الفهرس 
IntroductionOnly 14 pages are availabe for public view
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Abstract
Secret key and public key are fundamental building blocks in cryptography that
are used in numerous protocols. A trusted party is obligated to protect its secret
key from any unfaithful party and spread its public key through secure
communication channels to all other parties, this notation is called public key
encryption.
Node resources scale sub linearly in all respects (storage, disk IO,
computation, and bandwidth). So a need for a deeper concept arises. Our goal is
to implement a stateless consensus architecture, which means that all blocks can
be fully validated without any access to state. The motivation is that this will
allow validators to not keep any main chain state, lowering validator hardware
requirements and making it more accessible. One of the difficulties with this is
that the witness sizes required for this can be very substantial. We can reduce this
by using polynomial commitments, in which any number of data elements can be
proven using just a single group element as a witness. One such scheme relying
on sorted key-value lists. However, it introduces significant complexity in the
form of several layers of caching and needing permutation arguments to merge
those separate commitments.
This thesis aim is to address these two fundamental issues. First, we scale
threshold cryptosystems, which protect secret keys by dividing them up across
many parties. We discuss threshold signatures, verifiable secret sharing and
distributed key generation protocols that can scale to millions of participants. Our
protocols reduce execution time, depending on the scale. For example, at large
scales, we reduce time from tens of hours to tens of seconds. At the core of most
of our contributions lie new techniques for computing evaluation proofs in
constant-sized polynomial commitments. Specifically, we describe how to
decrease the time to calculate n proofs for a degree bound n polynomial from
𝛩(𝑛
2
) to 𝛩(𝑛 𝑙𝑜𝑔 𝑛), at the cost of increasing proof size from 𝛩(1) to 𝛩(𝑙𝑜𝑔 𝑛).