The paper "Threshold Cryptosystems" by Yvo Desmedt and Yair Frankel proposes practical, non-interactive public key systems for organizations, allowing the reuse of a shared secret key without revealing it to insiders or outsiders. The authors address the challenge of secure communication in a multi-level organizational structure, where messages must be encrypted and only certain individuals can decrypt them based on a security policy. They introduce a method where each organization has a single public key, and any message can be read by a subset of authorized individuals who collectively possess enough "shadows" of the secret key. The proposed system is based on the ElGamal public key cryptosystem, adapted to meet the specific requirements of threshold schemes. The key innovations include: 1. **Lagrange Interpolation for Modified Shadows**: Each participant receives a modified shadow of the secret key, which is calculated using Lagrange interpolation. These modified shadows are kept secret and used to perform partial calculations. 2. **Geometry-Based Threshold Scheme**: Another threshold scheme based on geometry is introduced, where each participant is given a plane equation whose intersection with other planes reveals the secret. This method ensures that only a subset of participants can reconstruct the secret. The paper also discusses enhancements to increase security and practicality, such as avoiding the use of Galois fields and implementing anonymity to prevent collusion among shadowholders. The security of the system is proven using the concept of zero-knowledge, ensuring that no extra information is leaked about the modified shadows. Finally, the authors highlight the practicality and non-interactive nature of their solution, which contrasts with earlier interactive solutions and provides a robust framework for secure communication in group-oriented societies.The paper "Threshold Cryptosystems" by Yvo Desmedt and Yair Frankel proposes practical, non-interactive public key systems for organizations, allowing the reuse of a shared secret key without revealing it to insiders or outsiders. The authors address the challenge of secure communication in a multi-level organizational structure, where messages must be encrypted and only certain individuals can decrypt them based on a security policy. They introduce a method where each organization has a single public key, and any message can be read by a subset of authorized individuals who collectively possess enough "shadows" of the secret key. The proposed system is based on the ElGamal public key cryptosystem, adapted to meet the specific requirements of threshold schemes. The key innovations include: 1. **Lagrange Interpolation for Modified Shadows**: Each participant receives a modified shadow of the secret key, which is calculated using Lagrange interpolation. These modified shadows are kept secret and used to perform partial calculations. 2. **Geometry-Based Threshold Scheme**: Another threshold scheme based on geometry is introduced, where each participant is given a plane equation whose intersection with other planes reveals the secret. This method ensures that only a subset of participants can reconstruct the secret. The paper also discusses enhancements to increase security and practicality, such as avoiding the use of Galois fields and implementing anonymity to prevent collusion among shadowholders. The security of the system is proven using the concept of zero-knowledge, ensuring that no extra information is leaked about the modified shadows. Finally, the authors highlight the practicality and non-interactive nature of their solution, which contrasts with earlier interactive solutions and provides a robust framework for secure communication in group-oriented societies.