Showing papers by "Johan Håstad published in 1998"

The security of individual RSA bits

Abstract: We study the security of individual bits in an RSA encrypted message E/sub N/(X). We show that given E/sub N/(X), predicting any single bit in x with only a non-negligible advantage over the trivial guessing strategy is (through a polynomial time reduction) as hard as breaking RSA. We briefly discuss a related result for bit security of the discrete logarithm.

Monotone Circuits for Connectivity Have Depth (log n ) 2- o (1)

Abstract: We prove that a monotone circuit of size nd recognizing connectivity must have depth $\Omega((\log n)^2/\log d)$. For formulas this implies depth $\Omega((\log n)^2/\log\log n)$. For polynomial-size circuits the bound becomes $\Omega((\log n)^2)$ which is optimal up to a constant.

Circuit Bottom Fan-in and Computational Power

Abstract: We investigate the relationship between circuit bottom fan-in and circuit size when circuit depth is fixed. We show that in order to compute certain functions, a moderate reduction in circuit bottom fan-in will cause significant increase in circuit size. In particular, we prove that there are functions that are computable by circuits of linear size and depth k with bottom fan-in 2 but require exponential size for circuits of depth k with bottom fan-in 1. A general scheme is established to study the trade-off between circuit bottom fan-in and circuit size. Based on this scheme, we are able to prove, for example, that for any integer c, there are functions that are computable by circuits of linear size and depth k with bottom fan-in $O(\log n)$ but that require exponential size for circuits of depth k with bottom fan-in c, and that for any constant $\epsilon > 0$, there are functions that are computable by circuits of linear size and depth k with bottom fan-in $\log n$ but that require superpolynomial size for circuits of depth k with bottom fan-in $O(\log^{1-\epsilon} n)$. A consequence of these results is that the three input read-modes of alternating Turing machines proposed in the literature are all distinct.

Fitting Points on the Real Line and Its Application to RH Mapping

Abstract: The MATRIX-TO-LINE problem is that of, given an n × n symmetric matrix D, finding an arrangement of n points on the real line such that the so obtained distances agree as well as possible with the by D specified distances, w.r.t. the max-norm. The MATRIX-TO-LINE problem has previously been shown to be NP-complete [11]. We show that it can be approximated within 2, but not within 4=3 unless P=NP. We also show tight bounds under a stronger assumption. We show that the MATRIX-TO-LINE problem cannot be approximated within 2 - δ unless 3-colorable graphs can be colored with ⌊4/δ⌋ colors in polynomial time. Currently, the best polynomial time algorithm colors a 3-colorable graph with O(n3/14) colors [4]. We apply our MATRIX-TO-LINE algorithm to a problem in computational biology, namely, the Radiation Hybrid (RH) problem, i.e., the algorithmic part of a physical mapping method called RH mapping. This gives us the first algorithm with a guaranteed convergence for the general RH problem.

Some Recent Strong Inapproximability Results

Abstract: The purpose of this talk is to give some idea of the recent progress in obtaining strong, and sometimes tight, inapproximability constants for NP-hard optimization problems. Tight results have been obtained for Max-Ek-Sat for k ≥ 3, maximizing the number of satisfied linear equations in an over-determined system of linear equations modulo a prime p and Set Splitting.