STP-LWE: A Variant of Learning with Error for a Flexible Encryption

We construct a flexible lattice based scheme based on semitensor product learning with errors (STP-LWE), which is a variant of learning with errors problem. We have proved that STP-LWE is hard when LWE is hard. Our scheme is proved to be secure against indistinguishable chosen message attacks, and it can achieve a balance between the security and efficiency in the hierarchical encryption systems. In addition, our scheme is almost as efficient as the dual encryption in GPV08

Key-Agreement with Perfect Completeness from Random Oracles

In the Random Oracle Model (ROM) all parties have oracle access to a common random function, and the parties are limited in the number of queries they can make to the oracle. The Merkle’s Puzzles protocol, introduced by Merkle [CACM ’78], is a key-agreement protocol in the ROM with a quadratic gap between the query complexity of the honest parties and the eavesdropper. This quadratic gap is known to be optimal, by the works of Impagliazzo and Rudich [STOC ’89] and Barak and Mahmoody [Crypto ’09]. When the oracle function is injective or a permutation, Merkle’s Puzzles has perfect completeness. That is, it is certain that the protocol results in agreement between the parties. However, without such an assumption on the random function, there is a small error probability, and the parties may end up holding different keys. This fact raises the question: Is there a key-agreement protocol with perfect completeness and super-linear security in the ROM? In this paper we give a positive answer to the above question, showing that changes to the query distribution of the parties in Merkle’s Puzzles, yield a protocol with perfect completeness and roughly the same security

Homomorphic Encryption and the Approximate GCD Problem

With the advent of cloud computing, everyone from Fortune 500 businesses to personal consumers to the US government is storing massive amounts of sensitive data in service centers that may not be trustworthy. It is of vital importance to leverage the benefits of storing data in the cloud while simultaneously ensuring the privacy of the data. Homomorphic encryption allows one to securely delegate the processing of private data. As such, it has managed to hit the sweet spot of academic interest and industry demand. Though the concept was proposed in the 1970s, no cryptosystem realizing this goal existed until Craig Gentry published his PhD thesis in 2009. In this thesis, we conduct a study of the two main methods for construction of homomorphic encryption schemes along with functional encryption and the hard problems upon which their security is based. These hard problems include the Approximate GCD problem (A-GCD), the Learning With Errors problem (LWE), and various lattice problems. In addition, we discuss many of the proposed and in some cases implemented practical applications of these cryptosystems. Finally, we focus on the Approximate GCD problem (A-GCD). This problem forms the basis for the security of Gentry\u27s original cryptosystem but has not yet been linked to more standard cryptographic primitives. After presenting several algorithms in the literature that attempt to solve the problem, we introduce some new algorithms to attack the problem

On Public Key Encryption from Noisy Codewords

Several well-known public key encryption schemes, including those of Alekhnovich (FOCS 2003), Regev (STOC 2005), and Gentry, Peikert and Vaikuntanathan (STOC 2008), rely on the conjectured intractability of inverting noisy linear encodings. These schemes are limited in that they either require the underlying field to grow with the security parameter, or alternatively they can work over the binary field but have a low noise entropy that gives rise to sub-exponential attacks. Motivated by the goal of efficient public key cryptography, we study the possibility of obtaining improved security over the binary field by using different noise distributions. Inspired by an abstract encryption scheme of Micciancio (PKC 2010), we consider an abstract encryption scheme that unifies all the three schemes mentioned above and allows for arbitrary choices of the underlying field and noise distributions. Our main result establishes an unexpected connection between the power of such encryption schemes and additive combinatorics. Concretely, we show that under the ``approximate duality conjecture from additive combinatorics (Ben-Sasson and Zewi, STOC 2011), every instance of the abstract encryption scheme over the binary field can be attacked in time $2^{O(\sqrt{n})}$, where $n$ is the maximum of the ciphertext size and the public key size (and where the latter excludes public randomness used for specifying the code). On the flip side, counter examples to the above conjecture (if false) may lead to candidate public key encryption schemes with improved security guarantees. We also show, using a simple argument that relies on agnostic learning of parities (Kalai, Mansour and Verbin, STOC 2008), that any such encryption scheme can be {\em unconditionally} attacked in time $2^{O(n/\log n)}$, where $n$ is the ciphertext size. Combining this attack with the security proof of Regev\u27s cryptosystem, we immediately obtain an algorithm that solves the {\em learning parity with noise (LPN)} problem in time $2^{O(n/\log \log n)}$ using only $n^{1+\epsilon}$ samples, reproducing the result of Lyubashevsky (Random 2005) in a conceptually different way. Finally, we study the possibility of instantiating the abstract encryption scheme over constant-size rings to yield encryption schemes with no decryption error. We show that over the binary field decryption errors are inherent. On the positive side, building on the construction of matching vector families (Grolmusz, Combinatorica 2000; Efremenko, STOC 2009; Dvir, Gopalan and Yekhanin, FOCS 2010), we suggest plausible candidates for secure instances of the framework over constant-size rings that can offer perfectly correct decryption

Decryption Failure Attacks on Post-Quantum Cryptography

This dissertation discusses mainly new cryptanalytical results related to issues of securely implementing the next generation of asymmetric cryptography, or Public-Key Cryptography (PKC).PKC, as it has been deployed until today, depends heavily on the integer factorization and the discrete logarithm problems.Unfortunately, it has been well-known since the mid-90s, that these mathematical problems can be solved due to Peter Shor's algorithm for quantum computers, which achieves the answers in polynomial time.The recently accelerated pace of R&D towards quantum computers, eventually of sufficient size and power to threaten cryptography, has led the crypto research community towards a major shift of focus.A project towards standardization of Post-quantum Cryptography (PQC) was launched by the US-based standardization organization, NIST. PQC is the name given to algorithms designed for running on classical hardware/software whilst being resistant to attacks from quantum computers.PQC is well suited for replacing the current asymmetric schemes.A primary motivation for the project is to guide publicly available research toward the singular goal of finding weaknesses in the proposed next generation of PKC.For public key encryption (PKE) or digital signature (DS) schemes to be considered secure they must be shown to rely heavily on well-known mathematical problems with theoretical proofs of security under established models, such as indistinguishability under chosen ciphertext attack (IND-CCA).Also, they must withstand serious attack attempts by well-renowned cryptographers both concerning theoretical security and the actual software/hardware instantiations.It is well-known that security models, such as IND-CCA, are not designed to capture the intricacies of inner-state leakages.Such leakages are named side-channels, which is currently a major topic of interest in the NIST PQC project.This dissertation focuses on two things, in general:1) how does the low but non-zero probability of decryption failures affect the cryptanalysis of these new PQC candidates?And 2) how might side-channel vulnerabilities inadvertently be introduced when going from theory to the practice of software/hardware implementations?Of main concern are PQC algorithms based on lattice theory and coding theory.The primary contributions are the discovery of novel decryption failure side-channel attacks, improvements on existing attacks, an alternative implementation to a part of a PQC scheme, and some more theoretical cryptanalytical results