Leonard M. Adleman

Two theorems on random polynomial time

The use of randomness in computation was first studied in abstraction by Gill [4]. In recent years its use in both practical and theoretical areas has become apparent. Strassen and Solovay [10]; Miller [7]; and Rabin [8] have used it to transform primality testing into a (for many purposes) tractible problem. We can see in retrospect that it was implicit in algorithms by Ber1ekamp [2], Lehmer [6], and Cippola [3] (1903!). Where the traditional method of polynomial reduction has been inapplicable, randomness has been used in demonstrating intractibility by Adleman and Manders [1], and in showing problems equivalent by Rabin [9]. In light of these developments and the insights they provide, a new examination of randomness is in order.

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 Sticker-Based Model for DNA Computation

We introduce a new model of molecular computation that we call the sticker model. Like many previous proposals it makes use of DNA strands as the physical substrate in which information is represented and of separation by hybridization as a central mechanism. However, unlike previous models, the stickers model has a random access memory that requires no strand extension and uses no enzymes; also (at least in theory), its materials are reusable. The paper describes computation under the stickers model and discusses possible means for physically implementing each operation. Finally, we go on to propose a specific machine architecture for implementing the stickers model as a microprocessor-controlled parallel robotic workstation. In the course of this development a number of previous general concerns about molecular computation (Smith, 1996; Hartmanis, 1995; Linial et al., 1995) are addressed. First, it is clear that general-purpose algorithms can be implemented by DNA-based computers, potentially solving a wide class of search problems. Second, we find that there are challenging problems, for which only modest volumes of DNA should suffice. Third, we demonstrate that the formation and breaking of covalent bonds is not intrinsic to DNA-based computation. Fourth, we show that a single essential biotechnology, sequence-specific separation, suffices for constructing a general-purpose molecular computer. Concerns about errors in this separation operation and means to reduce them are addressed elsewhere (Karp et al., 1995; Roweis and Winfree, 1999). Despite these encouraging theoretical advances, we emphasize that substantial engineering challenges remain at almost all stages and that the ultimate success or failure of DNA computing will certainly depend on whether these challenges can be met in laboratory investigations.

A subexponential algorithm for the discrete logarithm problem with applications to cryptography

In 1870 Bouniakowsky [2 J publ ished an algorithm to solve the congruence aX _ bMOD (q). While his algorithm contained several clever ideas useful for small numbers, its asymptotic complexity was O(q). Despite its long history, no fast algorithm has ever emerged for the Discrete Logarithm Problem and the best published method, due to Shanks [lOJ requires O(ql/2) in time and space. The problem has attracted renewed interest in recent years because of its use in cryptography [7 ], [15J,[19J. In particular, the security of the Diffie-Hellman Public Key Distribution Sy s t em [7 J II de pen d s c r ucia 11yon the d iff i c u1t Y 0 f com put i ng log a r i t hms MOD q II • We present a new algorithm for this problem which runs in RTIME better than O(qE) for all E > O.t While no effort is made to present the most efficient incarnation of tActually our algorithm runs in RTIME O(2(O(/10g(q)loglog(q))). RTIME denotes Random Time and refers to algorithms which may use random numbers in their processing. For example, the well known composite testing algorithms of Solovay &Strassen [21J, Miller [11J and Rabin [16J run in RTIME (0(log3(q))). For precise definitions see [1], [llJ and [9J.

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.

On Applying Molecular Computation to the Data Encryption Standard

Recently, Boneh, Dunworth, and Lipton (1996) described the potential use of molecular computation in attacking the United States Data Encryption Standard (DES). Here, we provide a description of such an attack using the sticker model of molecular computation. Our analysis suggests that such an attack might be mounted on a tabletop machine using approximately a gram of DNA and might succeed even in the presence of a large number of errors.

On taking roots in finite fields

How hard is it to decide if a is a quadratic residue modulo m? Few problems have received more attention [7]. When m is prime, the Legendre symbol, the Jacobi symbol and the Gaussian Law of Quadratic Reciprocity yield a polynomial time algorithm [14]. When m is composite, then the above result for primes together with the Chinese Remainder Theorem yields a polynomial time algorithm assuming m can be factored (thus this problem is probably notyorNP-complete). Finding XiS such that x2 = a MOD(m) when a is a quadratic residue is a far more complex problem. In [11] it was shown that finding the least x such that x2 =a MOD(m) is NP-complete (even if m is factored). The main result of this paper is:

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.

Solution of a Satisfiability Problem on a Gel-Based DNA Computer

We have succeeded in solving an instance of a 6-variable 11-clause 3-SAT problem on a gel-based DNA computer. Separations were performed using probes covalently bound to polyacrylamide gel. During the entire computation, DNA was retained within a single gel and moved via electrophoresis. The methods used appear to be readily automatable and should be suitable for problems of a significantly larger size.

Finding irreducible polynomials over finite fields

I. In troduc t ion Irreducible polynomials in Fp[X] are used to carry out the arithmetic in field extension of Fp. Computations in such extensions occur in coding theory [2], complexity theory [8] and cryptography [3] . Random polynomial time algorithms exist for finding irreducible polynomials of any degree over Fp [2, 8], and so as a practical matter the problem is solved. However, the deterministic complexity of the problem has yet to be established.