Explaining the role of Scripture in the economy of redemption as it relates to the theological and hermeneutical contributions of David Tracy, Hans Frei, Kevin Vanhoozer and Henri de Lubac

In this thesis, we focus on applying information-theoretic techniques to create more powerful cryptographic primitives. In particular, this work mainly focuses on lossy trapdoor functions as defined by Peikert and Waters [PW08], and lossy encryption [GOS06, KN08, PVW08, BHY09]. This work consists of three main parts.;In the first part of this work, Chapter 3 we present new and general constructions of lossy encryption schemes. By applying results from Eurocrypt '09, we obtain new general constructions of cryptosystems secure against a Selective Opening Adversaries (SOA). Although it was recognized almost twenty years ago that SOA security was important, it was not until the recent breakthrough works of Hofheinz and Bellare, Hofheinz and Yilek that any progress was made on this fundamental problem.;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. · Applying our constructions to well-known cryptosystems (such as Elgamal or Paillier), we obtain selective opening secure commitments and encryptions from the Decisional Diffie-Hellman (DDH), Decisional Composite Residuosity (DCR) and Quadratic Residuosity (QR) assumptions, that are either simpler or more efficient than existing constructions of Bellare, Hofheinz and Yilek. By applying our general results to the Paillier cryptosystem, we obtain the first cryptosystem to achieve simulation-based selective opening security from the DCR assumption. · We provide (indistinguishability and simulation-based) definitions of adaptive chosen-ciphertext security (CCA2) in the selective opening setting and describe the first encryption schemes that provide security in the sense of this definition. In the indistinguishability-based model, we notably achieve short ciphertexts under standard number theoretic assumptions. In the simulation-based security chosen-ciphertext attack scenario, we handle non-adaptive (i.e., CCA1) adversaries and describe the first encryption scheme which is simultaneously CCA1 and SOA-secure.;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 au 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 their work, Peikert and Waters gave constructions of LTDFs from the Decisional Diffie-Hellman (DDH) assumption and lattice assumptions. Boldyreva et al., and Rosen and Segev also gave (identical) efficient constructions of LTDFs from Paillier's Decisional Composite Residuosity (DCR) assumption. The concurrent, independent work of Freeman et al., gives constructions of LTDFs from the d-linear assumption, and slightly lossy trapdoor functions based on the Quadratic Residuosity (QR) assumption. To date, these remain the only known constructions of lossy trapdoor functions.;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 our results to the results of Boldyreva et al. we obtain a construction of deterministic encryption from smooth homomorphic hash proof systems. · Applying our results to the results of Rosen and Segev, we obtain a construction of correlated product secure functions from smooth homomorphic hash proof systems. · 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.;Our main result has a number of interesting corollaries: · Applying the results of Mol and Yilek (PKC '10), we also show that Statistical Oblivious 'Transfer for long strings implies correlated product secure functions and IND-CCA secure encryption. · Statistical OT for long strings a weak form of RDM security. Combining this with our results that lossy encryption is equivalent to OT, where if OT uses randomness shorter than the message so does Lossy Encryption. Thus, our main result also implies an injective one-way trapdoor function from any lossy encryption with short randomness. This is somewhat surprising since injective trapdoor functions are deterministic and, given the trapdoor, allow recovery of a complete inverse, while public-key encryptions are probabilistic and recover only the plaintext and not necessarily the randomness used in the encryption process. Our result corroborates the previous result of Bellare, Halevi, Sahai and Vadhan (CRYPTO '98) showing that IND-CPA secure encryption implies injective one-way trapdoor permutations in the random oracle model. We stress that in our work we do not make use of a random oracle.