We propose new and improved instantiations of lossy trapdoor functions (Peikert and Waters, STOC ’08), and correlation-secure trapdoor functions (Rosen and Segev, TCC ’09). Our constructions widen the set of number-theoretic assumptions upon which these primitives can be based, and are summarized as follows: • Lossy trapdoor functions based on the quadratic residuosity assumption. Our construction relies on modular squaring, and whereas previous such constructions were based on seemingly stronger assumptions, we present the first construction that is based solely on the quadratic residuosity assumption. We also present a generalization to higher order power residues. • Lossy trapdoor functions based on the composite residuosity assumption. Our construction guarantees essentially any required amount of lossiness, where at the same time the functions are more efficient than the matrix-based approach of Peikert and Waters. • Lossy trapdoor functions based on the d-Linear assumption. Our construction both simplifies the DDH-based construction of Peikert and Waters, and admits a generalization to the whole family of d-Linear assumptions without any loss of efficiency. • Correlation-secure trapdoor functions related to the hardness of syndrome decoding. Keywords: Public-key encryption, lossy trapdoor functions, correlation-secure trapdoor functions. An extended abstract of this work appears in Public Key Cryptography — PKC 2010, Springer LNCS 6056 (2010),
We propose new and improved instantiations of lossy trapdoor functions (Peikert and Waters, STOC ’08), and correlation-secure trapdoor functions (Rosen and Segev, TCC ’09). Our constructions widen the set of number-theoretic assumptions upon which these primitives can be based, and are summarized as follows: • Lossy trapdoor functions based on the quadratic residuosity assumption. Our construction relies on modular squaring, and whereas previous such constructions were based on seemingly stronger assumptions, we present the first construction that is based solely on the quadratic residuosity assumption. We also present a generalization to higher order power residues. • Lossy trapdoor functions based on the composite residuosity assumption. Our construction guarantees essentially any required amount of lossiness, where at the same time the functions are more efficient than the matrix-based approach of Peikert and Waters. • Lossy trapdoor functions based on the d-Linear assumption. Our construction both simplifies the DDH-based construction of Peikert and Waters, and admits a generalization to the whole family of d-Linear assumptions without any loss of efficiency. • Correlation-secure trapdoor functions related to the hardness of syndrome decoding. Keywords: Public-key encryption, lossy trapdoor functions, correlation-secure trapdoor functions. An extended abstract of this work appears in Public Key Cryptography — PKC 2010, Springer LNCS 6056 (2010),
In the first part of this work, Chapter 3 we present new and general constructions of lossy encryption schemes.
The Selective Opening problem is as follows: suppose an adversary receives n commitments (or encryptions) of (possibly) correlated messages, and now the adversary can choose n/2 of the messages, and receive de-commitments (or decryptions and the randomness used to encrypt them). Do the unopened commitments (encryptions) remain secure? A protocol achieving this type of security is called secure against a selective opening adversary (SOA). This question arises naturally in the context of Byzantine Agreement and Secure Multiparty Computation, where an active adversary is able to eavesdrop on all the wires, and then choose a subset of players to corrupt. Unfortunately, the traditional definitions of security (IND-CPA, IND-CCA) do not guarantee security in this setting. In this paper: · We formally define re-randomizable encryption and show that every re-randomizable encryption scheme gives rise to efficient encryptions secure against a selective opening adversary. (Very informally, an encryption is re-randomizable, if given any ciphertext, there is an efficient way to map it to an almost uniform re-encryption of the same underlying message). · We show that statistically-hiding 2-round Oblivious Transfer (OT) implies Lossy Encryption and so do smooth hash proof systems, as defined by Cramer-Shoup. Combining this with known results immediately shows that Private Information Retrieval and homomorphic encryption both imply Lossy Encryption, and thus Selective Opening Secure Public Key Encryption.
In the second part of this work, Chapter 4, we focus on constructing lossy trapdoor functions.
Lossy trapdoor functions (LTDFs) were introduced by Peikert and Waters in STOC '08. Since their introduction, lossy trapdoor functions have found many uses in cryptography. In the work of Peikert and Waters, lossy trapdoor functions were used to give an efficient construction of a chosen-ciphertext secure (IND-CCA2) cryptosystem. Lossy trapdoor functions were then shown to imply deterministic encryption by Boldyreva, Fehr and O'Neill in CRYPTO '08. In TCC '09, Rosen and Segev showed that lossy trapdoor functions are correlated product secure, meaning that they remain one-way even when evaluated on correlated inputs.
In this section we extend the notion of smooth hash proof systems as defined by Cramer and Shoup in Eurocrypt '02, to include an additional homomorphic property. We call this primitive smooth homomorphic hash proof systems . We show that smooth homomorphic projective hash proof systems include all Diverse Group Systems, as defined by Cramer and Shoup. Using this definition, we show that · Smooth homomorphic hash proof systems imply LTDFs. · Diverse group systems as defined in [CS02] imply LTDFs. These are the first known generic constructions of LTDFs. · Applying our generic construction the specific constructions of smooth hash proof systems given by Cramer and Shoup, we obtain the first construction of fully lossy trapdoor functions from the quadratic residuosity (QR) assumption. We also obtain a novel construction of LTDFs from Paillier's decisional composite residuosity (DCR) assumption. · Applying the black-box separation results of Rosen and Segev, we show that there is a black-box separation between smooth homomorphic hash proof systems and one-way trapdoor permutations.
In the third part of this work, Chapter 5, we examine in what situations we can construct lossy trapdoor functions from the seemingly weaker primitive lossy encryption.
Injective one-way trapdoor functions are one of the most fundamental cryptographic primitives. In this section we give a novel construction of injective trapdoor functions based on oblivious transfer for long strings. Our main result is to show that any 2-message statistically sender-private semi-honest oblivious transfer (OT) for strings longer than the sender randomness implies the existence of injective one-way trapdoor functions. This is perhaps surprising given the black box separation of injective one-way trapdoor functions from many common cryptographic protocols, e.g. IND-CCA encryption. As a tool for creating injective one-way trapdoor functions, we define a new notion of security for a public key encryption scheme called Randomness Dependent Message (RDM) security, and use it as a stepping stone for creating injective one-way trapdoor functions. (Abstract shortened by UMI.)
Lossy Trapdoor Functions (LTFs) was introduced by Peikert and Waters in 2008. The importance of the LTFs was justified by their numerous cryptographic applications, like the construction of injective one-way trapdoor functions, CCA-secure public-key encryption, etc. However, little research on application of LTFs to key-leakage resilient public-key encryption was done. In this article we introduce a new variant of LTFs featuring leakage-resilience, namely lrLTFs and give a realization of lrLTFs with leakage rate 1/Θ(κ) (where κ is the security parameter) under the Decisional Diffie-Hellman (DDH) assumption. We further improve the leakage rate to 1-o(1) over a composite-order group in which the Decisional Composite Residuosity (DCR) assumption holds. We also introduce a new notion of key-leakage attacks, which we call weak key-leakage attacks, for bridging the adaptive and non-adaptive key-leakage attacks in the setting of public-key cryptosystem. In this model, the leakage adversary only gets a part of public key before accessing to a leakage oracle. We show that lrLTFs imply public-key encryption schemes secure against chosen-ciphertext weak key-leakage attacks in a black-box sense.
Abstract. Lossy Trapdoor Functions (LTDFs), introduced by Peikert and Waters (STOC 2008) have been useful for building many cryptographic primitives. In particular, by using an LTDF that loses a (1 − 1/ω(log n)) fraction of all its input bits, it is possible to achieve CCA security using the LTDF as a black-box. Unfortunately, not all candidate LTDFs achieve such a high level of lossiness. In this paper we drastically improve upon previous results and show that an LTDF that loses only a non-negligible fraction of a single bit can be used in a black-box way to build numerous cryptographic primitives, including oneway injective trapdoor functions, CPA secure public-key encryption (PKE), and CCA-secure PKE. We then describe a novel technique for constructing such slightly-lossy LTDFs and give a construction based on modular squaring.
Lossy trapdoor functions (LTFs) were first introduced by Peikert and Waters (STOC’08). Since their introduction, lossy trapdoor functions have found numerous applications. They can be used as tools to construct important cryptographic primitives such as injective one-way trapdoor functions, chosen-ciphertext-secure public key encryptions, deterministic encryptions, et al. In this paper, we focus on the lossy trapdoor functions in the presence of continuous leakage. We introduce the new notion of updatable lossy trapdoor functions (ULTFs) and give their formal definition and security properties. Based on these, we extend the security model to the LTFs against continuous leakage when the evaluation algorithm is leakage resilient. Under the standard DDH assumption and DCR assumption, respectively, we show two explicit lossy trapdoor functions against continuous leakage in the standard model. In these schemes, using the technology of matrix kernel, the trapdoor can be refreshed at regular intervals and the adversaries can learn unbounded leakage information on the trapdoor along the whole system life. At the same time, we also show the performance of the proposed schemes compared with the known existing continuous leakage resilient lossy trapdoor functions.
We propose a general cryptographic primitive called lossy trapdoor functions (lossy TDFs), and we use it to develop new approaches for constructing several important cryptographic tools, including (injective) trapdoor functions, collision-resistant hash functions, oblivious transfer, and chosen ciphertext-secure cryptosystems (in the standard model). All of these constructions are simple, efficient, and black-box. We realize lossy TDFs based on a variety of cryptographic assumptions, including the hardness of the decisional Diffie–Hellman (DDH) problem and the hardness of the “learning with errors” problem (which is implied by the worst-case hardness of various lattice problems). Taken together, our results resolve some long-standing open problems in cryptography. They give the first injective TDFs based on problems not directly related to integer factorization and provide the first chosen ciphertext-secure cryptosystem based solely on worst-case complexity assumptions.
We initiate the study of one-wayness under correlated products. We are interested in identifying necessary and sufficient conditions for a function f and a distribution on inputs (x1,…,xk) so that the function (f(x1),…,f(xk)) is one-way. The main motivation of this study is the construction of public-key encryption schemes that are secure against chosen-ciphertext attacks (CCAs). We show that any collection of injective trapdoor functions that is secure under a very natural correlated product can be used to construct a CCA-secure public-key encryption scheme. The construction is simple, black-box, and admits a direct proof of security. It can be viewed as a simplification of the seminal work of Dolev, Dwork, and Naor [SIAM J. Comput., 30 (2000), pp. 391–437], while relying on a seemingly incomparable assumption. We provide evidence that security under correlated products is achievable by demonstrating that lossy trapdoor functions [Peikert and Waters, Proceedings of the 40th Annual ACM Symposium on Theory of Computing, 2008, pp. 187–196] yield injective trapdoor functions that are secure under the above-mentioned correlated product. Although we currently base security under correlated products on existing constructions of lossy trapdoor functions, we argue that the former notion is potentially weaker as a general assumption. Specifically, there is no fully black-box construction of lossy trapdoor functions from trapdoor functions that are secure under correlated products.