Ring-LWE:applications to cryptography and their efficient realization

© Springer International Publishing AG 2016. The persistent progress of quantum computing with algorithms of Shor and Proos and Zalka has put our present RSA and ECC based public key cryptosystems at peril. There is a flurry of activity in cryptographic research community to replace classical cryptography schemes with their post-quantum counterparts. The learning with errors problem introduced by Oded Regev offers a way to design secure cryptography schemes in the post-quantum world. Later for efficiency LWE was adapted for ring polynomials known as Ring-LWE. In this paper we discuss some of these ring-LWE based schemes that have been designed. We have also drawn comparisons of different implementations of those schemes to illustrate their evolution from theoretical proposals to practically feasible schemes

Post-contrast FLAIR imaging in a patient with posterior reversible encephalopathy syndrome (PRES).

We herein present a case of delayed enhancement of CSF on fluidattenuated inversion recovery (FLAIR) imaging in a patient with posterior reversible encephalopathy syndrome (PRES). In our case despite the settled clinical setting of PRES initial MR scan was negative and on repeated FLAIR imaging increased CSF signal intensity was more conspicuous than subtle cortical involvement

Improving Speed of Dilithium’s Signing Procedure

Dilithium is a round 2 candidate for digital signature schemes in NIST initiative for post-quantum cryptographic schemes. Since Dilithium is built upon the “Fiat Shamir with Aborts” framework, its signing procedure performs rejection sampling of its signatures to ensure they do not leak information about the secret key. Thus, the signing procedure is iterative in nature with a number of rejected iterations, which serve as unnecessary overheads hampering its overall performance. As a first contribution, we propose an optimization that reduces the computations in the rejected iterations through early-evaluation of the conditional checks. This allows to perform an early detection of the rejection condition and reject a given iteration as early as possible. We also incorporate a number of standard optimizations such as unrolling and inlining to further improve the speed of the signing procedure. We incorporate and evaluate our optimizations over the software implementation of Dilithium on both the Intel Core i5-4460 and ARM Cortex-M4 CPUs. As a second contribution, we identify opportunities to present a more refined evaluation of Dilithium’s signing procedure in several scenarios where pre-computations can be carried out. We also evaluate the performance of our optimizations and the memory requirements for the pre-computed intermediates in the considered scenarios. We could yield speed-ups in the range of 6% upto 35%, considering all the aforementioned scenarios, thus presenting the fastest software implementation of Dilithium till date

Design and Reliability Performance Evaluation of Network Coding Schemes for Lossy Wireless Networks

This thesis investigates lossy wireless networks, which are wireless communication networks consisting of lossy wireless links, where the packet transmission via a lossy wireless link is successful with a certain value of probability. In particular, this thesis analyses all-to-all broadcast in lossy wireless networks, where every node has a native packet to transmit to all other nodes in the network. A challenge of all-to-all broadcast in lossy wireless networks is the reliability, which is defined as the probability that every node in the network successfully obtains a copy of the native packets of all other nodes. In this thesis, two novel network coding schemes are proposed, which are the neighbour network coding scheme and the random neighbour network coding scheme. In the two proposed network coding schemes, a node may perform a bit-wise exclusive or (XOR) operation to combine the native packet of itself and the native packet of its neighbour, called the coding neighbour, into an XOR coded packet. The reliability of all-to-all broadcast under both the proposed network coding schemes is investigated analytically using Markov chains. It is shown that the reliability of all-to-all broadcast can be improved considerably by employing the proposed network coding schemes, compared with non-coded networks with the same link conditions, i.e. same probabilities of successful packet transmission via wireless channels. Further, the proposed schemes take the link conditions of each node into account to maximise the reliability of a given network. To be more precise, the first scheme proposes the optimal coding neighbour selection method while the second scheme introduces a tuning parameter to control the probability that a node performs network coding at each transmission. The observation that channel condition can have a significant impact on the performance of network coding schemes is expected to be applicable to other network coding schemes for lossy wireless networks

Number Not Used Once - Practical fault attack on pqm4 implementations of NIST candidates

In this paper, we demonstrate practical fault attacks over a number of lattice based schemes, in particular NewHope, Kyber, Frodo, Dilithium which are based on the hardness of the Learning with Errors (LWE) problem. One of the common traits of all the considered LWE schemes is the use of nonces as domain separators to sample the secret components of the LWE instance. We show that simple faults targeting the usage of nonce can result in a nonce-reuse scenario which allows key recovery and message recovery attacks. To the best of our knowledge, we propose the first practical fault attack on lattice-based Key encapsulation schemes secure in the CCA model. We perform experimental validation of our attack using Electromagnetic fault injection on reference implementations of the aforementioned schemes taken from the pqm4 library, a benchmarking and testing framework for post quantum cryptographic implementations for the ARM Cortex-M4. We use the instruction skip fault model, which is very practical and popular in microcontroller based implementations. Our attack requires to inject a very few number of faults (numbering less than 10 for recommended parameter sets) and can be repeated with a 100% accuracy with our Electromagnetic fault injection setup

An Experimentally Verified Attack on Full Grain-128 Using Dedicated Reconfigurable Hardware

In this paper we describe the first single-key attack which can recover the full key of the full version of Grain-128 for arbitrary keys by an algorithm which is significantly faster than exhaustive search (by a factor of about 238). It is based on a new version of a cube tester, which uses an improved choice of dynamic variables to eliminate the previously made assumption that ten particular key bits are zero. In addition, the new attack is much faster than the previous weak-key attack, and has a simpler key recovery process. Since it is extremely difficult to mathemat-ically analyze the expected behavior of such attacks, we implemented it on RIVYERA, which is a new massively parallel reconfigurable hardware, and tested its main components for dozens of random keys. These tests experimentally verified the correctness and expected complexity of the attack, by finding a very significant bias in our new cube tester for about 7.5 % of the keys we tested. This is the first time that the main compo-nents of a complex analytical attack are successfully realized against a full-size cipher with a special-purpose machine. Moreover, it is also the first attack that truly exploits the configurable nature of an FPGA-based cryptanalytical hardware

Short, Invertible Elements in Partially Splitting Cyclotomic Rings and Applications to Lattice-Based Zero-Knowledge Proofs

When constructing practical zero-knowledge proofs based on the hardness of the Ring-LWE or the Ring-SIS problems over polynomial rings $Z_p[X]/(X^n+1)$, it is often necessary that the challenges come from a set $\mathcal{C}$ that satisfies three properties: the set should be large (around $2^{256}$), the elements in it should have small norms, and all the non-zero elements in the difference set $\mathcal{C}-\mathcal{C}$ should be invertible. The first two properties are straightforward to satisfy, while the third one requires us to make efficiency compromises. We can either work over rings where the polynomial $X^n+1$ only splits into two irreducible factors modulo $p$, which makes the speed of the multiplication operation in the ring sub-optimal; or we can limit our challenge set to polynomials of smaller degree, which requires them to have (much) larger norms. In this work we show that one can use the optimal challenge sets $\mathcal{C}$ and still have the polynomial $X^n+1$ split into more than two factors. This comes as a direct application of our more general result that states that all non-zero polynomials with ``small\u27\u27 coefficients in the cyclotomic ring $Z_p[X]/(\Phi_m(X))$ are invertible (where ``small\u27\u27 depends on the size of $p$ and how many irreducible factors the $m^{th}$ cyclotomic polynomial $\Phi_m(X)$ splits into). We furthermore establish sufficient conditions for $p$ under which $\Phi_m(X)$ will split in such fashion. For the purposes of implementation, if the polynomial $X^n+1$ splits into $k$ factors, we can run FFT for $\log{k}$ levels until switching to Karatsuba multiplication. Experimentally, we show that increasing the number of levels from one to three or four results in a speedup by a factor of $\approx 2$ -- $3$. We point out that this improvement comes completely for free simply by choosing a modulus $p$ that has certain algebraic properties. In addition to the speed improvement, having the polynomial split into many factors has other applications -- e.g. when one embeds information into the Chinese Remainder representation of the ring elements, the more the polynomial splits, the more information one can embed into an element

No Place to Hide: Contactless Probing of Secret Data on FPGAs

Field Programmable Gate Arrays (FPGAs) have been the target of different physical attacks in recent years. Many different countermeasures have already been integrated into these devices to mitigate the existing vulnerabilities. However, there has not been enough attention paid to semi-invasive attacks from the IC backside due to the following reasons. First, the conventional semi-invasive attacks from the IC backside --- such as laser fault injection and photonic emission analysis --- cannot be scaled down without further effort to the very latest nanoscale technologies of modern FPGAs and programmable SoCs. Second, the more advanced solutions for secure storage, such as controlled Physically Unclonable Functions (PUFs), make the conventional memory-readout techniques almost impossible. In this paper, however, novel approaches have been explored: Attacks based on Laser Voltage Probing (LVP) and its derivatives, as commonly used in Integrated Circuit (IC) debug for nanoscale low voltage technologies, are successfully launched against a $60$ nanometer technology FPGA. We discuss how these attacks can be used to break modern bitstream encryption implementations. Our attacks were carried out on a Proof-of-Concept PUF-based key generation implementation. To the best of our knowledge this is the first time that LVP is used to perform an attack on secure ICs

Simple Lattice Trapdoor Sampling from a Broad Class of Distributions

International audienceAt the center of many lattice-based constructions is an algorithm that samples a short vector s, satisfying [A|AR − HG]s = t mod q where A, AR, H, G are public matrices and R is a trapdoor. Although the algorithm crucially relies on the knowledge of the trapdoor R to perform this sampling efficiently, the distribution it outputs should be independent of R given the public values. We present a new, simple algorithm for performing this task. The main novelty of our sampler is that the distribution of s does not need to be Gaussian, whereas all previous works crucially used the properties of the Gaussian distribution to produce such an s. The advantage of using a non-Gaussian distribution is that we are able to avoid the high-precision arithmetic that is inherent in Gaussian sampling over arbitrary lattices. So while the norm of our output vector s is on the order of √ n to n-times larger (the representation length, though, is only a constant factor larger) than in the samplers of Gentry, Peikert, Vaikuntanathan (STOC 2008) and Micciancio, Peikert (EUROCRYPT 2012), the sampling itself can be done very efficiently. This provides a useful time/output trade-off for devices with constrained computing power. In addition, we believe that the conceptual simplicity and generality of our algorithm may lead to it finding other applications