Some remarks on Lucas-Based cryptosystems

We review the well-known relation between Lucas sequences and exponentiation. This leads to the observation that certain public-key cryptosystems that are based on the use of Lucas sequences have some elementary properties their re-inventors were apparently not aware of. In particular, we present a chosen-message forgery for `LUC' [Smith, 1993] and [Smith and Lennon, 1993], and we show that `LUCELG' and `LUCDIF' [Smith, 1994] and [Smith and Skinner, 1994] are vulnerable to subexponential time attacks. This proves that various claims that were made about Lucas-based cryptosystems are incorrec

A method for computing Lucas sequences

AbstractMost of public-key cryptosystems rely on one-way functions, which can be used to encrypt and sign messages. Their encryption and signature operations are based on the computation of exponentiation. Recently, some public-key cryptosystems are proposed and based on Lucas functions, and the Lucas sequences are performed as S = V(d)modN. In this paper, we will transform the concept of addition chains for computing the exponentiation evaluations to the Lucas chains for computing the Lucas sequences. Theoretically, the shorter Lucas chain for d is generated, the less computation time for evaluating the value V(d) is required. Therefore, we proposed a heuristic algorithm for evaluating a shorter Lucas chain and then use it to compute the Lucas sequence with less modular multiplications

Security systems based on Gaussian integers : Analysis of basic operations and time complexity of secret transformations

Many security algorithms currently in use rely heavily on integer arithmetic modulo prime numbers. Gaussian integers can be used with most security algorithms that are formulated for real integers. The aim of this work is to study the benefits of common security protocols with Gaussian integers. Although the main contribution of this work is to analyze and improve the application of Gaussian integers for various public key (PK) algorithms, Gaussian integers were studied in the context of image watermarking as well. The significant benefits of the application of Gaussian integers become apparent when they are used with Discrete Logarithm Problem (DLP) based PK algorithms. In order to quantify the complexity of the Gaussian integer DLP, it is reduced to two other well known problems: DLP for Lucas sequences and the real integer DLP. Additionally, a novel exponentiation algorithm for Gaussian integers, called Lucas sequence Exponentiation of Gaussian integers (LSEG) is introduced and its performance assessed, both analytically and experimentally. The LSEG achieves about 35% theoretical improvement in CPU time over real integer exponentiation. Under an implementation with the GMP 5.0.1 library, it outperformed the GMP\u27s mpz_powm function (the particularly efficient modular exponentiation function that comes with the GMP library) by 40% for bit sizes 1000-4000, because of low overhead associated with LSEG. Further improvements to real execution time can be easily achieved on multiprocessor or multicore platforms. In fact, over 50% improvement is achieved with a parallelized implementation of LSEG. All the mentioned improvements do not require any special hardware or software and are easy to implement. Furthermore, an efficient way for finding generators for DLP based PK algorithms with Gaussian integers is presented. In addition to DLP based PK algorithms, applications of Gaussian integers for factoring-based PK cryptosystems are considered. Unfortunately, the advantages of Gaussian integers for these algorithms are not as clear because the extended order of Gaussian integers does not directly come into play. Nevertheless, this dissertation describes the Extended Square Root algorithm for Gaussian integers used to extend the Rabin Cryptography algorithm into the field of Gaussian integers. The extended Rabin Cryptography algorithm with Gaussian integers allows using fewer preset bits that are required by the algorithm to guard against various attacks. Additionally, the extension of RSA into the domain of Gaussian integers is analyzed. The extended RSA algorithm could add security only if breaking the original RSA is not as hard as factoring. Even in this case, it is not clear whether the extended algorithm would increase security. Finally, the randomness property of the Gaussian integer exponentiation is utilized to derive a novel algorithm to rearrange the image pixels to be used for image watermarking. The new algorithm is more efficient than the one currently used and it provides a degree of cryptoimmunity. The proposed method can be used to enhance most picture watermarking algorithms

An efficient probabilistic public-key cryptosystem over quadratic fields quotients

AbstractWe present a new probabilistic cryptosystem working in quadratic fields quotients. Computation in such objects can be done efficiently with Lucas sequences which help to design a fast system. The security of the scheme is based on the LUC problem and its semantic security on a new decisional problem. This system appears to be an alternative to schemes based on the RSA primitive and has a full computational cost smaller than the El Gamal EC cryptosystem

A public-key cryptosystem based on second order linear sequences

Based on Lucas functions, an improved version of the Diffie-Hellman distribution key scheme and to the ElGamal public key cryptosystem scheme are proposed, together with an implementation and computational cost. The security relies on the difficulty of factoring an RSA integer and on the difficulty of computing the discrete logarithm

Quantum Computing in the NISQ era and beyond

Noisy Intermediate-Scale Quantum (NISQ) technology will be available in the near future. Quantum computers with 50-100 qubits may be able to perform tasks which surpass the capabilities of today's classical digital computers, but noise in quantum gates will limit the size of quantum circuits that can be executed reliably. NISQ devices will be useful tools for exploring many-body quantum physics, and may have other useful applications, but the 100-qubit quantum computer will not change the world right away --- we should regard it as a significant step toward the more powerful quantum technologies of the future. Quantum technologists should continue to strive for more accurate quantum gates and, eventually, fully fault-tolerant quantum computing.Comment: 20 pages. Based on a Keynote Address at Quantum Computing for Business, 5 December 2017. (v3) Formatted for publication in Quantum, minor revision

Criptosistema de Lucas

En aquest treball es descriuen exhaustivament les funcions de Lucas i el mètode XTR (mètodes de criptogràfics sobre extessions de cossos finits), i s'analitzen tots els algorismes que s'han publicat fins ara per computar les funcions trapdoor que usen aquests dos mètodes en un ambient criptogràfic. També s'ha fet un recull de totes les aplicacions criptogràfiques que han derivat dels articles [3] i [18], introductors dels dos mètodes anteriors

Recurrent Sequences and Cryptography

Fibonacci numbers are defined as recursively as F_(n+1)=F_n+F_(n-1) with initial conditions F_1=F_2=1. Lucas numbers also enjoys the same recurrence relation but with different initial conditions L_1=1,L_2=3. Large prims are very useful in public key cryptography. Lucas numbers can also be exploited for these purpose