 \\
    = 8075 \cdot 367 + 16283 \cdot -18219=95−38⋅2=95−(133−95⋅1)⋅2=95−(133⋅2+95⋅−2)=95⋅3+133⋅−2=(8075+133⋅−60)⋅3+133⋅−2=8075⋅3+133⋅−180+133⋅−2=8075⋅3+133⋅−182=8075⋅3+(16283+8075⋅−2)⋅−182=8075⋅3+16283⋅−182+8075⋅364=8075⋅367+16283⋅−182
Therefore our equation au+bv=gcd(a,b)au + bv = gcd(a,b)au+bv=gcd(a,b)
 is as follows:
16283⋅−182+8075⋅367=1916283 \cdot -182 + 8075 \cdot 367 = 1916283⋅−182+8075⋅367=19
Crypto
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Modular Arithmetic
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