This paper introduces a general framework for describing and analyzing cryptographic protocols, called the UC (Universal Composition) framework. The framework allows specifying the security requirements of any cryptographic task in a unified and systematic way. It also ensures that the security of protocols is preserved under a general composition operation, called universal composition. This allows for modular design and analysis of complex cryptographic protocols from simpler building blocks. The framework guarantees that protocols maintain their security in any context, even when running concurrently with an unbounded number of arbitrary protocol sessions. The paper discusses the challenges of defining security for cryptographic protocols, particularly in capturing threats from the execution environment and ensuring modular design. It highlights that initial definitions of security for specific cryptographic tasks were insufficient in many contexts. The UC framework addresses these challenges by defining a general methodology for expressing security requirements and a general formal operation for composing protocols. This operation, called universal composition, ensures that security is preserved under composition, allowing protocols to remain secure even in complex and unpredictable environments. The paper presents a formal model of computation, including interactive Turing machines (ITMs), and discusses the model of protocol execution and UC-emulation. It defines the ideal process for carrying out a distributed computational task and shows how protocols can securely realize these tasks. The universal composition theorem is presented, which states that running a composed protocol is at least as secure as running the original protocol. The theorem guarantees that any coordinated attack on the composed protocol can be translated to an attack on the original protocol. The paper also discusses the implications of the composition theorem, including modularity and stronger security guarantees. It highlights that protocols that UC-realize some functionality are guaranteed to continue doing so in any protocol environment, even those not known in advance. The paper presents two models for the UC framework, one simpler and one more expressive, and discusses the trade-offs between simplicity and expressibility. It also reviews related work and previous versions of the paper, emphasizing the evolution of the UC framework over time.This paper introduces a general framework for describing and analyzing cryptographic protocols, called the UC (Universal Composition) framework. The framework allows specifying the security requirements of any cryptographic task in a unified and systematic way. It also ensures that the security of protocols is preserved under a general composition operation, called universal composition. This allows for modular design and analysis of complex cryptographic protocols from simpler building blocks. The framework guarantees that protocols maintain their security in any context, even when running concurrently with an unbounded number of arbitrary protocol sessions. The paper discusses the challenges of defining security for cryptographic protocols, particularly in capturing threats from the execution environment and ensuring modular design. It highlights that initial definitions of security for specific cryptographic tasks were insufficient in many contexts. The UC framework addresses these challenges by defining a general methodology for expressing security requirements and a general formal operation for composing protocols. This operation, called universal composition, ensures that security is preserved under composition, allowing protocols to remain secure even in complex and unpredictable environments. The paper presents a formal model of computation, including interactive Turing machines (ITMs), and discusses the model of protocol execution and UC-emulation. It defines the ideal process for carrying out a distributed computational task and shows how protocols can securely realize these tasks. The universal composition theorem is presented, which states that running a composed protocol is at least as secure as running the original protocol. The theorem guarantees that any coordinated attack on the composed protocol can be translated to an attack on the original protocol. The paper also discusses the implications of the composition theorem, including modularity and stronger security guarantees. It highlights that protocols that UC-realize some functionality are guaranteed to continue doing so in any protocol environment, even those not known in advance. The paper presents two models for the UC framework, one simpler and one more expressive, and discusses the trade-offs between simplicity and expressibility. It also reviews related work and previous versions of the paper, emphasizing the evolution of the UC framework over time.