### THERE ARE INFINITELY MANY PERRIN PSEUDOPRIMES

"... Abstract. We prove the existence of infinitely many Perrin pseudoprimes, as conjectured by Adams and Shanks in 1982. The theorem proven covers a general class of pseudoprimes based on recurrence sequences. We use ingredients of the proof of the infinitude many Carmichael numbers, along with zero-den ..."

Abstract
- Add to MetaCart

Abstract. We prove the existence of infinitely many Perrin pseudoprimes, as conjectured by Adams and Shanks in 1982. The theorem proven covers a general class of pseudoprimes based on recurrence sequences. We use ingredients of the proof of the infinitude many Carmichael numbers, along with zero-density estimates for Hecke L-functions. 1. Background In a 1982 paper [1], Adams and Shanks introduced a probable primality test based on third order recurrence sequences. The following is a version of that test. Consider sequences An = An(r, s) defined by the following relations: A−1 = s, A0 = 3, A1 = r, and An = rAn−1 − sAn−2 + An−3. Let f(x) = x 3 − rx 2 + sx − 1 be the associated polynomial and ∆ its discriminant. (Perrin’s sequence is An(0, −1).) Definition. The signature mod m of an integer n with respect to the sequence Ak(r, s) is the 6-tuple (A−n−1, A−n, A−n+1, An−1, An, An+1) mod m. Definitions. An integer n is said to have an S-signature if its signature mod n is congruent to (A−2, A−1, A0, A0, A1, A2). An integer n is said to have a Q-signature if its signature mod n is congruent to (A, s, B, B, r, C), where for some integer a with f(a) ≡ 0 mod n, A ≡ a −2 + 2a, B ≡ −ra 2 + (r 2 − s)a, and C ≡ a 2 + 2a −1. An integer n is said to have an I-signature if its signature mod n is congruent to (r, s, D ′ , D, r, s), where D ′ + D ≡ rs − 3 mod n and (D ′ − D) 2 ≡ ∆. Definition. A Perrin pseudoprime with parameters (r, s) is an odd composite n such that either

### Interpolation of Shifted-Lacunary Polynomials

, 2010

"... Given a “black box” function to evaluate an unknown rational polynomial f ∈ Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t ∈ Z>0, the shift α ∈ Q, the exponents 0 ≤ e1 ..."

Abstract
- Add to MetaCart

Given a “black box” function to evaluate an unknown rational polynomial f ∈ Q[x] at points modulo a prime p, we exhibit algorithms to compute the representation of the polynomial in the sparsest shifted power basis. That is, we determine the sparsity t ∈ Z>0, the shift α ∈ Q, the exponents 0 ≤ e1 <e2 < ·· · <et, and the coefficients c1,...,ct ∈ Q \{0} such that f(x) =c1(x − α) e1 + c2(x − α) e2 + ···+ ct(x − α) et. The computed sparsity t is absolutely minimal over any shifted power basis. The novelty of our algorithm is that the complexity is polynomial in the (sparse) representation size, which may be logarithmic in the degree of f. Our method combines previous celebrated results on sparse interpolation and computing sparsest shifts, and provides a way to handle polynomials with extremely high degree which are, in some sense, sparse in information.

### Then

"... Abstract. My research concerns explicit inequalities in elementary number theory. Specifically, I am interested in the distribution of twin primes, along with theoretical and computational techniques for putting explicit bounds on classes numbers such as twin primes. One important method I use to ex ..."

Abstract
- Add to MetaCart

Abstract. My research concerns explicit inequalities in elementary number theory. Specifically, I am interested in the distribution of twin primes, along with theoretical and computational techniques for putting explicit bounds on classes numbers such as twin primes. One important method I use to examine the density of twins is to look for bounds on Brun’s Constant. 1. Motivation

### Computational Number Theory and Algebra May 16, 2012 Lecture 9

"... In the last class, we mentioned that an irreducible polynomial of degree n over a finite field Fq can be used to generate the extension field Fqn. This gives us a method to construct large finite fields starting from small fields. To give you an example as to where such extension fields are useful, ..."

Abstract
- Add to MetaCart

In the last class, we mentioned that an irreducible polynomial of degree n over a finite field Fq can be used to generate the extension field Fqn. This gives us a method to construct large finite fields starting from small fields. To give you an example as to where such extension fields are useful, recall that in the Reed-Solomon encoding procedure, we need to use a finite field whose size is at least as large as the codeword length. On the other hand, in the list decoding phase we need to factor a bivariate polynomial. Given that bivariate factoring reduces to univariate factoring and that we only know of a deterministic poly-time factoring algorithm for low-characteristic finite fields, it makes sense to start with a small prime field and extend it suitably to a sufficiently large finite field. In today’s class, we will see how to generate an irreducible polynomial over a finite field in random polynomial time. The topics of discussion for today’s class are: • Generating irreducible polynomials over finite fields, • Miller-Rabin primality test. 1 Generating irreducible polynomials over finite fields We want to generate an irreducible polynomial of degree n over a finite field Fq. Recall from the last class that irreducibility of a given polynomial can be checked in deterministic polynomial time. Now, if we can show that the density of irreducible polynomials is sufficiently large then we can just pick a random polynomial of degree n and test if it is irreducible. This should yield an irreducible polynomial with high probability (provided the density is large). To make this idea formal, we need to estimate the density of irreducible polynomials of degree n over a finite field Fq.

### WHEN THE SIEVE WORKS

, 2012

"... Abstract. We are interested in classifying those sets of primes P such that when we sieve out the integers up to x by the primes in P c we are left with roughly the expected number of unsieved integers. In particular, we obtain the first general results for sieving an interval of length x with prime ..."

Abstract
- Add to MetaCart

Abstract. We are interested in classifying those sets of primes P such that when we sieve out the integers up to x by the primes in P c we are left with roughly the expected number of unsieved integers. In particular, we obtain the first general results for sieving an interval of length x with primes including some in ( √ x, x], using methods motivated by additive combinatorics. 1. Introduction and

### Sign Modules in Secure Arithmetic Circuits (A Full Version)

"... In 1994 [18], Feige, Killian, and Naor suggested a toy protocol of secure comparison, which takes secret input [x]7 and [y]7 between 0 and 2, using the modulo 7 arithmetic circuit. Because 0, 1, and 2 are quadratic residues while 5 and 6 are non-residues modulo 7, the protocol is done by securely ev ..."

Abstract
- Add to MetaCart

In 1994 [18], Feige, Killian, and Naor suggested a toy protocol of secure comparison, which takes secret input [x]7 and [y]7 between 0 and 2, using the modulo 7 arithmetic circuit. Because 0, 1, and 2 are quadratic residues while 5 and 6 are non-residues modulo 7, the protocol is done by securely evaluating the Legendre symbol of [x − y]7, which can be carried out very efficiently by O(1) secure multiplication gates. However, the extension regarding computation in large fields is undiscussed, and furthermore, whether it is possible to turn a toy comparison into a practically usable protocol is unknown. Motivated by these questions, in this paper, we study secure comparison-related problems using only the secure arithmetic black-box of a finite field, counting the cost by the number of secure multiplications. We observe that a specific type of quadratic patterns exists in all finite fields, and the existence of these patterns can be utilized to explore new solutions with sublinear complexities to several problems. First, we define sign modules as partial functions that simulate integer signs in an effective range using a polynomial number of arithmetic operations in a finite field. Let ℓ denote the bit-length of a finite field size. We show the existence of ⌊ℓ/5⌋-“effective ” sign modules in any finite field that has a sufficiently large characteristic. When ℓ is decided first, we further show (by a constructive proof) the existence of prime fields that contain an Ω(ℓ log ℓ)-“effective” sign module and propose an efficient polynomial-time randomized algorithm that finds concrete instances of sign modules. Then, based on one effective sign module in an odd prime field Zp and providing a binaryexpressed random number in Zp, prepared in the offline phase, we show that the computation of bitwise less-than can be improved from the best known result of O(ℓ) to O( ℓ log ℓ

### MATHEMATICAL ASPECTS OF SHOR’S ALGORITHM

"... Abstract. Given a large n-bits integer N < 2n, Shor’s algorithm finds with positive probability a factor of N after O(n2 log n log log n) quantum steps. We describe some of the mathematical aspects of Shor’s algorithm. We mainly follow a description due to M. Batty, S.L. Braunstein, A. J. Duncan ..."

Abstract
- Add to MetaCart

Abstract. Given a large n-bits integer N < 2n, Shor’s algorithm finds with positive probability a factor of N after O(n2 log n log log n) quantum steps. We describe some of the mathematical aspects of Shor’s algorithm. We mainly follow a description due to M. Batty, S.L. Braunstein, A. J. Duncan and S. Rees. 1.