## 3 Certificateless Cryptography  
Certificate-based cryptography, also known as Public Key Infrastructure (PKI) cryptography, is a method of secure communication that uses digital certificates to establish trust between parties. It is based on the concept of public key cryptography, in which each user has a public key and a private key. These are used for data encryption and signatures. A digital certificate is a digital document that contains a user's public key and a set of identifying information, such as the user's name, address, and other identifying information. The certificate is signed by a trusted third party known as a certificate authority (CA), who attests to the authenticity of the public key and the identity of the user. Certificateless cryptography (CLC) or certificateless public key cryptography (CL-PKC) is an alternative method of secure communication that does not rely on digital certificates. One of the main differences between certificate-based and CL-PKC is the way trust is established between parties. In a typical PKI, trust is established through the use of the CA who verifies the identity of users and attests to the authenticity of their public keys; thus, being able to issue the certificates. In CL-PKC, trust is established through the use of mathematical algorithms and protocols that eliminate the need for a trusted third party. The notion of CL-PKC was first discussed by [PERSON] and [PERSON] [18]. Primarily, CL-PKC is used for authentication and key agreement protocols and it eliminates the key escrow problem (KEP), as well as challenges with certificate management such as scalability and revocation. In a CL-PKC system, the _Key Generation Center_ (KGC) is the node thatgenerates a user's public and private keys partially, which the user then uses to complete the key-pair generation. The user typically seeds the KGC with some secret value or identity string, then called Identity-Based Cryptography (IBC), for further partial key generation. For this reason, the KGC will not contain any of the final keys and, thus, no certificates are needed as in traditional PKI. A conceptual depiction of a CL-PKC is provided in Figure 1.  
The KEP refers to a situation where a trusted third party, known as the key escrow agent, has access to the private keys of the users in a cryptographic system. This creates potential security and privacy concerns, as the key escrow agent can potentially decrypt or forge messages, making users vulnerable to unauthorized access [18]. In a traditional PKI, the KEP is not inherent, as users generate their own private keys. However, PKIs rely on certificates to bind public keys to user identities, which can be cumbersome to manage and validate. While IBC simplifies key management by eliminating the need for certificates, it introduces the KEP as the trusted authority has access to all users' private keys. CL-PKC combines the benefits of both PKI and IBC while eliminating the KEP by separating the key generation process between the user and the KGC. At the same time, CL-PKC avoids the complexity of certificate management in traditional PKI systems, offering a more efficient and secure solution for key management and authentication in cryptographic protocols. In certificate-based cryptography, a malicious third party can impersonate the CA and issue fake certificates, allowing them to intercept and read encrypted messages (e.g., DigiNotar [28]), whereas in CL-PKC, there is no central authority that can be impersonated, making it more difficult for an attacker to intercept and read encrypted messages. Additionally, CL-PKC is more efficient in terms of computation and communication, as it does not involve certificate management and certificate revocation. Hence, it reduces the computational and communication overhead and makes it suitable also for environments with low-powered IoT devices [29].