Ming-Deh A. Huang

Running time and program size for self-assembled squares

Recently Rothemund and Winfree [6] have considered the program size complexity of constructing squares by self-assembly. Here, we consider the time complexity of such constructions using a natural generalization of the Tile Assembly Model defined in [6]. In the generalized model, the Rothemund-Winfree construction of n \times n squares requires time &THgr;(n log n) and program size &THgr;(log n). We present a new construction for assembling n \times n squares which uses optimal time &THgr;(n) and program size &THgr;(\frac{log n}{log log n}). This program size is also optimal since it matches the bound dictated by Kolmogorov complexity. Our improved time is achieved by demonstrating a set of tiles for parallel self-assembly of binary counters. Our improved program size is achieved by demonstrating that self-assembling systems can compute changes in the base representation of numbers. Self-assembly is emerging as a useful paradigm for computation. In addition the development of a computational theory of self-assembly promises to provide a new conduit by which results and methods of theoretical computer science might be applied to problems of interest in biology and the physical sciences.

A subexponential algorithm for discrete logarithms over the rational subgroup of the jacobians of large genus hyperelliptic curves over finite fields

There are well known subexponential algorithms for finding discrete logarithms over finite fields. However, the methods which underlie these algorithms do not appear to be easily adaptable for finding discrete logarithms in the groups associated with elliptic curves and the Jacobians of hyperelliptic curves. This has led to the development of cryptographic systems based on the discrete logarithm problem for such groups [12, 7, 8]. In this paper a Subexponential algorithm is presented for finding discrete logarithms in the group of rational points on the Jacobians of large genus hyperelliptic curves over finite fields. We give a heuristic argument that under certain assumptions, there exists a c e ℜ>0 such that for all sufficiently large g e Z>0, for all odd primes p with log p ≤ (2g + 1).98, the algorithm computes discrete logarithms in the group of rational points on the Jacobian of a genus g hyperelliptic curve over GF(p) within expected time: Lp2g+1[1/2, c] where c ≤ 2.181.

Recognizing primes in random polynomial time

This paper is the first in a sequence of papers which will prove the existence of a random polynomial time algorithm for the set of primes. The techniques used are from arithmetic algebraic geometry and to a lesser extent algebraic and analytic number theory. The result complements the well known result of Strassen and Soloway that there exists a random polynomial time algorithm for the set of composites.

Function Field Sieve Method for Discrete Logarithms over Finite Fields

We present a function field sieve method for discrete logarithms over finite fields. This method is an analog of the number field sieve method originally developed for factoring integers. It is asymptotically faster than the previously known algorithms when applied to finite fields Fpn, where p6?n.

Primality testing and Abelian varieties over finite fields

Acknowledgement.- Overview of the algorithm and the proof of the main theorem.- Reduction of main theorem to three propositions.- Proof of proposition 1.- Proof of proposition 2.- Proof of proposition 3.

Counting Points on Curves over Finite Fields

We consider the problem of counting the number of points on a plane curve, defined by a homogeneous polynomialF(x,y,z) ?Fqx,y,z, which are rational over a ground field Fq. More precisely, we show that if we are given a projective plane curve C of degreen, and if C has only ordinary multiple points, then one can compute the number of Fq-rational points on C in randomized time (logq)?where ?=nO(1). Since our algorithm actually computes the characteristic polynomial of the Frobenius endomorphism on the Jacobian of C, it ,follows that we may also compute (1) the number of Fq-rational points on the smooth projective model of C, (2) the number of Fq-rational points on the Jacobian of C, and (3) the number of Fqm-rational points on C in any given finite extension Fqmof the ground field, each in a similar time bound.

Efficient Algorithms for the Riemann-Roch Problem and for Addition in the Jacobian of a Curve

We study the effective Riemann-Roch problem of computing a basis for the linear space L(D) associated with a divisor D on a plane curve C and its applications. Our presentation begins with an extension of Noethers algorithm for this problem to plane curves with singularities. Like the original, this algorithm has a worst-case complexity of ?(n!|D|), where n is the degree of the curve C.We next consider representations of divisors based on Chow forms. Using these, we produce a factorization-free polynomial-time algorithm for the effective Riemann-Roch problem, which improves the complexity of Noethers algorithm by an order of magnitude. We also present further improvements which, for curves of bounded degree, yield an algorithm with complexity which is linear in the size of the given divisor. Finally, we consider applications of this problem: to the arithmetic of the Jacobian of a plane curve, to the construction of rational functions on a curve with prescribed zeroes and poles, and to the construction of curves with prescribed intersections.

Riemann hypothesis and finding roots over finite fields

It is shown that assuming <italic>Generalized Riemann Hypothesis</italic>, the roots of ƒ(<italic>x</italic>) = O <italic>mod p</italic>, where <italic>p</italic> is a prime and f(x) is an integral Abilene polynomial can be found in <italic>deterministic</italic> polynomial time. The method developed for solving this problem is also applied to prime decomposition in Abelian number fields, and the following result is obtained: assuming Generalized Riemann Hypotheses, for Abelian number fields <italic>K</italic> of finite extension degree over the rational number field <italic>Q</italic>, the decomposition pattern of a prime <italic>p</italic> in <italic>K</italic>, i.e. the <italic>ramification index</italic> and the <italic>residue class degree</italic>, can be computed in deterministic polynomial time, providing <italic>p</italic> does not divide the extension degree of <italic>K</italic> over <italic>Q</italic>. It is also shown, as a theorem fundamental to our algorithm, that for <italic>q</italic>, <italic>p</italic> prime and <italic>m</italic> the order of <italic>p</italic> mod <italic>q</italic>, there is a <italic>q</italic>-th nonresidue in the finite field <italic>F<subscrpt>p<supscrpt>m</supscrpt></subscrpt></italic> that can be written as <italic>a</italic><subscrpt>o</subscrpt> + <italic>a</italic><subscrpt>1</subscrpt><italic>w</italic> + … + <italic>a</italic><subscrpt><italic>m</italic>-1</subscrpt><italic>w</italic><supscrpt><italic>m</italic>-1</supscrpt>, where |<italic>a</italic><subscrpt>1</subscrpt>| ≤ <italic>cq</italic><supscrpt>2</supscrpt> log<supscrpt>2</supscrpt>(<italic>pq</italic>), <italic>c</italic> is an absolute effectively computable constant, and 1, <italic>w</italic>, …, <italic>w</italic><supscrpt><italic>m</italic>-1</supscrpt> form a basis of <italic>F<subscrpt>p</subscrpt><supscrpt>m</supscrpt></italic> over <italic>F<subscrpt>p</subscrpt></italic>. More explicitly, <italic>w</italic> is a root of the q-th cyclotomic polynomial over <italic>F<subscrpt>p</subscrpt></italic>. This result partially generalizes, to finite field extensions over <italic>F<subscrpt>p</subscrpt></italic>, a classical result in number theory stating that assuming Generalized Riemann Hypothesis, the least <italic>q</italic>-th nonresidue mod <italic>p</italic> for <italic>p</italic>,<italic>q</italic> prime and <italic>q</italic> dividing <italic>p</italic> - t is bounded by <italic>c</italic> log<supscrpt>2</supscrpt><italic>p</italic>, where <italic>c</italic> is an absolute, effectively computable constant.

Counting Rational Points on Curves and Abelian Varieties over Finite Fields

We develop efficient methods for deterministic computations with semi-algebraic sets and apply them to the problem of counting rational points on curves and abelian varieties over finite fields. For abelian varieties of dimension g in projective N space over Fq, we improve Pilas result and show that the problem can be solved in O((log q)δ) time where δ is polynomial in g as well as in N. For hyperelliptic curves of genus g over Fq we show that the number of rational points on the curve and the number of rational points on its Jacobian can be computed in time \(O((\log q)^{O(g^6 )} )\)time.

Solving some graph problems with optimal or near-optimal speedup on mesh-of-trees networks

We present a systematic approach for solving graph problems under the network models. We illustrate this approach on the mesh-of-trees networks. It is known that under the CREW PRAM model, when a undirected graph of n nodes is given by an n by n adjacency matrix, the problems of finding minimum spanning forest, connected components, and biconnected components can all be solved with optimal speedup when the number of processors p ≤ n2/log2n. We show that for these problems, the same optimal speedup can be achieved even under the much more restrictive mesh-of-trees network. We also show that for the problem of finding directed spanning forest of arbitrary digraphs and the problem of testing strong connectivity of 1-reachable digraphs, near-optimal speedup can be achieved.