  This paper proposes a novel recursive polynomial commitment scheme (PCS) and
a new polynomial interactive oracle proof (PIOP) protocol, which compile into
efficient and transparent zk-SNARKs (zero-knowledge succinct non-interactive
arguments of knowledge). The Ethereum blockchain utilizes zero-knowledge
Rollups (ZKR) to improve its scalability (the ability to handle a large number
of transactions), and ZKR uses zk-SNARKs to validate transactions. The
currently used zk-SNARKs rely on a trusted setup ceremony, where a group of
participants uses secret information about transactions to generate the public
parameters necessary to verify the zk-SNARKs. This introduces a security risk
into Ethereum's system. Thus, researchers have been developing transparent
zk-SNARKs (which do not require a trusted setup), but those are not as
efficient as non-transparent zk-SNARKs, so ZKRs do not use them. In this
research, I developed LUMEN, a set of novel algorithms that generate
transparent zk-SNARKs that improve Ethereum's security without sacrificing its
efficiency. Various techniques were creatively incorporated into LUMEN,
including groups with hidden orders, Lagrange basis polynomials, and an
amortization strategy. I wrote mathematical proofs for LUMEN that convey its
completeness, soundness and zero-knowledgeness, and implemented LUMEN by
writing around $8000$ lines of Rust and Python code, which conveyed the
practicality of LUMEN. Moreover, my implementation revealed the efficiency of
LUMEN (measured in proof size, proof computation time, and verification time),
which surpasses the efficiency of existing transparent zk-SNARKs and is on par
with that of non-transparent zk-SNARKs. Therefore, LUMEN is a promising
solution to improve Ethereum's security while maintaining its efficiency.
