  In recent years, quantum key distribution (QKD) has evolved from a scientific
research field to a commercially viable security solution, supported by
mathematically formulated security proofs. However, since the knowledge
required for a full understanding of a security proof is scattered across
numerous publications, it has proven difficult to gain a comprehensive
understanding of each step involved in the process and their limitations
without considerable effort and attention to detail. Our paper aims to address
this issue by presenting an accessible and comprehensive security proof for the
finite-size 1-decoy (and 2-decoy) BB84 protocol in Renner's entropic
uncertainty relation framework. We extensively consolidate and unify concepts
from many works, thoroughly discussing the underlying assumptions and resolving
technical inconsistencies. This work can serve as a foundation for the
discussion of QKD security and for the identification of potential
vulnerabilities and device imperfections. Our step-by-step approach and
consistent notation assumes no prior exposure to security proofs, making it a
robust and comprehensible reference, while maintaining theoretical rigor.
Therefore, our contribution represents a significant advancement towards a
broader understanding of QKD security proofs.
