Euler's Equation

Euler's number, e, the constant 2.71828, is the base of the natural logarithms. Given n approaching infinity, Euler's number is the limit of:

\begin{align*}\displaystyle{\displaylines{(1 + 1/n)n}}\end{align*}

It's used frequently abroad across the sciences. It can also be elegantly expressed as an infinite series, like so:

\begin{align*} {\displaystyle e=\sum \limits _{n=0}^{\infty }{\frac {1}{n!}}=1+{\frac {1}{1}}+{\frac {1}{1\cdot 2}}+{\frac {1}{1\cdot 2\cdot 3}}+\cdots .} \end{align*}

Separately, the imaginary unit i, \({\displaystyle {\sqrt {-i}}}\), represents the imaginary solution to the quadratic equation, x² + 1 = 0. The value can also be used to extend real numbers to complex numbers.

And π is pi, the irrational number we all know and love, roughly approximate to 3.14159, representing the ratio of the circle's circumference to its diameter.

While it isn't absolutely understood, we can join the three numbers in a seemingly bizarre proof that just works.

\( {\displaystyle e^{i\pi }=-1} \)

Abelian Groups

Commutative groups, those groups in which operand order does not change an equation's result, form Abelian groups that commute: "7 × 3 = 3 × 7". When this condition is not satisfied, we say the expression is non-commutative.

Cyclic groups are a special case of commutative Abelian groups—sets that are monogenous—generated by a single element—and invertible with a single operation.

Consider a set that, if we iterated over every other element with a particular operation, we'd be able to derive all of the elements of the set.

For a finite cyclic group, let G be the group, n be the size of the set, and e be the identity element, such that gⁱ = g^j whenever i ≡ j (mod n); like so.

G = {e, g, g², ... , gⁿ⁻¹}

The commutative property also holds over the additive group of ℤ, or the integers, which are isomorphic to any infinite cyclic group. Similarly, the additive group of ℤ/nℤ, integers modulo n, is isomorphic to the finite cyclic group of order n.

Since all cyclic groups commute, they are all abelian groups, and all finitely produced abelian groups are the direct products of cyclic groups.

For example, the powers of 10 form an infinite subset G = {…, 0.001, 0.01, 0.1, 1, 10, 100, 1000, …} over the rational numbers.

With 10 as a generator, set G is a multiplicative cyclic group. For any element a of the group, one can derive log₁₀ a.

Our set contains 10 and 100. The product \(10^1 \cdot 10^2\) is equivalent to \(10^{1+2} = 1000\). Every cyclic group G is Abelian because if \({\displaystyle x}, {\displaystyle y}\) are in \({\displaystyle G}\), then:

\( {\displaystyle xy=a^{m}a^{n}=a^{m+n}=a^{n}a^{m}=yx}\)

This homomorphic property is relevant in cryptography. It's also useful for computing commitments. For example, we can perform operations to verify information, like so. Let m be a message and r be a random value:

\( {\displaystyle C(m_{1},r_{1})\cdot C(m_{2},r_{2})=C(m_{1}+m_{2},r_{1}+r_{2})}\)

That is to say, with this special property, one could compute and verify the sums of values without knowing the actual values being committed.

WHOIS, TLS, and Recon

While doing reconnaissance against web applications, I wanted to speed up the process of finding new attack surfaces that some subdomain tools might miss.

Revisiting Wireless Security

One of my earlier memories with computers was as a teen, just as WiFi was becoming popular. Back then, internet service was often horrible. But I quickly picked up an Alfa antenna from Ebay and learned that with a few Debian packages, it was possible to defeat the encryption of nearby access points.