Public Key

The public portion of a key pair to an address which can be used to verify signatures made with the private portion of the key pair.
From this deep look into Public Keys by FlatOutCrypto (9-10-2019):

"How does it work?

DLTs would not function with symmetric key algorithms, for obvious reasons. Bitcoin was created in order to allow two parties to trust each other without the need to trust a third party. If the key is the same for both locking and unlocking, then if Alice wished to send some BTC to Bob she would have to trust that Bob would not unlock her account and steal everything. This is unworkable.

Asymmetric key algorithms, however, provide two keys – a public and a private key. This means that Bob can provide his public key to Alice to receive BTC safe in the knowledge that she will not be able to unlock his account – that is what Bob’s private key is for. Furthermore, if Alice wanted to send information to Bob and Bob wanted to make sure the information was really coming from Alice, then Alice could give her public key to Bob. She can then sign messages using her private key, which Bob is then able to verify using her public key.

For this to be a safe and efficient method of transaction:

No-one can be able to find out the private key despite having access to the public key
It is easy to create the public key when the private key is known

These are known as the one-way and trapdoor functions respectively. The difference between the two is that a trapdoor function is reversible, whereas a one-way function is (as the name suggests) not. These functions are enabled by two types of solutions – the factorization of large numbers and the discrete logarithm problem.

Cryptography implementations such as RSA work on the basis that it is extremely difficult and time consuming to factor large numbers. An incentivized RSA challenge began in 1991, yet of the more than 50 numbers laid out, just 20 (with the smallest decimal digits) have been solved to date, highlighting the computational resources required to crack encryption relying on large number factorization.

The trapdoor function of ECC, however, rests on what is known as the discrete logarithm problem rather than factoring large numbers. The discrete logarithm problem is, given the same length of inputs, harder to solve than factorization, and this is what allows ECC to maintain the same levels of security as prior solutions whilst using smaller keys. A more detailed look at ECC and the discrete logarithm problem can be found here.

Both the factorization of large numbers and the discrete logarithm problem are fundamental to public key cryptography. They are what enable us to use public keys without worrying that a malicious actor will be able to find out private key based on said public key.

What are they used for?

This public key cryptography enables DLTs to achieve a number of things including:

Authenticity: When Alice uses her private key to sign a Bitcoin transaction, the recipient knows that it has come from her and her alone
Confidentiality: Information encrypted with Alice’s public key can only be decrypted by using her private key – meaning that no-one can access Alice’s wallet bar her
Non-repudiation: If Alice signs and sends a message, she can not claim at a later date that she did not send it

These points rest upon the assumption that Alice has not lost or had her private key stolen."