On formal verification of arithmetic-based cryptographic primitives

Cryptographic primitives are fundamental for information security: they are used as basic components for cryptographic protocols or public-key cryptosystems. In many cases, their security proofs consist in showing that they are reducible to computationally hard problems. Those reductions can be subtle and tedious, and thus not easily checkable. On top of the proof assistant Coq, we had implemented in previous work a toolbox for writing and checking game-based security proofs of cryptographic primitives. In this paper we describe its extension with number-theoretic capabilities so that it is now possible to write and check arithmetic-based cryptographic primitives in our toolbox. We illustrate our work by machine checking the game-based proofs of unpredictability of the pseudo-random bit generator of Blum, Blum and Shub, and semantic security of the public-key cryptographic scheme of Goldwasser and Micali.Comment: 13 page

Revocation in Publicly Verifiable Outsourced Computation

The combination of software-as-a-service and the increasing use of mobile devices gives rise to a considerable difference in computational power between servers and clients. Thus, there is a desire for clients to outsource the evaluation of complex functions to an external server. Servers providing such a service may be rewarded per computation, and as such have an incentive to cheat by returning garbage rather than devoting resources and time to compute a valid result. In this work, we introduce the notion of Revocable Publicly Verifiable Computation (RPVC), where a cheating server is revoked and may not perform future computations (thus incurring a financial penalty). We introduce a Key Distribution Center (KDC) to efficiently handle the generation and distribution of the keys required to support RPVC. The KDC is an authority over entities in the system and enables revocation. We also introduce a notion of blind verification such that results are verifiable (and hence servers can be rewarded or punished) without learning the value. We present a rigorous definitional framework, define a number of new security models and present a construction of such a scheme built upon Key-Policy Attribute-based Encryption.

Securing computation against continuous leakage

30th Annual Cryptology Conference, Santa Barbara, CA, USA, August 15-19, 2010. ProceedingsWe present a general method to compile any cryptographic algorithm into one which resists side channel attacks of the only computation leaks information variety for an unbounded number of executions. Our method uses as a building block a semantically secure subsidiary bit encryption scheme with the following additional operations: key refreshing, oblivious generation of cipher texts, leakage resilience re-generation, and blinded homomorphic evaluation of one single complete gate (e.g. NAND). Furthermore, the security properties of the subsidiary encryption scheme should withstand bounded leakage incurred while performing each of the above operations. We show how to implement such a subsidiary encryption scheme under the DDH intractability assumption and the existence of a simple secure hardware component. The hardware component is independent of the encryption scheme secret key. The subsidiary encryption scheme resists leakage attacks where the leakage is computable in polynomial time and of length bounded by a constant fraction of the security parameter.Israel Science Foundation (710267)United States-Israel Binational Science Foundation (710613)National Science Foundation (U.S.) (6914349)Weizmann KAMAR Gran

Secret-Sharing for NP

A computational secret-sharing scheme is a method that enables a dealer, that has a secret, to distribute this secret among a set of parties such that a "qualified" subset of parties can efficiently reconstruct the secret while any "unqualified" subset of parties cannot efficiently learn anything about the secret. The collection of "qualified" subsets is defined by a Boolean function. It has been a major open problem to understand which (monotone) functions can be realized by a computational secret-sharing schemes. Yao suggested a method for secret-sharing for any function that has a polynomial-size monotone circuit (a class which is strictly smaller than the class of monotone functions in P). Around 1990 Rudich raised the possibility of obtaining secret-sharing for all monotone functions in NP: In order to reconstruct the secret a set of parties must be "qualified" and provide a witness attesting to this fact. Recently, Garg et al. (STOC 2013) put forward the concept of witness encryption, where the goal is to encrypt a message relative to a statement "x in L" for a language L in NP such that anyone holding a witness to the statement can decrypt the message, however, if x is not in L, then it is computationally hard to decrypt. Garg et al. showed how to construct several cryptographic primitives from witness encryption and gave a candidate construction. One can show that computational secret-sharing implies witness encryption for the same language. Our main result is the converse: we give a construction of a computational secret-sharing scheme for any monotone function in NP assuming witness encryption for NP and one-way functions. As a consequence we get a completeness theorem for secret-sharing: computational secret-sharing scheme for any single monotone NP-complete function implies a computational secret-sharing scheme for every monotone function in NP

Circular and leakage resilient public-key encryption under subgroup indistinguishability (or: Quadratic residuosity strikes back)

30th Annual Cryptology Conference, Santa Barbara, CA, USA, August 15-19, 2010. ProceedingsThe main results of this work are new public-key encryption schemes that, under the quadratic residuosity (QR) assumption (or Paillier’s decisional composite residuosity (DCR) assumption), achieve key-dependent message security as well as high resilience to secret key leakage and high resilience to the presence of auxiliary input information. In particular, under what we call the subgroup indistinguishability assumption, of which the QR and DCR are special cases, we can construct a scheme that has: • Key-dependent message (circular) security. Achieves security even when encrypting affine functions of its own secret key (in fact, w.r.t. affine “key-cycles” of predefined length). Our scheme also meets the requirements for extending key-dependent message security to broader classes of functions beyond affine functions using previous techniques of Brakerski et al. or Barak et al. • Leakage resiliency. Remains secure even if any adversarial low-entropy (efficiently computable) function of the secret key is given to the adversary. A proper selection of parameters allows for a “leakage rate” of (1 − o(1)) of the length of the secret key. • Auxiliary-input security. Remains secure even if any sufficiently hard to invert (efficiently computable) function of the secret key is given to the adversary. Our scheme is the first to achieve key-dependent security and auxiliary-input security based on the DCR and QR assumptions. Previous schemes that achieved these properties relied either on the DDH or LWE assumptions. The proposed scheme is also the first to achieve leakage resiliency for leakage rate (1 − o(1)) of the secret key length, under the QR assumption. We note that leakage resilient schemes under the DCR and the QR assumptions, for the restricted case of composite modulus product of safe primes, were implied by the work of Naor and Segev, using hash proof systems. However, under the QR assumption, known constructions of hash proof systems only yield a leakage rate of o(1) of the secret key length.Microsoft Researc

Efficient noninteractive certification of RSA moduli and beyond

In many applications, it is important to verify that an RSA public key (N; e) speci es a permutation over the entire space ZN, in order to prevent attacks due to adversarially-generated public keys. We design and implement a simple and e cient noninteractive zero-knowledge protocol (in the random oracle model) for this task. Applications concerned about adversarial key generation can just append our proof to the RSA public key without any other modi cations to existing code or cryptographic libraries. Users need only perform a one-time veri cation of the proof to ensure that raising to the power e is a permutation of the integers modulo N. For typical parameter settings, the proof consists of nine integers modulo N; generating the proof and verifying it both require about nine modular exponentiations. We extend our results beyond RSA keys and also provide e cient noninteractive zero- knowledge proofs for other properties of N, which can be used to certify that N is suitable for the Paillier cryptosystem, is a product of two primes, or is a Blum integer. As compared to the recent work of Auerbach and Poettering (PKC 2018), who provide two-message protocols for similar languages, our protocols are more e cient and do not require interaction, which enables a broader class of applications.https://eprint.iacr.org/2018/057First author draf

On the semantic security of functional encryption schemes

Functional encryption (FE) is a powerful cryptographic primitive that generalizes many asymmetric encryption systems proposed in recent years. Syntax and security definitions for FE were proposed by Boneh, Sahai, and Waters (BSW) (TCC 2011) and independently by O’Neill (ePrint 2010/556). In this paper we revisit these definitions, identify several shortcomings in them, and propose a new definitional approach that overcomes these limitations. Our definitions display good compositionality properties and allow us to obtain new feasibility and impossibility results for adaptive token-extraction attack scenarios that shed further light on the potential reach of general FE for practical applications.ENIAC Joint UndertakingFundação para a Ciência e a Tecnologia (FCT

Spatial and spatio-temporal patterns in a cell-haptotaxis model

We investigate a cell-haptotaxis model for the generation of spatial and spatio-temporal patterns in one dimension. We analyse the steady state problem for specific boundary conditions and show the existence of spatially hetero-geneous steady states. A linear analysis shows that stability is lost through a Hopf bifurcation. We carry out a nonlinear multi-time scale perturbation procedure to study the evolution of the resulting spatio-temporal patterns. We also analyse the model in a parameter domain wherein it exhibits a singular dispersion relation

Non-malleable encryption: simpler, shorter, stronger

In a seminal paper, Dolev et al. [15] introduced the notion of non-malleable encryption (NM-CPA). This notion is very intriguing since it suffices for many applications of chosen-ciphertext secure encryption (IND-CCA), and, yet, can be generically built from semantically secure (IND-CPA) encryption, as was shown in the seminal works by Pass et al. [29] and by Choi et al. [9], the latter of which provided a black-box construction. In this paper we investigate three questions related to NM-CPA security: 1. Can the rate of the construction by Choi et al. of NM-CPA from IND-CPA be improved? 2. Is it possible to achieve multi-bit NM-CPA security more efficiently from a single-bit NM-CPA scheme than from IND-CPA? 3. Is there a notion stronger than NM-CPA that has natural applications and can be achieved from IND-CPA security? We answer all three questions in the positive. First, we improve the rate in the scheme of Choi et al. by a factor O(λ), where λ is the security parameter. Still, encrypting a message of size O(λ) would require ciphertext and keys of size O(λ2) times that of the IND-CPA scheme, even in our improved scheme. Therefore, we show a more efficient domain extension technique for building a λ-bit NM-CPA scheme from a single-bit NM-CPA scheme with keys and ciphertext of size O(λ) times that of the NM-CPA one-bit scheme. To achieve our goal, we define and construct a novel type of continuous non-malleable code (NMC), called secret-state NMC, as we show that standard continuous NMCs are not enough for the natural “encode-then-encrypt-bit-by-bit” approach to work. Finally, we introduce a new security notion for public-key encryption that we dub non-malleability under (chosen-ciphertext) self-destruct attacks (NM-SDA). After showing that NM-SDA is a strict strengthening of NM-CPA and allows for more applications, we nevertheless show that both of our results—(faster) construction from IND-CPA and domain extension from one-bit scheme—also hold for our stronger NM-SDA security. In particular, the notions of IND-CPA, NM-CPA, and NM-SDA security are all equivalent, lying (plausibly, strictly?) below IND-CCA securit