# Richard PinchInstitute of Mathematics and its Applications

Richard Pinch

D.Phil.

## About

51

Publications

3,451

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409

Citations

Introduction

Additional affiliations

October 1993 - September 1998

**Queens' College Cambridge**

Position

- Fellow

Description

- Teaching and research in mathematics

October 1989 - September 1993

October 1985 - September 1989

**Emmanuel College**

Position

- Fellow

Description

- Teaching and research in Mathematics

## Publications

Publications (51)

This presentation is an exposition of an application of the theory of recurrence relations to enumerating strings over an alphabet with a forbidden factor (consecutive substring). As an illustration we examine the case of binary strings with a forbidden factor of k consecutive symbols 1 for given k, using generating function techniques that deserve...

We describe some primality tests based on quadratic rings and discuss the absolute pseudoprimes for these tests.

There are 38975 Fermat pseudoprimes (base 2) up to 1011, 101629 up to 1012 and 264239 up to 1013: we describe the calculations and give some statistics. The numbers were generated by a variety of strategies, the most important
being a back-tracking search for possible prime factorisations, and the computations checked by a sieving technique.

We extend our previous computations to show that there are 1401644 Carmichael numbers up to $10^{18}$. As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a ``large prime variation''. We present further statistics on the distribution of Carmichael numbers.

In this note we describe two algorithms for calculating the expres- sions which find the roots of one irreducible polynomial over a finite field in terms of the roots of another such polynomial.

We show that the problem of computing the distance of a given permutation from a subgroup $H$ of $S_n$ is in general NP-complete, even under the restriction that $H$ is elementary Abelian of exponent 2. The problem is shown to be polynomial-time equivalent to a problem related to finding a maximal partition of the edges of an Eulerian directed grap...

We answer a number of questions relating to the pseudo-Smarandache function Z(n). We show that the ratio of consecutive values $Z(n+1)/Z(n)$ and $Z(n-1)/Z(n)$ are unbounded; that $Z(2n)/Z(n)$ is unbounded; that $n/Z(n)$ takes every integer value infinitely often; and that the series $\sum_n 1/Z(n)^\alpha$ is convergent for any $\alpha > 1$.

We extend our previous computations to show that there are 585355 Carmichael numbers up to $10^{17}$. As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a ``large prime variation''. We present further statistics on the distribution of Carmichael numbers.

For original paper see E. Dawson and C.-K. Wu, ibid., vol.33,
no.14, pp.1210-11 (1997). Dawson and Wu proposed a key agreement scheme
based on generalised matrix inverses. They suggest that for a key length
of kn, using matrices of size k×m and m×n, with
k⩽m⩽n the security parameter should be at least 2<sup>(m-k)n
</sup>. We give an attack...

We extend our previous computations to show that there are 246683 Carmichael numbers up to $10^{16}$. As before, the numbers were generated by a back-tracking search for possible prime factorisations together with a ``large prime variation''. We present further statistics on the distribution of Carmichael numbers.

We list the elliptic curves defined over $Q(\sqrt 5)$ with good reduction away from 2. There are 368 isomorphism classes. We give a global minimal model for each class.

A number $n$ is said to be economical if the prime power factorisation of $n$ can be written with no more digits than $n$ itself. We show that under a plausible hypothesis, related to the twin prime conjecture, there are arbitrarily long sequences of consecutive economial numbers, and exhibit such a sequence of length 9.

We show that the inadvertent use of a Carmichael number instead of a prime factor in the modulus of an RSA cryptosystem is likely to make the system fatally vulnerable, but that such numbers may be detected.

. A protocol for computationally secure "on-line" secret-sharing is presented,based on the intractability of the Diffie--Hellman problem, in which the participants" shares can be reused.Introduction.Cachin [1] presents a protocol for "on-line" secret sharing with general accessstructures, with shares as short as the secret and in which participants...

We study the distribution of linear recurrent sequences modulo p
n
for prime p when the auxiliary polynomial is irreducible and the period is maximal. We show that such a sequence takes each possible value equally often up to an error of order pn/2.

The authors show that the attack of Hastad on broadcast messages
using the RSA cryptosystem extends to the case where the public keys are
different, and that the attack is also applicable to the LUC system,
with the same security parameters

The author shows that the attack of Wiener on RSA cryptosystems with a short deciphering exponent extends to systems using other groups such as elliptic curves, and LUC

Piveteau [see ibid., vol. 29, p. 2185, 1993] proposes a digital
signature scheme with message recovery. We have p a large prime, such
that computation of discrete logarithms module p is intractable, and a
message represented as a nonzero integer m module p Nyberg and Rueppei
have found independently and generalised the scheme described by
Piveteau....

There are 105212 Carmichael numbers up to {10^{15}} : we describe the calculations. The numbers were generated by a back-tracking search for possible prime factorizations, and the computations checked by searching selected ranges of integers directly using a sieving technique, together with a "large-prime variation".

The sequence of consecutive integer squares has constant second difference 2. We list every such sequence of squares containing a term less than $1000^2$.

We describe the primality testing algorithms in use in some popular computer algebrasystems, and give some examples where they break down in practice.1 IntroductionIn recent years, fast primality testing algorithms have been a popular subject of research andsome of the modern methods are now incorporated in computer algebra systems (CAS) asstandard...

Cambridge Core - Discrete Mathematics Information Theory and Coding - Communication Theory - by Charles M. Goldie

We study the possible periods of linear recurrent sequences modulo p n for prime p. The maximum possible period depends on the degree and ramification index of the p-adic field generated by the roots of the auxiliary polynomial of the recurrence: in most cases, this is sufficient to determine the maximum period completely. In any case, the maximum...

In this paper we list the elliptic curves defined over Q(√− 3) with good reduction away from the prime dividing 3. As in [8] and [9] a discriminant estimate is used to show that such a curve must have a subgroup of order 3 defined over Q(√−3).(Received May 06 1986)

Bollobás and Erdös[1] have posed the problem:
If a is irrational, show that for 1 ≤ i < j ≤ p the number of integers t with 1 ≤ t ≤ p such that {( t–i ) ² a } < d and {( t–j ) ² a } < d , 0 < d < 1, is d ² p + o(p) uniformly in i, j .

For a ε R, we define a subset V of R ⁿ to be a-convex if x, y ε V implies
Clearly V is (1 — a )-convex iff it is a -convex: V is convex iff it is a -convex for all a ε [0, 1], and any set is 1-convex. We define the a-convex hull of V to be the intersection of all a -convex subsets of R ⁿ containing V. Put D = 0,1} and D(a) the a -convex hull of D...

In this paper we list the elliptic curves defined over Q √ − 1, Q√ −2 or Q√ − 3 which have good reduction away from 2. The possible invariants of such curves are given in Table 1, and their minimal equations in Tables 2, 3 and 4. These extend (and agree with) results of Ogg[4] and Stroeker [10], by a different method.(Received January 20 1984)

In this paper we describe a method for finding integer solutions of simultaneous Pellian equations, that is, integer triples ( x , y , z ) satisfying equations of the form
where the coefficients a , b , c , d , f are integers and we assume that a , c , and ac are not square.

In [1], Vanstone and Zuccherato proposed a public-key elliptic curve cryptosystem in which the public key consists of an integer N and an elliptic curve E defined over the ring Z=NZ. Here N is a product of two secret primes p and q, each of special form, and the order of E modulo N is smooth. We present three attacks, each of which factors the modu...

this paper, two more O(n

We show that a quadratic polynomial symmetric about an integer point which is not a perfect square can not take more than ve distinct integer square values about the centre of symmetry. 0. Introduction. Allinson 1] asks whether a quadratic polynomial symmetric about an integer point which is not a perfect square can take more than ve integer square...

Work carried out at the Mathematical Institute. Thesis (D. Phil.)--University of Oxford, 1982. Includes bibliographical references.

## Projects

Projects (3)

To compute tables of Carmichael numbers and other classes of pseudoprimes. To gather evidence for conjectures on their distribution, and furnish counterexamples.