This thesis presents a comprehensive examination of the BLS (Boneh-Lynn Shacham) signature scheme, with particular focus on its aggregation properties and security foundations. We begin by establishing the mathematical
groundwork, exploring elliptic curves over prime fields and their essential
properties. We then delve into the theory of elliptic curve pairings, examining crucial concepts such as torsion points, embedding degrees, and rational
functions, culminating in the construction of the Weil pairing. Building upon
these foundations, we present the BLS signature scheme and demonstrate
how it leverages these mathematical structures to create efficient digital signatures. We extend the basic scheme to support signature aggregation, allowing multiple signatures to be combined and verified as one, while addressing
potential vulnerabilities such as the rogue key attack. Finally, we provide a
rigorous security proof that reduces the security of our scheme to the computational hardness of the co-CDH problem, establishing its cryptographic
soundness.
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