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

Primality Testing

This tutorial describes the Miller-Rabin method for testing the primality of large integers. The method is illustrated by a Pascal algorithm. The performance of the algorithm was measured on a Computing Surface

On cyclotomic primality tests

In 1980, L. Adleman, C. Pomerance, and R. Rumely invented the first cyclotomicprimality test, and shortly after, in 1981, a simplified and more efficient versionwas presented by H.W. Lenstra for the Bourbaki Seminar. Later, in 2008, ReneSchoof presented an updated version of Lenstra\u27s primality test. This thesis presents adetailed description of the cyclotomic primality test as described by Schoof, along withsuggestions for implementation. The cornerstone of the test is a prime congruencerelation similar to Fermat\u27s \little theorem that involves Gauss or Jacobi sumscalculated over cyclotomic fields. The algorithm runs in very nearly polynomial time.This primality test is currently one of the most computationally efficient tests and isused by default for primality proving by the open source mathematics systems Sageand PARI/GP. It can quickly test numbers with thousands of decimal digits

Primality Tests on Commutator Curves

Das Thema dieser Dissertation sind effiziente Primzahltests. Zunächst wird die Kommutatorkurve eingeführt, die durch einen skalaren Parameter in der zweidimensionalen speziellen linearen Gruppe bestimmt wird. Nach Erforschung der Grundlagen dieser Kurve wird sie in verschiedene Pseudoprimzahltests (z.B. Fermat-Test, Solovay-Strassen-Test) eingebunden. Als wichtigster Pseudoprimzahltest ist dabei der Kommutatorkurventest zu nennen. Es wird bewiesen, dass dieser Test nach einer festen Anzahl von Probedivisionen (alle Primzahlen kleiner 80) das Ergebnis 'wahr' für eine zusammengesetzte Zahl mit einer Wahrscheinlichkeit ausgibt, die kleiner als 1/16 ist. Darüberhinaus wird bewiesen, dass der Miller-Primzahltest unter der Annahme der Korrektheit der Erweiterten Riemannschen Hypothese zur Überprüfung einer Zahl n nur noch für alle Primzahlbasen kleiner als 3/2*ln(n)^2 durchgeführt werden muss. Im Beweis des Primzahltests von G. L. Miller konnte dabei die Notwendigkeit der Erweiterten Riemannschen Hypothese auf nur noch ein Schlüssellemma eingegrenzt werden.This thesis is about efficient primality tests. First, the commutator curve which is described by one scalar parameter in the two-dimensional special linear group will be introduced. After fundamental research of of this curve, it will be included into different compositeness tests (e.g. Fermat's test, Solovay-Strassen test). The most important commutator test is the Commutator Curve Test. Besides, it will be proved that this test after a fixed number of trial divisions (all prime numbers up to 80) returns the result 'true' for a composite number with a probability less than 1/16. Moreover, it will be shown that Miller's test to check a number n only has to be carried out for all prime bases less than 3/2*ln(n)^2. This happens under the assumption that the Extended Riemann Hypothesis is true. The necessity of the Extended Riemann Hypothesis to prove the primality test of G. L. Miller can be reduced to a single key lemma

Fooling primality tests on smartcards

We analyse whether the smartcards of the JavaCard platform correctly validate primality of domain parameters. The work is inspired by the paper Prime and prejudice: primality testing under adversarial conditions, where the authors analysed many open-source libraries and constructed pseudoprimes fooling the primality testing functions. However, in the case of smartcards, often there is no way to invoke the primality test directly, so we trigger it by replacing (EC)DSA and (EC)DH prime domain parameters by adversarial composites. Such a replacement results in vulnerability to Pohlig-Hellman style attacks, leading to private key recovery. Out of nine smartcards (produced by five major manufacturers) we tested, all but one have no primality test in parameter validation. As the JavaCard platform provides no public primality testing API, the problem cannot be fixed by an extra parameter check, %an additional check before the parameters are passed to existing (EC)DSA and (EC)DH functions, making it difficult to mitigate in already deployed smartcards

Comparison between the RSA cryptosystem and elliptic curve cryptography

In the globalization era, cryptography becomes more popular and powerful; in fact it is very important in many areas (i.e. mathematics, computer science, networks, etc). This thesis provides an overview and comparison between the RSA cryptosystem and elliptic curve cryptography, which both focus on sending and receiving messages. The basic theories of the RSA cryptosystem and elliptic curve cryptography are explored. The RSA cryptosystem and elliptic curve cryptography theories are quite similar but elliptic curve cryptography is more complicated. The idea of the RSA cryptosystem is based on three popular theorems which are Euler's Theorem, Fermat's Little Theorem and the Chinese Remainder Theorem. This discussion shows that the reliability and strong security of the RSA cryptosystem depends on the degree of dif- ficulty of integer factorization. Therefore, methods for integer factorization are discussed. In addition I show how the security of elliptic curve cryptography depends on the apparent difficulty of solving the elliptic curve discrete logarithm problem (ECDLP)