Dave Bayer

Improving the Efficiency and Reliability of Digital Time-Stamping

To establish that a document was created after a given moment in time, it is necessary to report events that could not have been predicted before they happened. To establish that a document was created before a given moment in time, it is necessary to cause an event based on the document, which can be observed by others. Cryptographic hash functions can be used both to report events succinctly, and to cause events based on documents without revealing their contents. Haber and Stornetta have proposed two schemes for digital time-stamping which rely on these principles [HaSt 91].

Ribbons and their canonical embeddings

We study double structures on the projective line and on certain other varieties, with a view to having a nice family of degenerations of curves and K3 surfaces of given genus and Clifford index. Our main interest is in the canonical embeddings of these objects, with a view toward Greens Conjecture on the free resolutions of canonical curves. We give the canonical embeddings explicitly, and exhibit an approach to determining a minimal free resolution.

Karmarkar's linear programming algorithm and Newton's method

This paper describes a full-dimensional version of Karmarkars linear programming algorithm, theprojective scaling algorithm, which is defined for any linear program in ℝn having a bounded, full-dimensional polytope of feasible solutions. If such a linear program hasm inequality constraints, then it is equivalent under an injective affine mappingJ:ℝn→ℝm to Karmarkars original algorithm for a linear program in ℝm havingm—n equality constraints andm inequality constraints. Karmarkars original algorithm minimizes a potential functiong(x), and the projective scaling algorithm is equivalent to that version of Karmarkars algorithm whose step size minimizes the potential function in the step direction.The projective scaling algorithm is shown to be a global Newton method for minimizing a logarithmic barrier function in a suitable coordinate system. The new coordinate system is obtained from the original coordinate system by a fixed projective transformationy = Ф(x) which maps the hyperplaneHopt ={x:〈c, x〉 =c0} specified by the optimal value of the objective function to the “hyperplane at infinity”. The feasible solution set is mapped underФ to anunbounded polytope. LetfLB(y) denote the logarithmic barrier function associated to them inequality constraints in the new coordinate system. It coincides up to an additive constant with Karmarkars potential function in the new coordinate system. Theglobal Newton method iteratey* for a strictly convex functionf(y) defined on a suitable convex domain is that pointy* that minimizesf(y) on the search ray {y+λvN(y): λ≧0} wherevN(y) =−(∇2f(y))−1(∇f(y)) is the Newtons method vector. If {x(k)} are a set of projective scaling algorithm iterates in the original coordinate system andy(k) =Ф(x(k)) then {y(k)} are a set of global Newton method iterates forfLB(y) and conversely.Karmarkars algorithm with step size chosen to minimize the potential function is known to converge at least at a linear rate. It is shown (by example) that this algorithm does not have a superlinear convergence rate.

Reverse search for monomial ideals

We give a set of multidegrees that support all the numerical information for a monomial ideal that can be reverse searched and hence is parallelizable and has space complexity that is polynomial in the size of the input. Our approach uses a new definition of closed sets for simplicial complexes that may be useful in other contexts.