Functional encryption for threshold functions (or fuzzy IBE) from lattices

Cryptosystems based on the hardness of lattice problems have recently acquired much importance due to their average-case to worst-case equivalence, their conjectured resistance to quantum cryptanalysis, their ease of implementation and increasing practicality, and, lately, their promising potential as a platform for constructing advanced functionalities. In this work, we construct “Fuzzy” Identity Based Encryption from the hardness of the Learning With Errors (LWE) problem. We note that for our parameters, the underlying lattice problems (such as gapSVP or SIVP) are assumed to be hard to approximate within supexponential factors for adversaries running in subexponential time. We give CPA and CCA secure variants of our construction, for small and large universes of attributes. All our constructions are secure against selective-identity attacks in the standard model. Our construction is made possible by observing certain special properties that secret sharing schemes need to satisfy in order to be useful for Fuzzy IBE. We also discuss some obstacles towards realizing lattice-based attribute-based encryption (ABE).

Crypto topics and applications II

In this chapter we continue the exposition of crypto topics that was begun in the previous chapter. This chapter covers secret sharing, threshold cryptography, signature schemes, and finally quantum key distribution and quantum cryptography. As in the previous chapter, we have focused only on the essentials of each topic. We have selected in the bibliography a list of representative items, which can be consulted for further details. First we give a synopsis of the topics that are discussed in this chapter. Secret sharing is concerned with the problem of how to distribute a secret among a group of participating individuals, or entities, so that only predesignated collections of individuals are able to recreate the secret by collectively combining the parts of the secret that were allocated to them. There are numerous applications of secret-sharing schemes in practice. One example of secret sharing occurs in banking. For instance, the combination to a vault may be distributed in such a way that only specified collections of employees can open the vault by pooling their portions of the combination. In this way the authority to initiate an action, e.g., the opening of a bank vault, is divided for the purposes of providing security and for added functionality, such as auditing, if required. Threshold cryptography is a relatively recently studied area of cryptography. It deals with situations where the authority to initiate or perform cryptographic operations is distributed among a group of individuals. Many of the standard operations of single-user cryptography have counterparts in threshold cryptography. Signature schemes deal with the problem of generating and verifying electronic) signatures for documents.Asubclass of signature schemes is concerned with the shared-generation and the sharedverification of signatures, where a collaborating group of individuals are required to perform these actions. A new paradigm of security has recently been introduced into cryptography with the emergence of the ideas of quantum key distribution and quantum cryptography. While classical cryptography employs various mathematical techniques to restrict eavesdroppers from learning the contents of encrypted messages, in quantum cryptography the information is protected by the laws of physics.

Cheating immune secret sharing

We consider secret sharing with binary shares. This model allows us to use the well developed theory of cryptographically strong boolean functions. We prove that for given secret sharing, the average cheating probability over all cheating and original vectors, i.e., ρ ¯= 1 n ⋅ 2 −n ∑ n c=1 ∑ α∈Vn ρ c,α , satisfies ρ ¯⩾ 1 2 , and the equality holds ⇔ ρc,α satisfies ρc,α = 1/2 for every cheating vector δc and every original vector α. In this case the secret sharing is said to be cheating immune. We further establish a relationship between cheating-immune secret sharing and cryptographic criteria of boolean functions. This enables us to construct cheating-immune secret sharing.

Cheating prevention in secret sharing over GF(pt)

The work investigates cheating prevention in secret sharing. It is argued that cheating is immune against cheating if the cheaters gain no advantage over honest participants by submitting invalid shares to the combiner. This work addresses the case when shares and the secret are taken from GF(pt). Two models are considered. The first one examines the case when cheaters consistently submit always invalid shares. The second modeldeal s with cheaters who submit a mixture of valid and invalid shares. For these two models, cheating immunity is defined, properties of cheating immune secret sharing are investigated and their constructions are given.

Constructions of cheating immune secret sharing

The work addresses the problem of cheating prevention in secret sharing. Two cheating scenarios are considered. In the first one, the cheaters always submit invalid shares to the combiner. In the second one, the cheaters collectively decide which shares are to be modified so the combiner gets a mixture of valid and invalid shares from the cheaters. The secret scheme is said to be k-cheating immune if any group of k cheaters has no advantage over honest participants. The paper investigates cryptographic properties of the defining function of secret sharing so the scheme is k-cheating immune. Constructions of secret sharing immune against k cheaters are given.

Nonlinear secret sharing immune against cheating

The paper investigates the design of secret sharing that is immune against cheating (as defined by the Tompa-Woll attack). We examine secret sharing with binary shares and secrets. Bounds on the probability of successful cheating are given for two cases. The first case relates to secret sharing based on bent functions and results in a non-perfect scheme. The second case considers perfect secret sharing built on highly nonlinear balanced Boolean functions.

On cheating immune secret sharing

The paper addresses the cheating prevention in secret sharing. We consider secret sharing with binary shares. The secret also is binary. This model allows us to use results and constructions from the well developed theory of cryptographically strong boolean functions. In particular, we prove that for given secret sharing, the average cheating probability over all cheating vectors and all original vectors, i.e., 1/n 2n ∑c=1...n ∑α∈V n ρc,α , denoted by ρ, satisfies ρ ≥ ½, and the equality holds if and only if ρc,α satisfies ρc,α= ½ for every cheating vector δc and every original vector α. In this case the secret sharing is said to be cheating immune. We further establish a relationship between cheating-immune secret sharing and cryptographic criteria of boolean functions.This enables us to construct cheating-immune secret sharing.

A multiple-control fuzzy vault

We introduce multiple-control fuzzy vaults allowing generalised threshold, compartmented and multilevel access structure. The presented schemes enable many useful applications employing multiple users and/or multiple locking sets. Introducing the original single control fuzzy vault of Juels and Sudan we identify several similarities and differences between their vault and secret sharing schemes which influence how best to obtain working generalisations. We design multiple-control fuzzy vaults suggesting applications using biometric credentials as locking and unlocking values. Furthermore we assess the security of our obtained generalisations for insider/ outsider attacks and examine the access-complexity for legitimate vault owners.

An ideal and robust threshold RSA

We present a novel implementation of the threshold RSA. Our solution is conceptually simple, and leads to an easy design of the system. The signing key is shared in additive form, which is desirable for collaboratively performing cryptographic transformations, and its size, at all times, is logn, where n is the RSA modulus. That is, the system is ideal.

On secure multi-party computation in black-box groups

We study the natural problem of secure n-party computation (in the passive, computationally unbounded attack model) of the n-product function f G (x 1,...,x n ) = x 1 ·x 2 ⋯ x n in an arbitrary finite group (G,·), where the input of party P i is x i  ∈ G for i = 1,...,n. For flexibility, we are interested in protocols for f G which require only black-box access to the group G (i.e. the only computations performed by players in the protocol are a group operation, a group inverse, or sampling a uniformly random group element). Our results are as follows. First, on the negative side, we show that if (G,·) is non-abelian and n ≥ 4, then no ⌈n/2⌉-private protocol for computing f G exists. Second, on the positive side, we initiate an approach for construction of black-box protocols for f G based on k-of-k threshold secret sharing schemes, which are efficiently implementable over any black-box group G. We reduce the problem of constructing such protocols to a combinatorial colouring problem in planar graphs. We then give two constructions for such graph colourings. Our first colouring construction gives a protocol with optimal collusion resistance t < n/2, but has exponential communication complexity O(n*2t+1^2/t) group elements (this construction easily extends to general adversary structures). Our second probabilistic colouring construction gives a protocol with (close to optimal) collusion resistance t < n/μ for a graph-related constant μ ≤ 2.948, and has efficient communication complexity O(n*t^2) group elements. Furthermore, we believe that our results can be improved by further study of the associated combinatorial problems.

Ideal threshold schemes from MDS codes

We observe that MDS codes have interesting properties that can be used to construct ideal threshold schemes. These schemes permit the combiner to detect cheating, identify cheaters and recover the correct secret. The construction is later generalised so the resulting secret sharing is resistant against the Tompa-Woll cheating.

Ideal threshold schemes from orthogonal arrays

The work investigates the design of ideal threshold secret sharing in the context of cheating prevention. We showed that each orthogonal array is exactly a defining matrix of an ideal threshold scheme. To prevent cheating, defining matrices should be nonlinear so both the cheaters and honest participants have the same chance of guessing of the valid secret. The last part of the work shows how to construct nonlinear secret sharing based on orthogonal arrays.

Batch Verification of Validity of Bids in Homomorphic E-auction

Bid opening in e-auction is efficient when a homomorphic secret sharing function is employed to seal the bids and homomorphic secret reconstruction is employed to open the bids. However, this high efficiency is based on an assumption: the bids are valid (e.g., within a special range). An undetected invalid bid can compromise correctness and fairness of the auction. Unfortunately, validity verification of the bids is ignored in the auction schemes employing homomorphic secret sharing (called homomorphic auction in this paper). In this paper, an attack against the homomorphic auction in the absence of bid validity check is presented and a necessary bid validity check mechanism is proposed. Then a batch cryptographic technique is introduced and applied to improve the efficiency of bid validity check.

Experimenting with an intrusion detection system for encrypted networks

Network-based Intrusion Detection Systems (NIDSs) analyse network traffic to detect instances of malicious activity. Typically, this is only possible when the network traffic is accessible for analysis. With the growing use of Virtual Private Networks (VPNs) that encrypt network traffic, the NIDS can no longer access this crucial audit data. In this paper, we present an implementation and evaluation of our approach proposed in Goh et al. (2009). It is based on Shamir's secret-sharing scheme and allows a NIDS to function normally in a VPN without any modifications and without compromising the confidentiality afforded by the VPN.

Intrusion detection framework for encrypted networks

Network-based Intrusion Detection Systems (NIDSs) monitor network traffic for signs of malicious activities that have the potential to disrupt entire network infrastructures and services. NIDS can only operate when the network traffic is available and can be extracted for analysis. However, with the growing use of encrypted networks such as Virtual Private Networks (VPNs) that encrypt and conceal network traffic, a traditional NIDS can no longer access network traffic for analysis. The goal of this research is to address this problem by proposing a detection framework that allows a commercial off-the-shelf NIDS to function normally in a VPN without any modification. One of the features of the proposed framework is that it does not compromise on the confidentiality afforded by the VPN. Our work uses a combination of Shamir’s secret-sharing scheme and randomised network proxies to securely route network traffic to the NIDS for analysis. The detection framework is effective against two general classes of attacks – attacks targeted at the network hosts or attacks targeted at framework itself. We implement the detection framework as a prototype program and evaluate it. Our evaluation shows that the framework does indeed detect these classes of attacks and does not introduce any additional false positives. Despite the increase in network overhead in doing so, the proposed detection framework is able to consistently detect intrusions through encrypted networks.