A Non-commutative Cryptosystem Based on Quaternion Algebras

We propose BQTRU, a non-commutative NTRU-like cryptosystem over quaternion algebras. This cryptosystem uses bivariate polynomials as the underling ring. The multiplication operation in our cryptosystem can be performed with high speed using quaternions algebras over finite rings. As a consequence, the key generation and encryption process of our cryptosystem is faster than NTRU in comparable parameters. Typically using Strassen's method, the key generation and encryption process is approximately $16/7$ times faster than NTRU for an equivalent parameter set. Moreover, the BQTRU lattice has a hybrid structure that makes inefficient standard lattice attacks on the private key. This entails a higher computational complexity for attackers providing the opportunity of having smaller key sizes. Consequently, in this sense, BQTRU is more resistant than NTRU against known attacks at an equivalent parameter set. Moreover, message protection is feasible through larger polynomials and this allows us to obtain the same security level as other NTRU-like cryptosystems but using lower dimensions.Comment: Submitted for possible publicatio

NTRU software implementation for constrained devices

The NTRUEncrypt is a public-key cryptosystem based on the shortest vector problem. Its main characteristics are the low memory and computational requirements while providing a high security level. This document presents an implementation and optimization of the NTRU public-key cryptosys- tem for constrained devices. Speci cally the NTRU cryptosystem has been implemented on the ATMega128 and the ATMega163 microcontrollers. This has turned in a major e ort in order to reduce the consumption of memory and op- timize the computational resources. The di erent resulting optimizations have been compared and evaluated throught the AVR Studio 4 [1]. The nal outcome has also been compared with other published public-key cryptosystems as RSA or ECC showing the great performance NTRUEncrypt is able to deliver at a surprising very low cost

Notes on Lattice-Based Cryptography

Asymmetrisk kryptering er avhengig av antakelsen om at noen beregningsproblemer er vanskelige å løse. I 1994 viste Peter Shor at de to mest brukte beregningsproblemene, nemlig det diskrete logaritmeproblemet og primtallsfaktorisering, ikke lenger er vanskelige å løse når man bruker en kvantedatamaskin. Siden den gang har forskere jobbet med å finne nye beregningsproblemer som er motstandsdyktige mot kvanteangrep for å erstatte disse to. Gitterbasert kryptografi er forskningsfeltet som bruker kryptografiske primitiver som involverer vanskelige problemer definert på gitter, for eksempel det korteste vektorproblemet og det nærmeste vektorproblemet. NTRU-kryptosystemet, publisert i 1998, var et av de første som ble introdusert på dette feltet. Problemet Learning With Error (LWE) ble introdusert i 2005 av Regev, og det regnes nå som et av de mest lovende beregningsproblemene som snart tas i bruk i stor skala. Å studere vanskelighetsgraden og å finne nye og raskere algoritmer som løser den, ble et ledende forskningstema innen kryptografi. Denne oppgaven inkluderer følgende bidrag til feltet: - En ikke-triviell reduksjon av Mersenne Low Hamming Combination Search Problem, det underliggende problemet med et NTRU-lignende kryptosystem, til Integer Linear Programming (ILP). Særlig finner vi en familie av svake nøkler. - En konkret sikkerhetsanalyse av Integer-RLWE, en vanskelig beregningsproblemvariant av LWE, introdusert av Gu Chunsheng. Vi formaliserer et meet-in-the-middle og et gitterbasert angrep for denne saken, og vi utnytter en svakhet ved parametervalget gitt av Gu, for å bygge et forbedret gitterbasert angrep. - En forbedring av Blum-Kalai-Wasserman-algoritmen for å løse LWE. Mer spesifikt, introduserer vi et nytt reduksjonstrinn og en ny gjetteprosedyre til algoritmen. Disse tillot oss å utvikle to implementeringer av algoritmen, som er i stand til å løse relativt store LWE-forekomster. Mens den første effektivt bare bruker RAM-minne og er fullt parallelliserbar, utnytter den andre en kombinasjon av RAM og disklagring for å overvinne minnebegrensningene gitt av RAM. - Vi fyller et tomrom i paringsbasert kryptografi. Dette ved å gi konkrete formler for å beregne hash-funksjon til G2, den andre gruppen i paringsdomenet, for Barreto-Lynn-Scott-familien av paringsvennlige elliptiske kurver.Public-key Cryptography relies on the assumption that some computational problems are hard to solve. In 1994, Peter Shor showed that the two most used computational problems, namely the Discrete Logarithm Problem and the Integer Factoring Problem, are not hard to solve anymore when using a quantum computer. Since then, researchers have worked on finding new computational problems that are resistant to quantum attacks to replace these two. Lattice-based Cryptography is the research field that employs cryptographic primitives involving hard problems defined on lattices, such as the Shortest Vector Problem and the Closest Vector Problem. The NTRU cryptosystem, published in 1998, was one of the first to be introduced in this field. The Learning With Error (LWE) problem was introduced in 2005 by Regev, and it is now considered one of the most promising computational problems to be employed on a large scale in the near future. Studying its hardness and finding new and faster algorithms that solve it became a leading research topic in Cryptology. This thesis includes the following contributions to the field: - A non-trivial reduction of the Mersenne Low Hamming Combination Search Problem, the underlying problem of an NTRU-like cryptosystem, to Integer Linear Programming (ILP). In particular, we find a family of weak keys. - A concrete security analysis of the Integer-RLWE, a hard computational problem variant of LWE introduced by Gu Chunsheng. We formalize a meet-in-the-middle attack and a lattice-based attack for this case, and we exploit a weakness of the parameters choice given by Gu to build an improved lattice-based attack. - An improvement of the Blum-Kalai-Wasserman algorithm to solve LWE. In particular, we introduce a new reduction step and a new guessing procedure to the algorithm. These allowed us to develop two implementations of the algorithm that are able to solve relatively large LWE instances. While the first one efficiently uses only RAM memory and is fully parallelizable, the second one exploits a combination of RAM and disk storage to overcome the memory limitations given by the RAM. - We fill a gap in Pairing-based Cryptography by providing concrete formulas to compute hash-maps to G2, the second group in the pairing domain, for the Barreto-Lynn-Scott family of pairing-friendly elliptic curves.Doktorgradsavhandlin

Aspects of hardware methodologies for the NTRU public-key cryptosystem

Cryptographic algorithms which take into account requirements for varying levels of security and reduced power consumption in embedded devices are now receiving additional attention. The NTRUEncrypt algorithm has been shown to provide certain advantages when designing low power and resource constrained systems, while still providing comparable security levels to higher complexity algorithms. The research presented in this thesis starts with an examination of the general NTRUEncrypt system, followed by a more practical examination with respect to the IEEE 1363.1 draft standard. In contrast to previous research, the focus is shifted away from specific optimizations but rather provides a study of many of the recommended practices and suggested optimizations with particular emphasis on polynomial arithmetic and parameter selection. Various methods are examined for storing, inverting and multiplying polynomials used in the system. Recommendations for algorithm and parameter selection are made regarding implementation in software and hardware with respect to the resources available. Although the underlying mathematical principles have not been significantly questioned, stable recommended practices are still being developed for the NTRUEncrypt system. As a further complication, recommended optimizations have come from various researchers and have been split between hardware and software implementations. In this thesis, a generic VHDL model is presented, based on the IEEE 1363.1 draft standard, which is designed for adaptation to software or hardware implementation while providing flexibility for changes in recommended practices

NTRU software implementation for constrained devices

The NTRUEncrypt is a public-key cryptosystem based on the shortest vector problem. Its main characteristics are the low memory and computational requirements while providing a high security level. This document presents an implementation and optimization of the NTRU public-key cryptosys- tem for constrained devices. Speci cally the NTRU cryptosystem has been implemented on the ATMega128 and the ATMega163 microcontrollers. This has turned in a major e ort in order to reduce the consumption of memory and op- timize the computational resources. The di erent resulting optimizations have been compared and evaluated throught the AVR Studio 4 [1]. The nal outcome has also been compared with other published public-key cryptosystems as RSA or ECC showing the great performance NTRUEncrypt is able to deliver at a surprising very low cost

Survey on Lightweight Primitives and Protocols for RFID in Wireless Sensor Networks

The use of radio frequency identification (RFID) technologies is becoming widespread in all kind of wireless network-based applications. As expected, applications based on sensor networks, ad-hoc or mobile ad hoc networks (MANETs) can be highly benefited from the adoption of RFID solutions. There is a strong need to employ lightweight cryptographic primitives for many security applications because of the tight cost and constrained resource requirement of sensor based networks. This paper mainly focuses on the security analysis of lightweight protocols and algorithms proposed for the security of RFID systems. A large number of research solutions have been proposed to implement lightweight cryptographic primitives and protocols in sensor and RFID integration based resource constraint networks. In this work, an overview of the currently discussed lightweight primitives and their attributes has been done. These primitives and protocols have been compared based on gate equivalents (GEs), power, technology, strengths, weaknesses and attacks. Further, an integration of primitives and protocols is compared with the possibilities of their applications in practical scenarios

NFC Secure Payment and Verification Scheme with CS E-Ticket

As one of the most important techniques in IoT, NFC (Near Field Communication) is more interesting than ever. NFC is a short-range, high-frequency communication technology well suited for electronic tickets, micropayment, and access control function, which is widely used in the financial industry, traffic transport, road ban control, and other fields. However, NFC is becoming increasingly popular in the relevant field, but its secure problems, such as man-in-the-middle-attack and brute force attack, have hindered its further development. To address the security problems and specific application scenarios, we propose a NFC mobile electronic ticket secure payment and verification scheme in the paper. The proposed scheme uses a CS E-Ticket and offline session key generation and distribution technology to prevent major attacks and increase the security of NFC. As a result, the proposed scheme can not only be a good alternative to mobile e-ticket system but also be used in many NFC fields. Furthermore, compared with other existing schemes, the proposed scheme provides a higher security

A Lightweight Implementation of NTRUEncrypt for 8-bit AVR Microcontrollers

Introduced in 1996, NTRUEncrypt is not only one of the earliest but also one of the most scrutinized lattice-based cryptosystems and a serious contender in NIST’s ongoing Post-Quantum Cryptography (PQC) standardization project. An important criterion for the assessment of candidates is their computational cost in various hardware and software environments. This paper contributes to the evaluation of NTRUEncrypt on the ATmega class of AVR microcontrollers, which belongs to the most popular 8-bit platforms in the embedded domain. More concretely, we present AvrNtru, a carefully-optimized implementation of NTRUEncrypt that we developed from scratch with the goal of achieving high performance and resistance to timing attacks. AvrNtru complies with version 3.3 of the EESS#1 specification and supports recent product-form parameter sets like ees443ep1, ees587ep1, and ees743ep1. A full encryption operation (including mask generation and blinding- polynomial generation) using the ees443ep1 parameters takes 834,272 clock cycles on an ATmega1281 microcontroller; the decryption is slightly more costly and has an execution time of 1,061,683 cycles. When choosing the ees743ep1 parameters to achieve a 256-bit security level, 1,539,829 clock cycles are cost for encryption and 2,103,228 clock cycles for decryption. We achieved these results thanks to a novel hybrid technique for multiplication in truncated polynomial rings where one of the operands is a sparse ternary polynomial in product form. Our hybrid technique is inspired by Gura et al’s hybrid method for multiple-precision integer multiplication (CHES 2004) and takes advantage of the large register file of the AVR architecture to minimize the number of load instructions. A constant-time multiplication in the ring specified by the ees443ep1 parameters requires only 210,827 cycles, which sets a new speed record for the arithmetic component of a lattice-based cryptosystem on an 8-bit microcontroller