Matrix Product Chain Representation of an Addition Chain

I am looking for references to anything interesting thats know about matrix product chains that take the vector $ \{1\}$ to another vector $ \{n\}$ (the end result of an addition chain). Each matrix operation adds two entries together to form a new entry and chooses to keep or discard each entry used in the sum. The maximal dimension of the vector at an arbitrary step is tightly bound by the small step count of the addition chain. It’s the number of temporary variables the addition chain needs to be computed. So we have: $ \{n\} = M_{l(n)}..M_1M_2\{1\}$ .

Splitting the matrix product into two:

$ P = M_1M_2…M_p$ .

$ Q = M_{p+1}M_{p+2}…M_{l(n)}$ .

Gives a pair of problems. One an addition sequence problem and the other a vector addition chain problem. This is used for the fastest known algorithm for computing optimal addition chains. This is my area of interest.

When I try searching for matrix product chains I am swamped by the standard dynamic programming exercise to compute a matrix product efficiently. So can’t find anything interesting on matrix product chains at all.

Why are block header bits necessary? (Valid difficulty is already implied by chain history)

Difficulty or target is implied by chain history, so why does it need to be explicit in the header? I suppose it exposes miner-intended-difficulty, but I don’t see why that would be relevant without chain context.

So it seemingly represents redundant data in the header, unless there are any historical reasons for this design choice?

I am just wondering what are the methods which can hide all the transactions and also able to validate transactions in an block chain?

I am new to block chain and was looking at how block chain work mainly proof of stake based . They check validity by going through all the transactions which is something concerning as everyone can see the transactions. So what are ways to hide all transactions and also make everyone able to check validity of new block. This question is not only for block chain relm but also for cryptography.

Is it possible to create an application on a block chain without a native coin

Hello guys I’ve been pondering on some idea of creating a forex broker which will work on a block chain but i have some worries which I’m confused about…

  1. How can transactions go on in my broker if there is no coin to carry out mining activities what will the miners or validators earn as a reward of their services..

  2. Is it possible to use the blockchain only as a service of storage of information .. however this still brings us back to the first question what then will be the criteria to validate the information before storing in the blockchain…

  3. Is it possible to do run a block chain service in line with a fiat trading platform??