Let’s start with the basics.  We’ll get to elliptic curves, after I fondle yours for awhile while giving a conceptual overview of public-key crypto, symmetric crypto, hash functions, etc.

Also, I have changed my mind about the spanking.  I will spank you just for fun.

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Public-key ciphers

A public-key cipher uses two different keys:  A public key which is not secret (as the name implies), and a private key (also sometimes called the secret key).  These two respective keys are related by a mathematical function which lets messages encrypted to the public key be decrypted by the private key; also, by similar means, the private key can produce forgery-resistant digital signatures which can be verified using the public key.

The advantage of a public-key cipher is that the public key can be—well, it can be made public.

Bitcoin uses an elliptic-curve cryptography (ECC) cipher with curve parameters called “secp256k1”.  Remember what I said about the mathematical function which relates the public key to the private key.  Here, the function is based on something called the elliptic curve discrete logarithm problem (ECDLP).  Don’t worry about what that is right now.

A typical Bitcoin transaction must be bear a verifiable digital signature from a private key.  The corresponding public key is the basis for the type of Bitcoin address with which most users interact:  Old P2PKH (Pay To Public Key Hash; addresses starting with a “1”), or with Segwit, P2WPKH (Pay To Witness Public Key Hash).  The public key is not used directly to form the address:  Rather, it is hashed, and the hash of the public key (plus metadata) makes your Bitcoin address.

Symmetric ciphers

A symmetric cipher uses the same key for both encryption and decryption.  Obviously, this key must be kept secret!  The most commonly used symmetric cipher today is AES; you have probably heard of that.

Hashes

A hash is a one-way compression function which takes an arbitrary-length input, and produces therefrom a fixed-length output.  Take for example the SHA-256 function.  You can feed into it any arbitrary amount of sexy bits and love-bytes; and its output will always be 256 bits (that is, 32 bytes (that is more precisely, 32 octets)).

There are many types of hash functions used for different purposes in computer science.  Many, if not most hash functions are unsuitable for cryptography.  A cryptographically secure hash function must have certain properties which make it infeasible for attackers to do such things as finding two different inputs which produce the same arbitrary hash (called a collision), or finding an input which produces a certain particular hash (called a preimage attack).

I believe I do not exaggerate when I say that hashes are the single most important building block of Bitcoin.  Hashes are everywhere!  They are used to build the Merkle trees which tie together all the substantive contents of a block; they are used to produce a Merkle chain of one block to the next, i.e. this “blockchain” thing you keep hearing about; a partial preimage attack is the basis of block generation, a/k/a mining...