In this chapter we describe some of the recent progress in lattice-based cryptography. Lattice-based cryptographic constructions hold a great promise for post-quantum cryptography, as they enjoy very strong security proofs based on worst-case hardness, relatively efficient implementations, as well as great simplicity. In addition, lattice-based cryptography is believed to be secure against quantum computers. Our focus here

Abstract. The prospect of outsourcing an increasing amount of data storage and management to cloud services raises many new privacy concerns for individuals and businesses alike. The privacy concerns can be satisfactorily addressed if users encrypt the data they send to the cloud. If the encryption scheme is homomorphic, the cloud can still perform meaningful computations on the data, even though it is encrypted. In fact, we now know a number of constructions of fully homomorphic encryption schemes that allow arbitrary computation on encrypted data. In the last two years, solutions for fully homomorphic encryption have been proposed and improved upon, but it is hard to ignore the elephant in the room, namely efficiency – can homomorphic encryption ever be efficient enough to be practical? Certainly, it seems that all known fully homomorphic encryption schemes have a long way to go before they can be used in practice. Given this state of affairs, our contribution is two-fold. First, we exhibit a number of real-world applications, in the medical, financial, and the advertising domains, which require only that the encryption scheme is “somewhat ” homomorphic. Somewhat homomorphic encryption schemes, which support a limited number of homomorphic operations, can be much faster, and more compact than fully homomorphic encryption schemes. Secondly, we show a proof-of-concept implementation of the recent somewhat homomorphic encryption scheme of Brakerski and Vaikuntanathan, whose security relies on the “ring learning with errors ” (Ring LWE) problem. The system is very efficient, and has reasonably short ciphertexts. Our unoptimized implementation in magma enjoys comparable efficiency to even optimized pairing-based schemes with the same level of security and homomorphic capacity. We also show a number of application-specific optimizations to the encryption scheme, most notably the ability to convert between different message encodings in a ciphertext.

Abstract. We demonstrate how the framework that is used for creating efficient number-theoretic ID and signature schemes can be transferred into the setting of lattices. This results in constructions of the most efficient to-date identification and signature schemes with security based on the worst-case hardness of problems in ideal lattices. In particular, our ID scheme has communication complexity of around 65, 000 bits and the length of the signatures produced by our signature scheme is about 50, 000 bits. All prior lattice-based identification schemes required on the order of millions of bits to be transferred, while all previous lattice-based signature schemes were either stateful, too inefficient, or produced signatures whose lengths were also on the order of millions of bits. The security of our identification scheme is based on the hardness of finding the approximate shortest vector to within a factor of Õ(n2) in the standard model, while the security of the signature scheme is based on the same assumption in the random oracle model. Our protocols are very efficient, with all operations requiring Õ(n) time. We also show that the technique for constructing our lattice-based schemes can be used to improve certain number-theoretic schemes. In particular, we are able to shorten the length of the signatures that are produced by Girault’s factoring-based digital signature scheme ([10, 11, 31]). 1

Abstract. RSA-FDH and many other schemes secure in the Random-Oracle Model (ROM) require a hash function with output size larger than standard sizes. We show that the random-oracle instantiations proposed in the literature for such cases are weaker than a random oracle, including the proposals by Bellare and Rogaway from 1993 and 1996, and the ones implicit in IEEE P1363 and PKCS standards: for instance, we obtain a practical preimage attack on BR93 for 1024-bit digests (with complexity less than 2 30). Next, we study the security impact of hash function defects for ROM signatures. As an extreme case, we note that any hash collision would suffice to disclose the master key in the ID-based cryptosystem by Boneh et al. from FOCS ’07, and the secret key in the Rabin-Williams signature for which Bernstein proved tight security at EUROCRYPT ’08. We also remark that collisions can be found as a precomputation for any instantiation of the ROM, and this violates the security definition of the scheme in the standard model. Hence, this gives an example of a natural scheme that is proven secure in the ROM but that in insecure for any instantiation by a single function. Interestingly, for both of these schemes, a slight modification can prevent these attacks, while preserving the ROM security result. We give evidence that in the case of RSA and Rabin/Rabin-Williams, an appropriate PSS padding is more robust than all other paddings known. 1

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We give direct constructions of pseudorandom function (PRF) families based on conjectured hard lattice problems and learning problems. Our constructions are asymptotically efficient and highly parallelizable in a practical sense, i.e., they can be computed by simple, relatively small low-depth arithmetic or boolean circuits (e.g., in NC 1 or even TC 0). In addition, they are the first low-depth PRFs that have no known attack by efficient quantum algorithms. Central to our results is a new “derandomization ” technique for the learning with errors (LWE) problem which, in effect, generates the error terms deterministically. 1 Introduction and Main Results The past few years have seen significant progress in constructing public-key, identity-based, and homomorphic cryptographic schemes using lattices, e.g., [Reg05, PW08, GPV08, Gen09, CHKP10, ABB10a] and many more. Part of their appeal stems from provable worst-case hardness guarantees (starting with the seminal work of Ajtai [Ajt96]), good asymptotic efficiency and parallelism, and apparent resistance to quantum

Abstract. We provide an alternative method for constructing lattice-based digital signatures which does not use the “hash-and-sign ” methodology of Gentry, Peikert, and Vaikuntanathan (STOC 2008). Our resulting signature scheme is secure, in the random oracle model, based on the worst-case hardness of the Õ(n1.5)-SIVP problem in general lattices. The secret key, public key, and the signature size of our scheme are smaller than in all previous instantiations of the hash-and-sign signature, and our signing algorithm is also quite simple, requiring just a few matrix-vector multiplications and rejection samplings. We then also show that by slightly changing the parameters, one can get even more efficient signatures that are based on the hardness of the Learning With Errors problem. Our construction naturally transfers to the ring setting, where the size of the public and secret keys can be significantly shrunk, which results in the most practical to-date provably secure signature scheme based on lattices. 1

Abstract. Building upon a famous result due to Ajtai, we propose a sequence of lattice bases with growing dimension, which can be expected to be hard instances of the shortest vector problem (SVP) and which can therefore be used to benchmark lattice reduction algorithms. The SVP is the basis of security for potentially post-quantum cryptosystems. We use our sequence of lattice bases to create a challenge, which may be helpful in determining appropriate parameters for these schemes.

This report describes the SWIFFTX hash function. It is part of our submission package to the SHA-3 hash function competition. The SWIFFTX compression functions have a simple and mathematically elegant design. This makes them highly amenable to analysis and optimization. In addition, they enjoy two unconventional features: Asymptotic proof of security: it can be formally proved that finding a collision in a randomly-chosen compression function from the SWIFFTX family is at least as hard as finding short vectors in cyclic/ideal lattices in the worst case. High parallelizability: the compression function admits efficient implementations on modern microprocessors. This can be achieved even without relying on multi core capabilities, and is obtained through a novel cryptographic use of the Fast Fourier Transform (FFT). The main building block of SWIFFTX is the SWIFFT family of compression functions, presented in the 2008 workshop on Fast Software Encryption (Lyubashevsky et al., FSE’08). Great care was taken in making sure that SWIFFTX does not inherit the major shortcoming of SWIFFT – linearity – while preserving its provable collision resistance. The SWIFFTX compression function maps 2048 input bits to 520 output bits. The mode of operation that we employ is HAIFA (Biham and Dunkelman, 2007), resulting in a hash function that accepts inputs of any length up to 2 64 − 1 bits, and produces message digests of the SHA-3

Abstract. In this paper we make a first feasibility analysis for implementing lattice reduction algorithms on GPU using CUDA, a programming framework for NVIDIA graphics cards. The enumeration phase of the BKZ lattice reduction algorithm is chosen as a good candidate for massive parallelization on GPU. Given the nature of the problem we gain large speedups compared to previous CPU implementations. Our implementation saves more than 50 % of the time in high lattice dimensions. Among other impacts, this result influences the security of lattice based cryptosystems.