Using smoothness to achieve parallelism

Currently, there is a great deal of excitement about research in the area of parallel algorithms. Considerable progress has been made in many areas: graph theory, combinatorics, matrix theory, numericcal analysis,) etc. For many problems NC algorithms have been found (e.g. maximal independent set [KW], [GS], sorting [AKS], permutation grqup membership [Lu], CFL recognition [Ru], matrix arithmetic [C&l, [PRl]). For many other problems proofs of P-completeness have been obtained (e.g., the circuit value problem [La], unification [DKM], linear pr0grammin.g [DLR], roaximum network flow [GSS], lexicographically first maximal clique [Co]). Regrettably, the situation in computational number theory is less satisfactory. While NC algorithms have been discovered for lthe basic arithmetic operations [Sal, [Re], [BCH], and progress has been made on problems involving polynomials [Eb], [PR2], [BGH], the parallel complexity of such fundamental problems as integer gcds and modular exponentiation has remained open since first being raised in a paper by Cook [Co]. Indeed, it seems possible that the prospects for parallelism in computational number theory rest on these problems: if integer gcd

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