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The Pseudosquares Prime Sieve
"... Abstract. We present the pseudosquares prime sieve, which finds all primes up to n. Define p to be the smallest prime such that the pseudosquare Lp>n/(π(p)(log n) 2); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p)logn) space. It uses t ..."
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Abstract. We present the pseudosquares prime sieve, which finds all primes up to n. Define p to be the smallest prime such that the pseudosquare Lp>n/(π(p)(log n) 2); here π(x) is the prime counting function. Our algorithm requires only O(π(p)n) arithmetic operations and O(π(p)logn) space. It uses the pseudosquares primality test of Lukes, Patterson, and Williams. Under the assumption of the Extended Riemann Hypothesis, we have p ≤ 2(log n) 2, but it is conjectured that p ∼ 1 log nlog log n. Thus, log2 the conjectured complexity of our prime sieve is O(n log n) arithmetic operations in O((log n) 2) space. The primes generated by our algorithm are proven prime unconditionally. The best current unconditional bound known is p ≤ n 1/(4√e−ɛ) 1.132, implying a running time of roughly n using roughly n 0.132 space. Existing prime sieves are generally faster but take much more space, greatly limiting their range (O(n / log log n)operationswithn 1/3+ɛ space, or O(n) operationswithn 1/4 conjectured space). Our algorithm found all 13284 primes in the interval [10 33,10 33 +10 6] in about 4 minutes on a1.3GHzPentiumIV. We also present an algorithm to find all pseudosquares Lp up to n in sublinear time using very little space. Our innovation here is a new, spaceefficient implementation of the wheel datastructure. 1
On the Security of a Williams Based Public Key Encryption Scheme
"... Abstract. In 1984, H.C. Williams introduced a public key cryptosystem whose security is as intractable as factorization. Motivated by some strong and interesting cryptographic properties of the intrinsic structure of this scheme, we present a practical modification thereof that has very strong secur ..."
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Abstract. In 1984, H.C. Williams introduced a public key cryptosystem whose security is as intractable as factorization. Motivated by some strong and interesting cryptographic properties of the intrinsic structure of this scheme, we present a practical modification thereof that has very strong security properties. We establish, and prove, a generalization of the “solesamplability ” paradigm of ZhengSeberry (1993) which is reminiscent of the plaintextawareness concept of Bellare et. al. The assumptions that we make are both welldefined and reasonable. In particular, we do not model the functions as random oracles. In essence, the proof of security is based on the factorization problem of any large integer n = pq and Canetti’s “oracle hashing ” construction introduced in 1997. Another advantage of our system is that we do not rely on any special structure of the modulus n = pq, nor do we require any specific form of the primes p and q. As our main result we establish a model which implies security attributes even stronger than semantic security against chosen ciphertext attacks.
King’s Buildings
"... For Lucas sequences of the first kind (un)n≥0 and second kind (vn)n≥0 defined as usual by un = (α n − β n)/(α − β), vn = α n + β n, where α and β are either integers or conjugate quadratic integers, we describe the sets {n ∈ N: n divides un} and {n ∈ N: n divides vn}. Building on earlier work, parti ..."
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For Lucas sequences of the first kind (un)n≥0 and second kind (vn)n≥0 defined as usual by un = (α n − β n)/(α − β), vn = α n + β n, where α and β are either integers or conjugate quadratic integers, we describe the sets {n ∈ N: n divides un} and {n ∈ N: n divides vn}. Building on earlier work, particularly that of Somer, we show that the numbers in these sets can be written as a product of a socalled basic number, which can only be 1, 6 or 12, and particular primes, which are described explicitly. Some properties of the set of all primes that arise in this way is also given, for each kind of sequence. 1
The Fibonacci Quarterly 44(2006), no.2, 145153. EXPANSIONS AND IDENTITIES CONCERNING LUCAS SEQUENCES
"... Abstract. In the paper we obtain some new expansions and combinatorial identities concerning Lucas sequences. 1. Introduction. ..."
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Abstract. In the paper we obtain some new expansions and combinatorial identities concerning Lucas sequences. 1. Introduction.
The Fibonacci Quarterly 44(2006), no.2, 121130. PRIMALITY TESTS FOR NUMBERS OF THE FORM k · 2 m ± 1
"... Abstract. Let k, m ∈ Z, m ≥ 2, 0 < k < 2 m and 2 k. In the paper we give a general primality criterion for numbers of the form k · 2 m ± 1, which can be viewed as a generalization of the LucasLehmer test for Mersenne primes. In particular, for k = 3, 9 we obtain explicit primality tests, which are ..."
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Abstract. Let k, m ∈ Z, m ≥ 2, 0 < k < 2 m and 2 k. In the paper we give a general primality criterion for numbers of the form k · 2 m ± 1, which can be viewed as a generalization of the LucasLehmer test for Mersenne primes. In particular, for k = 3, 9 we obtain explicit primality tests, which are simpler than current known results. We also give a new primality test for Fermat numbers and criteria for 9 · 2 4n+3 ± 1, 3 · 2 20n+6 ± 1 or 3 · 2 36n+6 ± 1 to be twin primes. 1. Introduction. For nonnegative integers n, the numbers Fn = 22n +1 are called the Fermat numbers. In 1878 Pepin showed that Fn(n ≥ 1) is prime if and only if 3 (Fn−1)/2 ≡ −1 (mod Fn). For primes p, let Mp = 2 p − 1. The famous LucasLehmer test states that Mp is a
A CRITERION FOR POLYNOMIALS TO BE CONGRUENT TO THE PRODUCT OF LINEAR POLYNOMIALS (mod p)
, 2004
"... Abstract. Let {un} be defined by u1−m = · · · = u−1 = 0, u0 = 1 and un+a1un−1+ · · ·+ amun−m = 0 (m> 2, n> 1). In this paper we show that the congruence x m + a1x m−1 + · · ·+am ≡ 0 (mod p) has m distinct solutions if and only if up−m ≡ · · · ≡ up−2 ≡ 0 (mod p) and up−1 ≡ 1 (mod p), where ..."
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Abstract. Let {un} be defined by u1−m = · · · = u−1 = 0, u0 = 1 and un+a1un−1+ · · ·+ amun−m = 0 (m> 2, n> 1). In this paper we show that the congruence x m + a1x m−1 + · · ·+am ≡ 0 (mod p) has m distinct solutions if and only if up−m ≡ · · · ≡ up−2 ≡ 0 (mod p) and up−1 ≡ 1 (mod p), where p is a prime such that p> m and p am. 1. Introduction. In [2] the author extended Lucas series to general linear recurring sequences by defining {un(a1,..., am)} as follows: u1−m = · · · = u−1 = 0, u0 = 1,