We propose a new general primitive called lossy trapdoor functions (lossy TDFs), and realize it under a variety of different number theoretic assumptions, including hardness of the decisional Diffie-Hellman (DDH) problem and the worst-case hardness of standard lattice problems. Using lossy TDFs, we develop a new approach for constructing many important cryptographic primitives, including standard trapdoor functions, CCA-secure cryptosystems, collisionresistant hash functions, and more. All of our constructions are simple, efficient, and black-box. Taken all together, these results resolve some long-standing open problems in cryptography. They give the first known (injective) trapdoor functions based on problems not directly related to integer factorization, and provide the first known CCA-secure cryptosystem based solely on worst-case lattice assumptions.

Abstract. This paper considers two questions in cryptography. Cryptography Secure Against Memory Attacks. A particularly devastating side-channel attack against cryptosystems, termed the “memory attack”, was proposed recently. In this attack, a significant fraction of the bits of a secret key of a cryptographic algorithm can be measured by an adversary if the secret key is ever stored in a part of memory which can be accessed even after power has been turned off for a short amount of time. Such an attack has been shown to completely compromise the security of various cryptosystems in use, including the RSA cryptosystem and AES. We show that the public-key encryption scheme of Regev (STOC 2005), and the identity-based encryption scheme of Gentry, Peikert and Vaikuntanathan (STOC 2008) are remarkably robust against memory attacks where the adversary can measure a large fraction of the bits of the secret-key, or more generally, can compute an arbitrary function of the secret-key of bounded output length. This is done without increasing the size of the secret-key, and without introducing any

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 (CCA). We show that any collection of injective trapdoor functions that is secure under very natural correlated products 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. We provide evidence that security under correlated products is achievable by demonstrating that any collection of lossy trapdoor functions, a powerful primitive introduced by Peikert and Waters (STOC ’08), yields a collection of injective trapdoor functions that is secure under the above mentioned natural correlated products. Although we eventually base security under correlated products on 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.

We strengthen the foundations of deterministic public-key encryption via definitional equivalences and standard-model constructs based on general assumptions. Specifically we consider seven notions of privacy for deterministic encryption, including six forms of semantic security and an indistinguishability notion, and show them all equivalent. We then present a deterministic scheme for the secure encryption of uniformly and independently distributed messages based solely on the existence of trapdoor one-way permutations. We show a generalization of the construction that allows secure deterministic encryption of independent high-entropy messages. Finally we show relations between deterministic and standard (randomized) encryption.

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

Abstract. Public-key encryption schemes rely for their IND-CPA security on per-message fresh randomness. In practice, randomness may be of poor quality for a variety of reasons, leading to failure of the schemes. Expecting the systems to improve is unrealistic. What we show in this paper is that we can, instead, improve the cryptography to offset the lack of possible randomness. We provide public-key encryption schemes that achieve IND-CPA security when the randomness they use is of high quality, but, when the latter is not the case, rather than breaking completely, they achieve a weaker but still useful notion of security that we call IND-CDA. This hedged public-key encryption provides the best possible security guarantees in the face of bad randomness. We provide simple RO-based ways to make in-practice IND-CPA schemes hedge secure with minimal software changes. We also provide non-RO model schemes relying on lossy trapdoor functions (LTDFs) and techniques from deterministic encryption. They achieve adaptive security by establishing and exploiting the anonymity of LTDFs which we believe is of independent interest. 1

We initiate the cryptographic study of order-preserving symmetric encryption (OPE), a primitive suggested in the database community by Agrawal et al. (SIGMOD ’04) for allowing efficient range queries on encrypted data. Interestingly, we first show that a straightforward relaxation of standard security notions for encryption such as indistinguishability against chosen-plaintext attack (IND-CPA) is unachievable by a practical OPE scheme. Instead, we propose a security notion in the spirit of pseudorandom functions (PRFs) and related primitives asking that an OPE scheme look “as-random-as-possible ” subject to the order-preserving constraint. We then design an efficient OPE scheme and prove its security under our notion based on pseudorandomness of an underlying blockcipher. Our construction is based on a natural relation we uncover between a random order-preserving function and the hypergeometric probability distribution. In particular, it makes black-box use of an efficient sampling algorithm for the latter. 1

In this work, we show that strong leakage resilience for cryptosystems with advanced functionalities can be obtained quite naturally within the methodology of dual system encryption, recently introduced by Waters. We demonstrate this concretely by providing fully secure IBE, HIBE, and ABE systems which are resilient to bounded leakage from each of many secret keys per user, as well as many master keys. This can be realized as resilience against continual leakage if we assume keys are periodically updated and no (or logarithmic) leakage is allowed during the update process. Our systems are obtained by applying a simple modification to previous dual system encryption constructions: essentially this provides a generic tool for making dual system encryption schemes leakage-resilient. 1

Abstract. Format-preserving encryption (FPE) encrypts a plaintext of some specified format into a ciphertext of identical format—for example, encrypting a valid credit-card number into a valid creditcard number. The problem has been known for some time, but it has lacked a fully general and rigorous treatment. We provide one, starting off by formally defining FPE and security goals for it. We investigate the natural approach for achieving FPE on complex domains, the “rank-then-encipher ” approach, and explore what it can and cannot do. We describe two flavors of unbalanced Feistel networks that can be used for achieving FPE, and we prove new security results for each. We revisit the cycle-walking approach for enciphering on a non-sparse subset of an encipherable domain, showing that the timing information that may be divulged by cycle walking is not a damaging thing to leak. 1

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.