Results 1 
4 of
4
Finding strong pseudoprimes to several bases. II

, 2003
"... Define ψm to be the smallest strong pseudoprime to all the first m prime bases. If we know the exact value of ψm, we will have, for integers n<ψm, a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the ψm are known for 1 ≤ m ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
Define ψm to be the smallest strong pseudoprime to all the first m prime bases. If we know the exact value of ψm, we will have, for integers n<ψm, a deterministic efficient primality testing algorithm which is easy to implement. Thanks to Pomerance et al. and Jaeschke, the ψm are known for 1 ≤ m ≤ 8. Upper bounds for ψ9,ψ10 and ψ11 were first given by Jaeschke, and those for ψ10 and ψ11 were then sharpened by the first author in his previous paper (Math. Comp. 70 (2001), 863–872). In this paper, we first follow the first author’s previous work to use biquadratic residue characters and cubic residue characters as main tools to tabulate all strong pseudoprimes (spsp’s) n < 1024 to the first five or six prime bases, which have the form n = pq with p, q odd primes and q − 1= k(p−1),k =4/3, 5/2, 3/2, 6; then we tabulate all Carmichael numbers < 1020, to the first six prime bases up to 13, which have the form n = q1q2q3 with each prime factor qi ≡ 3 mod 4. There are in total 36 such Carmichael numbers, 12 numbers of which are also spsp’s to base 17; 5 numbers are spsp’s to bases 17 and 19; one number is an spsp to the first 11 prime bases up to 31. As a result the upper bounds for ψ9,ψ10 and ψ11 are lowered from 20 and 22decimaldigit numbers to a 19decimaldigit number: ψ9 ≤ ψ10 ≤ ψ11 ≤ Q11 = 3825 12305 65464 13051 (19 digits) = 149491 · 747451 · 34233211. We conjecture that ψ9 = ψ10 = ψ11 = 3825 12305 65464 13051, and give reasons to support this conjecture. The main idea for finding these Carmichael numbers is that we loop on the largest prime factor q3 and propose necessary conditions on n to be a strong pseudoprime to the first 5 prime bases. Comparisons of effectiveness with Arnault’s, Bleichenbacher’s, Jaeschke’s, and Pinch’s methods for finding (Carmichael) numbers with three prime factors, which are strong pseudoprimes to the first several prime bases, are given.
FINDING C3STRONG PSEUDOPRIMES
, 2004
"... Let q1 <q2 <q3 be odd primes and N = q1q2q3. Put d =gcd(q1−1,q2 − 1,q3 − 1) and hi = qi−1,i=1, 2, 3. d Then we call d the kernel, the triple (h1,h2,h3) the signature, andH = h1h2h3 the height of N, respectively. We call N a C3number if it is a Carmichael number with each prime factor qi≡3 m ..."
Abstract
 Add to MetaCart
Let q1 <q2 <q3 be odd primes and N = q1q2q3. Put d =gcd(q1−1,q2 − 1,q3 − 1) and hi = qi−1,i=1, 2, 3. d Then we call d the kernel, the triple (h1,h2,h3) the signature, andH = h1h2h3 the height of N, respectively. We call N a C3number if it is a Carmichael number with each prime factor qi≡3 mod 4. If N is a C3number and a strong pseudoprime to the t bases bi for 1 ≤ i ≤ t, we call NaC3spsp(b1,b2,...,bt). Since C3numbers have probability of error 1/4 (the upper bound of that for the RabinMiller test), they often serve as the exact values or upper bounds of ψm (the smallest strong pseudoprime to all the first m prime bases). If we know the exact value of ψm, we will have, for integers n<ψm, a deterministic efficient primality testing algorithm which is easy to implement. In this paper, we first describe an algorithm for finding C3spsp(2)’s, to a given limit, with heights bounded. There are in total 21978 C3spsp(2)’s < 1024 with heights < 109. We then give an overview of the 21978 C3spsp(2)’s and tabulate 54 of them, which are C3spsp’s to the first 8 prime bases up to 19; three numbers are spsp’s to the first 11 prime bases up to 31. No C3spsp’s < 1024 to the first 12 prime bases with heights < 109 were found. We conjecture that there exist no C3spsp’s < 1024 to the first 12 prime bases with heights ≥ 109 and so that
Title: Homomorphic cryptosystems for electronic voting
"... This bachelor's degree thesis deals with homomorphic publickey cryptography, or in other words cryptosystems with special addition properties. Such cryptosystems are widely used in real life situations, for instance to make electronic voting secure. Therefore, this thesis will walk you throug ..."
Abstract
 Add to MetaCart
This bachelor's degree thesis deals with homomorphic publickey cryptography, or in other words cryptosystems with special addition properties. Such cryptosystems are widely used in real life situations, for instance to make electronic voting secure. Therefore, this thesis will walk you through how Mathematics can lie behind day to day life. In Chapter 1 we introduce a few basic algebra results and other key concepts that will be used later. In Chapters 2 and 3 we describe and discuss the algorithms and properties of the two cryptosystems which are considered to be the best ones for evoting: Paillier and JoyeLibert. We have implemented the two schemes from scratch. We conclude the thesis in Chapter 4, by comparing running times of the two abovementioned cryptosystems, in simulations of reallife evoting systems, with up to tens of thousands of voters, and different levels of security. Through these simulations, we discern the situations where each of the two cryptosystems is preferable. vi To my love Carla for supporting my particularities for six years. To my parents for the energy that made me stronger. To the future that awaits me. ii