Lattice-Free Simplexes in Dimension 4

We use a numerical approach to discover lattice free simplexes in dimension 4 with width at least 3. We follow the methodologies of Mori, Morrison, and Morrison and use a theoretical result proven by Barille, Bernardi, Borisov, and Kantor to conjecture a complete list of empty-lattice simplexes in dimension 4. Similar work was done by Haase and Ziegler, however, using a different approach we were able to both produce more evidence for the conjecture and provide an explicit list of distinct empty-lattice simplexes in dimension 4

Design and development of a quantum circuit to solve the information set decoding problem

LAUREA MAGISTRALENegli ultimi anni i crittosistemi basati su codici lineari sono stati oggetto di studi sempre più approfonditi data la loro maggior resistenza ad attacchi tramite calcolatori quantistici. La sicurezza di questo tipo di crittosistemi si basa sulla difficoltà di ricavare il valore di una parola di codice corretta a partire da una affetta da errore dato un codice lineare con una struttura apparentemente casuale. In questo lavoro abbiamo progettato e implementato diversi circuiti quantistici in grado di risolvere il problema noto come Information Set Decoding, che è attualmente il più efficace tipo di attacco a tali crittosistemi. Basati sull'algoritmo di Grover, gli algoritmi quantistici proposti si sono dimostrati in grado di identificare l'errore originale con un'elevata percentuale di affidabilità, durante la loro validazione tramite simulatore di calcolatore quantistico. Abbiamo esplorato due tipi di attacchi diversi: il primo, basato su un algoritmo di ricerca esaustiva tradizionale, è puramente quantistico; il secondo, basato sull'algoritmo di Lee-Brickell, è un algoritmo ibrido classico-quantistico. In entrambi i casi, sono state utilizzate e comparate modalità di esecuzione diverse, dimostrando come un'attenta preparazione dello stato iniziale del sistema possa ridurre drasticamente il numero di iterazioni rispetto all'utilizzo di una versione base dell'algoritmo di Grover. In questo lavoro abbiamo inoltre fornito una misura quantitativa della complessità di calcolo di entrambi gli algoritmi proposti in termini di numero di quantum gates e numero complessivo di qubit.Cryptosystems based on linear codes are gaining momentum due to their stronger resistance to quantum attacks. They rely on the hardness of finding a minimum-weight codeword in a large linear code with an apparently random structure. In this work we designed and implemented several quantum circuits to specifically solve the Information Set Decoding problem, which is currently the most effective attack against code-based cryptoschemes. Relying on Grover's algorithm, the proposed algorithms were shown capable of effectively recover the original error vector simulating the computation of a quantum computer. Both an exhaustive search and a variant of Lee-Brickell's algorithm are proposed, with the former relying only on a quantum circuit and the latter using a hybrid classic-quantum approach. In both cases, two variants have been analyzed and compared, showing how a proper preparation of the initial state of the system can drastically reduce the number of iterations with respect to the uniform superposition of the classic Grover's algorithm. We provide, for the proposed algorithms, a quantitative evaluation of their computational complexity in terms of the number of involved quantum gates and required storage in qubits

Quantum circuits for information set decoding : quantum cryptanalysis of code-based cryptosystems

DOTTORATOL’avvento del calcolo quantistico rappresenta una profonda sfida alla sicurezza dei sistemi crittografici basati su chiavi pubbliche ampiamente utilizzati. Tali sistemi fanno affidamento sulla complessità computazionale di operazioni come la fattorizzazione di grandi numeri interi o la risoluzione di logaritmi discreti. Per affrontare questa sfida, istituzioni di grande prestigio come l’ufficio nazionale di standard e tecnologia degli Stati Uniti (NIST), l’associazione Cinese per la ricerca crittografica (CACR) e l’istituto Europeo per le norme delle telecomunicazioni (ETSI), sono impegnate nella formulazione di primitive crittografiche in grado di resistere sia agli attacchi classici che a quelli quantistici. Questi innovativi sistemi crittografici, noti collettivamente come crittosistemi post-quantistici, sono al centro degli sforzi di standardizzazione. Tra i principali contendenti in questo sforzo di standardizzazione emergono i crittosistemi basati su codici lineari, che basano la loro sicurezza sulla complessità computazionale del problema di decodifica della sindrome (SDP). Il SDP è definito come il compito di recuperare un vettore di errori a partire dalla matrice di controllo di parità di un codice di correzione di errori lineare a blocchi generato casualmente, e della sindrome dell’errore calcolata attraverso la stessa matrice. Dal punto di vista classico, la tecnica più efficace per risolvere il SDP è il metodo di decodifica dell’insieme di informazioni (ISD), che mostra una complessità esponenziale rispetto ai parametri dei crittosistemi. D’altra parte, le attuali soluzioni quantistiche per il SDP non superano l’accelerazione quadratica offerta dall’adattamento dell’algoritmo di Grover alla tecnica ISD e forniscono solo stime asintotiche dei costi computazionali, nascondendo potenziali fattori costanti e polinomiali non trascurabili. Il fulcro di questo studio ruota intorno alla valutazione precisa della complessità computazionale dei risolutori quantistici per il SDP, adattata ai parametri dei codici proposti per la crittografia post-quantistica. La ricerca svolta mostra circuiti quantistici progettati per modelli di calcolo universali basati su porte logiche quantistiche, che si basano sui fondamenti delle tecniche ISD classiche proposte da Prange, Lee e Brickell. L’analisi si estende sia a soluzioni quantistiche complete per il SDP che a metodologie ibride che suddividono efficacemente il carico computazionale tra risorse di calcolo classico e quantistico. Nel corso dello studio, è emersa chiaramente l’efficacia dell’approccio derivante dalla proposta di Prange alla tecnica ISD, in grado di ottenere un miglioramento sostanziale dell’efficienza computazionale. In particolare, si mostra una riduzione sia della profondità dei circuiti quantistici che della metrica profondità per larghezza da 212 a 224. Sorprendentemente, i risultati rivelano che i miglioramenti ottenuti tramite l’approccio ispirato alle idee di Lee e Brickell, che sono state materiliazzati come un algoritmo ibrido classico-quantistico, sono più modesti, variando da 210 a 220 per gli stessi parametri crittografici, contrariamente alle aspettative basate sulle controparti classiche, in cui l’approccio di Lee e Brickell è più efficiente di quello di Prange. Tuttavia, l’approccio ibrido riduce significativamente la dimensione e la profondità dei circuiti quantistici, rendendo le stime più realistiche e agevolando l’esecuzione parallela su piattaforme di calcolo quantistiche separate. L’analisi quantitativa dei costi computazionali porta a una conclusione significativa: tutti i crittosistemi basati su codici esaminati da istituzioni di grande prestigio come il NIST, in particolare BIKE, HQC e Classic McEliece, superano inequivocabilmente la soglia predefinita per la complessità computazionale. In altre parole, questi crittosistemi si rivelano computazionalmente più esigenti rispetto ai corrispondenti cifrari simmetrici con chiavi di dimensioni adeguate. Tuttavia, lo studio rivela una vulnerabilità critica nel crittosistema Classic McEliece. La parallelizzazione di questo algoritmo su diverse unità di elaborazione quantistiche erode la sua sicurezza, portandola al di sotto della soglia di sicurezza mirata di un fattore di 16. Un contributo accessorio di questa ricerca è la creazione di un insieme di circuiti quantistici capaci di risolvere comuni problemi algebrici e algoritmici, tra cui l’eliminazione di Gauss-Jordan su campi finiti, la classificazione di stringhe binarie e il calcolo del peso di Hamming, che possono rappresentare un notevole interesse indipendente nel campo del calcolo quantistico.The emergence of quantum computing represents a profound challenge to the security of widely-adopted public-key cryptographic systems, which rely on the computational complexity of tasks such as factoring large integers or solving discrete logarithms. To confront this challenge, esteemed organizations like the U.S. National Institute of Standards and Technology (NIST), the Chinese Association for Cryptologic Research (CACR), and the European Telecommunications Standards Institute (ETSI) are actively engaged in the formulation of cryptographic primitives capable of withstanding both classical and quantum attacks. These novel cryptographic systems, collectively termed post-quantum cryptosystems, are at the forefront of standardization efforts. Among the leading contenders in this standardization endeavor, linear code-based cryptosystems, deriving their strength from the computational complexity of the Syndrome Decoding Problem (SDP), have gained significant recognition. The SDP is defined as the task of retrieving an error vector when provided with the parity check matrix of a randomly generated linear block error correction code and the syndrome of the error, as computed through the same matrix. Classically, the most effective technique for solving the SDP is the Information Set Decoding (ISD) method, which, notably, exhibits exponential complexity with respect to the parameters of the cryptosystems. Current quantum approaches to the SDP, on the other hand, do not surpass the quadratic speedup offered by adapting Grover’s algorithm to the ISD technique, and provide only asymptotic estimates of their computational cost, potentially hiding non-trivial constant and polynomial factors. The central focus of this study revolves around the precise computational complexity evaluation of quantum solvers for the SDP, tailored to cryptography-grade code parameters. Our approach introduces quantum circuits designed for universal quantum gate-based computing models, that are build upon the foundations laid by classic ISD techniques. Our scrutiny extends to both complete quantum solutions to the SDP and hybrid methodologies that effectively partition the computational load between classical and quantum computing resources. In our investigation, the approach stemming from Prange’s approach to the ISD technique stands out, as it displays a substantial enhancement in computational efficiency. Notably, it leads to a reduction in both the depth of quantum circuits and the depth-times-width metric by factors ranging from 212 to 224 applicable to concrete cryptography-grade parameters. Surprisingly, our findings reveal that the gains achieved through the approach inspired by Lee and Brickell’s ideas, which materialize as a hybrid classical-quantum algorithm, are somewhat modest. These enhancements range from 210 to 220 for the same cryptographic parameters, a result contrary to expectations based on classical counterparts, where Lee and Brickell’s approach prevails over Prange’s one. However, the hybrid approach substantially reduces the size and depth of quantum circuits, rendering the estimates more realistic and facilitating parallel execution on separate quantum computing platforms. Our quantitative analysis of computational costs brings forth a significant conclusion: all code-based cryptoschemes under the scrutiny of esteemed organizations such as NIST, particularly BIKE, HQC, and McEliece, unequivocally surpass the predefined threshold for computational hardness. Put simply, they prove to be computationally more demanding than the task of breaking a corresponding symmetric cipher with appropriately-sized key lengths. Furthermore, a critical vulnerability in the Classic McEliece cryptoscheme is unveiled. Parallelizing this algorithm across multiple quantum processing units erodes its security, plunging it below the targeted security threshold by a factor of 16. An ancillary contribution of this research is the development of a set of quantum circuits capable of solving common algebraic and algorithmic problems, including Gauss-Jordan Elimination over finite fields, bit string sorting, and Hamming weight computation, which may be of independent interest in the field of quantum computing.DIPARTIMENTO DI ELETTRONICA, INFORMAZIONE E BIOINGEGNERIAComputer Science and Engineering35SILVANO, CRISTINAPIRODDI, LUIG

The Impact Of Using Selected Art Therapy Approaches On Students Diagnosed On The Autism Spectrum (As) Or With Behavioral Disabilities (Bd) In An Elementary School Art Classroom.

Students with autism and behavior disorders frequently have a difficult time communicating with their teachers and peers. Art can serve as a vehicle for communication while supporting strengths and encouraging self-esteem. Yet few art teachers have the specialized training needed to work with students on the spectrum, and we are often times left to our own intuition and devices on how to deal with these students. These students deserve greater individualized education, yet this can present a unique challenge to any classroom art teacher who does not have the training to deal with students who come from these special populations. With greater emphasis placed on core subjects such as math, science and reading, there is even less time to have students fully engage in art making activities that may help these students reach their full human potential. This proposal investigates the potential benefits of using selected art therapy approaches within art education classes, and its impact on students diagnosed on the Autism Spectrum (AS) or with Behavioral Disabilities (BD) in an elementary school setting. Through a multiple case study methodology, this proposal investigated the use of a variety of art therapy approaches being developed for use among the AS and BD population in an art classroom setting. Participants in this study were given an additional art class where art therapy approaches were implemented into the projects within the elementary art classrooms. Art therapy fundamentals included Color Psychology, Line Quality and Placement, all of which were used to help decode hidden messages that the artwork may have contained.The purpose of this paper was to discuss how art therapy, when integrated into an art education classroom, could benefit students diagnosed with Autism Spectrum or Behavioral Disabilities. The findings appear to suggest that when certain art therapy approaches are used with students with AS and BD, there is noticeable improvement in student behaviors, student learning outcomes and student creativity

Fagus sylvatica (European Beech) ID #1191

Location: Wakehurst Condition: Fair Age Class: Maturehttps://digitalcommons.salve.edu/bio140_arboretum/1007/thumbnail.jp

A Quantum Circuit to Execute a Key-Recovery Attack Against the DES and 3DES Block Ciphers

Quantum computing enabled cryptanalytic techniques are able to concretely reduce the security margin of existing cryptographic primitives. While this reduction is only polynomial for symmetric cryptosystems, it still provides a reduction in their security margin. In this work, we propose a detailed quantum circuit designed to cryptanalyze both the Data Encryption Standard (DES) cryptosystem, and its successor Triple-DES (3DES), currently standardized in ISO/IEC 18033-3, and still widely employed in satellite data and bank card encryption. To do so, we introduce the first quantum circuit implementation of the 8 substitution tables (a.k.a. S-boxes), applying a bitslicing strategy, which is currently the most efficient classical combinatorial circuit design in terms of number of two inputs Boolean gates. Secondly, we present the complete quantum circuits required to attack both DES and 3DES leveraging Grover’s algorithm. We provide finite regime, closed form equations, delineating the circuits complexities in terms of the number of qubits, gates, depth and number of qubits multiplied by depth. The complexity analysis is based on two distinct gate sets: a NOT-CNOT-Toffoli (NCT) extended with the Hadamard gate; and the fault-tolerant Clifford+T. Finally, akin to the classical attack to the 3DES, we introduce a meet-in-the-middle strategy relying on an exponential amount of Quantum Random Access Memory. Our findings show that the 3DES with keying option 2, the most widely employed variant of 3DES, can be attacked with a circuit depth of approximately 2^{67} and less than a thousand qubits. This is close to the 2^{64} value suggested by NIST for the depth achievable sequentially by a single quantum computer in a decade. Our technique can be further sped up parallelizing the approach onto multiple devices, pointing to the practicality of cryptanalyzing 3DES in such a scenario

Improving the Efficiency of Quantum Circuits for Information Set Decoding

The NIST Post-Quantum standardization initiative, that entered its fourth round, aims to select asymmetric cryptosystems secure against attacker equipped with a quantum computer. Code-based cryptosystems are a promising option for Post-Quantum Cryptography (PQC), as neither classical nor quantum algorithms provide polynomial time solvers for its underlying hard problems. Indeed, to provide sound alternatives to lattice-based cryptosystems, NIST advanced all round 3 code-based cryptosystems to round 4. We present a complete implementation of a quantum circuit based on the Information Set Decoding (ISD) strategy, the best known one against code-based cryptosystems, providing quantitative measures for the security margin achieved with respect to the quantum-accelerated key recovery on AES, targeting both the current state-of-the-art approach and the NIST estimates. Our work improves the state-of-the-art, reducing the circuit depth from 2¹⁹ to 2³⁰ for all the parameters of the NIST selected cryptosystems. We further analyse recently proposed optimizations, showing that the overhead introduced by their implementation overcomes their asymptotic advantages. Finally, we address the concern brought forward in the latest NIST report on the parameters choice for the McEliece cryptosystem, showing that the parameter choice yields a computational effort which is slightly below the required target level