We present a mathematical construct which provides a cryptographic protocol to verifiably shuffle a sequence of k modular integers, and discuss its application to secure, universally verifiable, multi-authority election schemes. The output of the shuffle operation is another sequence of k modular integers, each of which is the same secret power of a corresponding input element, but the order of elements in the output is kept secret. Though it is a trivial matter for the "shuffler" (who chooses the permutation of the elements to be applied) to compute the output from the input, the construction is important because it provides a linear size proof of correctness for the output sequence (i.e. a proof that it is of the form claimed) that can be checked by an arbitrary verifiers. The complexity of the protocol improves on that of Furukawa-Sako[16] both measured by number of exponentiations and by overall size.The protocol is shown to be honest-verifier zeroknowledge in a special case, and is computational zeroknowledge in general. On the way to the final result, we also construct a generalization of the well known Chaum-Pedersen protocol for knowledge of discrete logarithm equality [10], [7]. In fact, the generalization specializes exactly to the Chaum-Pedersen protocol in the case k = 2. This result may be of interest on its own.An application to electronic voting is given that matches the features of the best current protocols with significant efficiency improvements. An alternative application to electronic voting is also given that introduces an entirely new paradigm for achieving Universally Verifiable elections.

Given a univariate polynomial f(z) of degree n with complex coefficients, whose real and imaginary parts can be expressed as a ratio of two integers less than 2/sup m/ in magnitude, the root problem is to find all the roots of f(z) up to specified precision 2/sup -/spl mu//. Assuming the arithmetic model for computation, we provide, for any /spl epsiv/>0, an algorithm which has complexity O(n/sup 1+/spl epsiv// log b), where b=m+/spl mu/. This improves on the previous best known algorithm for the problem which has complexity O(n/sup 2/ log b). We claim it that it follows from the fact that we can bound the precision required in all the arithmetic computations, that the complexity of our algorithm in the Boolean model of computation is O(n/sup 2+/spl epsiv//(n+b) log/sup 2/ b log log b).>

Given a polynomial p(z) od degree n with integer coefficients, whose absolute values are bounded above by 2/sup m/, and a specified integer mu , it is shown that the problem of determining all roots of p with error less than 2/sup - mu / is in the parallel complexity class NC. To do this, an algorithm that runs on at most POLY(n+m+ mu ) processors with a parallel time complexity of O(log/sup 3/(n+m+ mu )) is constructed. This algorithm extends the algorithm of M. Ben-Or et al. (SIAM J. Comput., vol.17, p.1081-92, 1988) by removing the severe restriction that all the roots of p(z) should be real. >

In general, two quadric surfaces intersect in a nonsingular quartic space curve. Under special circumstances, however, this intersection may “degenerate” into a quartic with a double point, or a composite of lines, conics, and twisted cubics whose degrees, counted over the complex projective domain, sum to four. Such degenerate forms are important since they occur with surprising frequency in practice and, unlike the generic case, they admit rational parameterizations. Invoking concepts from classical algebraic geometry, we formulate the condition for a degenerate intersection in terms of the vanishing of a polynomial expression in the quadric coefficients. When this is satisfied, we apply a multivariate polynomial factorization algorithm to the projecting cone of the intersection curve. Factors of this cone which correspond to intersection components “at infinity” may be removed a priori. A careful examination of the remaining cone factors then facilitates the identification and parameterization of the various real, affine intersection elements that may arise: isolated points, lines, conics, cubics, and singular quartics. The procedure is essentially automatic (avoiding the tedium of case-by-case analyses), encompasses the full range of quadric forms, and is amenable to implementation in exact (symbolic) arithmetic.

In the synthesis of digital circuits, one encounters the problem of identifying blocks which have been designed, so that there is no replication in the expensive effort of generating the physical layout of these blocks. We present a model for the synthesis of combinational logic into complex MOS circuits and present a ranking and unranking procedure to characterize the layout of each complex MOS circuit.