How do certificates add data to public key and how is this implemented into TLS?

I want to create a self signed PKI for a couple servers I am running. I am finding tutorials with copy paste commands from openssl, and hand waving explanations that describe the general purpose of signing certificates, or 20 page papers on the algorithms used. However, I am really trying to understand the way certificates are added on from the basic public and private key.

I would really like to understand how this works, on a level in between the mathematics and the general "it's used for validation" explanations - ideally - precisely defined inputs and outputs but I don't care how the algorithm works to produce the output. Here is what I understand:

1. I create a root private key (random base 64 number) (rootkey.key)
2. I create a root public certificate with that rootkey (rootcert.crt), this contains the inverse of the rootkey, i.e. public key (q1)
3. I create a server private key (random base 64 number) (serverkey.key)
4. I sign the server private key with the rootkey to create (servercert.crt). (q2)

Now when I establish TLS connection with the server

1. I send my tls request with version number.
2. Server sends servercert.crt
3. I check servercert.crt against rootcert.crt (which I have on client side) (q3)
4. If it's trusted then we can continue with exchanging symmetric encryption keys

(q1) A certificate is a string of pseudorandom digits. What is the mapping between the public key + some strings of domain name, email etc. and these digits? My first thought is this is encrypted with the private key, but then, how could you decrypt since the public key is inside?

(q2) what is happening here? Is this the same process, using rootkey, as step 2?

(q3) What am I doing here? decrypting the server.crt with the public key from the rootcert.crt makes sense as server.crt would be encrypted with the rootkey, but how do I get that root public key?

I think fundamentally it boils down to what is the structure of the certificate. Not just what it contains written in ASN.1 as I read on wikipedia, but mapping this to the pseudorandom digits in the file itself. While writing this I narrowed my question a lot, very useful process, but I still cannot figure this out exactly.

1. No, a certificate is not a string of pseudorandom digits. It's an ASN.1 DER structure, with contents defined in RFC 5280 and its errata. The public key isn't necessarily an "inverse" of the private key, that would only be even vaguely correct for RSA keys. Just say the public key is "mathematically related to" the private key.

2. No, signing is an entirely unrelated process from generating a public key. That process is quite complicated, but is described in section 8 of RFC 3447 .

3. Certificates aren't encrypted. They contain a public key, some identity information (a Common Name, possibly other fields), some validity information (a date range when it's valid, possibly extensions defining what it can be used for), information to identify the signer, and a signature over all that information.

In your creation steps you missed the step of making a Certificate Signing Request (CSR) after making a public key. That CSR's TBS section is what gets signed, not just the public key.

The ASN.1 JavaScript Decoder can provide a very nice visual breakdown of the ASN.1 structure of a certificate, CSR, public key, or private key. Obviously don't enter a real private key you'd use there, just generate one to see how the encoding works.

A TLS handshake concludes with unilateral authentication of the server. For (EC)DHE-only key exchange, the server sends a Certificate message followed by a CertificateVerify message, the former contains a certificate (along with its certificate chain) for authentication, and the latter contains a signature (constructed with the private key corresponding to the public key in the certificate) over a hash of the handshake's transcript, thereby, proving possession of the private key used for signing, hence, identifying the server.

The certificate chain comprises DER-encoded X.509v3 certificates (unless an alternative was negotiated). The server's certificate must appear first and every subsequent certificate should certify the previous one (i.e., every subsequent certificate should contain a signature - using the private key corresponding to the certificate's public key - over the previous certificate's public key), hence, the list is a certificate chain.

Regarding your questions, you're creating (q2) and verifying (q3) the certificate.

Further details: https://arxiv.org/abs/1904.02148 (Section 2.5).