Is hashing an optional step to use with public key cryptography for blockchain

Question

I am confused by the public key cryptography versions that I've read so far. Are the steps optional, dependable on the types of blockchain or they are referring to different methods?

Version 1 =========

Alice to Bob

Step 1. Content > Hash > Digest > Encrypt with Alice's private key > Digital Signature

Step 2. Digital Signature > Encrypt with Alice's private key + Bob's public key > Digest

Step 3. Send digest to Bob

In the above steps, Alice uses private key TWICE. First to encrypt the content's digest. Second time encrypt together with Bob's public key.

Is hashing optional?

Is the hash generated automatically once it is encrypted with Alice's private key?

Or encrypting with Alice's private key step is optional?

Is step 2 optional too? or are all the above steps automated?

Bob received

Step 1. Digest > Alice's public key > Content

Step 2. Digest > Bob's private key > Content

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Version 2 =========

Alice to Bob

Step1. Alice generates the keys pair

Step2. Alice gives Bob her public key

Step3. Alice encrypts content with her own private key

In the above steps, Alice didn't hash the content first.

Bob received

Step1. Bob decrypts with Alice's public key

What if someone else has Alice's public key, aren't they able to decrypt it too?

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Version 3 =========

Alice to Bob

Step1. Alice encrypts with Bob's public key

Bob received

Step1. Bob decrypts with his own private key

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Answer 1

Questions like this cannot be answered without specifying the digital signature scheme you're using.

In general

In general, a digital signature scheme consists of 3 algorithms:

• KeyGen() -> (PrivKey, PubKey): generates a private key and a corresponding public key.
• Sign(PrivKey,Message) -> Signature: signs a message with a given private key.
• Verify(PubKey,Message,Signature) -> Bool: verifies a signature against a public key and a message.

What the data types involved are depends on the specific scheme.

ECDSA

In ECDSA, the digital signature scheme used in Bitcoin, the private key is a number, the public key is a point on an elliptic curve, the message is a byte array, and the signature is a pair of two numbers.

ECDSA does require a hashing step. In fact, without the hashing step, it is completely broken.

However, nowhere is there any encryption involved. There is simply a signing algorithm, and a verifying algorithm.

RSA

The mistake you're making, calling signing "encrypting with the private key" is a very common one, because it is often how digital signature schemes are explained.

It is true for some specific signature schemes, but not all. Most notably it is true for RSA. The private and public keys are interchangeable there, so indeed, by swapping them, the encryption scheme turns into a signature scheme.