Zero-knowledge proofs are a central field of study in modern cryptography, enabling a prover to convince a verifier that a statement is true without revealing any other information. Over the last decade, they have moved from being mostly theoretical protocols to practical tools. Despite recent progress, efficient real-world deployment remains challenging. Proving can be computationally demanding, memory can easily become a bottleneck, and seemingly minor algebraic design decisions can have major performance consequences. The choice of elliptic curve, scalar field, and polynomial representation of constraints directly influences the scalability of the system. In particular, whether computations are native to the proving field, or they must be emulated in a non-native field, can significantly affect the circuit size and hence, proving cost. A classical example of this problem is encoding cryptographic hash functions in arithmetic circuits [GKRRS19]. This leaves a considerable gap between what is possible in theory and what is practical in deployed systems. This PhD project aims to study zero-knowledge proof systems from both an algebraic and an implementation-oriented perspective. The candidate will investigate how structural choices, like curve selection, field compatibility, and the embedding of primitives, influence prover and verifier complexity. Circuit design, arithmetization, and parameter selection are also studied trying to assess trade-offs between prover time, verifier time, proof size, and memory usage. The scope also includes the potential design of new or adapted proof constructions when current approaches prove inadequate for specific constraints. The primary aim is to let practical bottlenecks and deployment requirements guide the design process to improve on existing schemes. Digital identity infrastructures form the main application context, since they combine strong privacy requirements with demanding usability and efficiency constraints. In such systems, holders of credentials should be able to prove assertions about themselves for e.g., age verification or entitlement to voting rights, without disclosing more information than strictly necessary [LKWL25][FS24]. Additional scenarios, such as end-to-end verifiable electronic voting [HKLR24], content provenance [DCB24], and verifiable machine learning inference [XLFWZJSZ25] serve as test cases to expose different measures of performance limitation. Electronic voting poses large-scale verification and aggregation challenges, while machine learning inference stresses circuit size and proving cost. Overall, the goal is to connect algebraic structure with practical performance and to develop optimization strategies that make zero-knowledge proofs efficient and deployable in real-world systems. References [DCB24] VerITAS: Verifying Image Transformations at Scale. Trisha Datta, Binyi Chen, Dan Boneh. Cryptology ePrint Archive, Paper 2024/1066. https://eprint.iacr.org/2024/1066 [FS24] Ano