Only inputs that have been declared beforehand can be assigned. This section specifies the properties of the protocol to be generated. The proof goal can be specified by arbitrarily nesting atoms using Boolean And and Or operators. Furthermore, the compiler supports k-out-of-n- threshold compositions [49] based on Shamir secret sharing [50] cf. Homomorphisms that appear in multiple atoms can be defined globally in this optional section. Describing homomorphisms in PSL is a natural translation from their mathematical notation consisting of name, domain, co-domain, and the mapping function.

Finally, the atoms used in ProtocolComposition are specified. Note that, in general, this value cannot be automatically determined by the compiler. It may depend, for example, on the size of the special exponent of the homomorphism, whose factorization might not be available. The compiler then automatically infers the required number of repetitions of each atom from this specification. The compiler automatically transforms the proof goal in order to reduce its complexity.

By introspecting the predicates, further optimizations could be implemented easily. The internal operation of the PVT is sketched in Fig.

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As inputs, two files are given: the pro- tocol specification a PSL file that was fed as input to the compiler, and the protocol implementation description that was produced by the compiler a PIL file. The PVT first checks 1 the syntactic correctness of the files and their semantic consistency e. Then, the information required for the construction of the soundness proof is extracted 2. This information essen- tially consists of the proof goal description from the PSL file and the code for the verifier in the implementation file.

In particular, the former includes the defini- tion of the concrete homomorphisms being used in the protocol, and information about the algebraic properties of group elements and homomorphisms3. The reason for the verification toolbox only considering the verifier code is that by definition [2] the soundness of the protocol essentially concerns providing guarantees for the verifier, regardless of whether the prover is honestly executing the protocol or not.

The exception is the final algebraic verification that is performed on the last response from the prover, which deter- mines whether the proof should be accepted. The theoretical soundness proof that we construct essentially establishes that this algebraic check is correct with respect to the proof goal, i. The soundness proof is then generated in three steps: a An adequate proof template is selected from those built into the tool 3. If no adequate template exists, the user is notified and the process terminates.

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If the proof assistant successfully finishes, then we have a formal proof of the theoretical sound- ness of the protocol. The process is fully automatic and achieving this was a major challenge to our design. As can be seen in Fig. In order to achieve automatic validation of the generated proofs, it was necessary to construct a library of general lemmata and theorems in HOL that capture, not only the properties of the algebraic constructions that are used in ZK-PoK protocols, but also the generic provable security stepping stones required to es- tablish the theoretical soundness property.

By relying on a set of existing libraries such as this, development time was greatly shortened, and we were able to create a proof environment in which we can express proof goals in a notation that is very close to the standard mathematical notation adopted in cryptography papers. No verification is carried out of the executable code generated from the PIL file. This is a program correctness problem rather than a theoretical security problem, and must be addressed using different techniques not covered here.

We next detail the most important aspects of our approach. Proof strategy. Proving the soundness property of the ZK-PoK protocols pro- duced by the compiler essentially means proving that the success probability of a malicious prover in cheating the verifier is bounded by the intended knowledge error. As all spe- cial homomorphisms used in cryptography fall into one of two easily recognizable classes, the verification toolbox is able to automatically find a pseudo-preimage for any concrete homomorphism that it encounters without human interaction. A central stepping stone in formally proving the existence of an efficient knowl- edge extractor is the following lemma which actually proves Theorem 1 that we have formalized in HOL.

Given a special homomorphism and two accepting protocol transcripts for a ZK- PoK of an atom, we prove the existence of a knowledge extractor by ensuring that we are able instantiate Lemma 2.

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If multiple predi- cates are combined by And, the verification tool defines as proof goal the exis- tence of a knowledge extractor for each and all of them separately: one needs to show that the witness for each predicate can be extracted independently from the other predicates. In case of Or proofs i. First, for each atom, an Isabelle theorem proves the existence of a knowledge extractor. In a second step, it is then shown that the assumptions of at least one of these theorems are satisfied i. The HOL theory file produced by the Protocol Verification Toolbox is typical, in the sense that it contains a set of auxiliary lemmata that are subsequently used as simplification rules, and a final lemma with the goal to be proved.

The purpose of the auxiliary lemmata is to decompose the final goal into simpler and easy to prove subgoals. They allow a systematic proof strategy that, because it is modularized, can handle proof goals of arbitrary complexity. Let G and H be commutative groups, where G rep- resents the group of integers.

A typical proof is then structured as follows. As we have embedded in our tool the domain specific knowledge to generate pseudo-preimages for the class of protocols that we formally verify, we can intro- duce another explicit pseudo-preimage as an hypothesis in our proof, e. At this point we can instantiate the formalization of Lemma 2, and complete the proof for the above theorem, which implies the existence of a knowledge extractor.

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## Cryptography - Wikipedia

Kunz-Jacques, S. In: Yung, M. PKC Springer, Heidelberg Bangerter, E. In: Vaudenay, S. Schnorr, C. Journal of Cryptology 4, — Pedersen, T. In: Feigenbaum, J. In: Pfitzmann, B.

Lipmaa, H. In: Laih, C. Paulson, L. Volume of LNCS. Springer MacKenzie, P. ACM, New York Malkhi, D. Underlying security is often a cryptographic system. Developers often have to design and implement systems which involve an element of security, and cryptography is usually underpinning such systems. Managers often manage security systems, and, again, those systems rely on strong cryptography. This module will explore how modern cryptography works, how it is implemented, and how it is usually incorporated into protocols. Apply cryptography and related functions, such as hashing and digital signatures, to the achievement of security services.