On the Foundations of Key Exchange

Key exchange protocols allow two parties to agree on a shared secret over an untrusted channel. A huge number of scientific works on key exchange have been published since the discovery of the first key exchange protocol, designed in 1976 by Diffie and Hellman [DH76]. The most prominent of these works is the game-based security model by Bellare and Rogaway [BR93], published in 1993 and, today, known as the "golden standard" for the security of key exchange protocols. The main purpose of key exchange protocols is to establish keys for a symmetric-key protocol such as a secure channel. Ideally, if both, the key exchange protocol and the channel protocol, are secure, then we would hope that they can be securely composed. While this is true for simulation-based security models [Can01, KT11, BPW03], game-based models usually lack composition theorems altogether. That is unfortunate, as they are often more suitable for real-life protocols such as TLS. Our first two composition theorems address the composition of BR-secure key exchange protocols with arbitrary symmetric-key protocols. We formally establish that an additional condition is necessary, namely, the key exchange requires a public session matching algorithm. Maybe surprisingly, many important key agreement protocols are not BR-secure due to the popular technique of explicit key confirmation. We devise a more flexible yet sufficiently strong model for this class of protocols and prove that it is closed under reductions. Overall, we devellop a tool for the modular analysis of real-life protocols and exemplify the use of our framwork on a profile of the TLS protocol. As in the case of most practical cryptography, information-theoretic security for key exchange protocols is out of reach, i.e., impossible in a formal sense. Therefore, protocols such as TLS rely on computational assumptions, e.g., the Diffie-Hellmann assumption or the hardness of factoring numbers. As the security of protocols is tightly related to the underlying complexity assumptions, researchers have been striving for simpler and weaker assumptions. The holy grail in this area is to base key agreement, one-way functions or simply any type of cryptography on the mere assumptions that NP does not equal P [FF93, BT03, AGGM06]. While positive results in this area remain elusive, there has been some recent progress on negative results [HMX10, PTV11], showing that cryptographic primitives such as hash functions cannot be based on NP-hardness unless coAM is contained in NP. In this thesis, we provide two oracle results that show that, via relativizing techniques, these negative results do not carry over to key agreement and regular one-way functions. In particular, we give an oracle relative to which the intersection of NP and coNP is easy while infinitely many often secure key agreement exists; and we give an oracle relative to which languages in the intersection of AM and coAM is easy while regular function families exists that are infinitely many often one-way.