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Low-order subdomain basis methods of moments provide little help when highly accurate electromagnetic computations are required. High order modeling, on the other hand, can fill the need for enhanced accuracy but at the expense of greater implementation cost. To supplement such methods, this paper presents Nyström techniques, both easily implemented and highly accurate, relevant to TM scattering by arbitrarily shaped smooth infinite curved strips. The analysis takes full account of both the singular nature of the kernels and the singularities of the solution at the edges; as a result, the proposed solutions are exponentially converging. In addition, by eliminating inner product integrals, closed form analytical expressions are obtained for all matrix elements; thus our algorithms have very low implementation and computational cost. Detailed numerical examples and case studies amply demonstrate the efficiency, stability, and extremely high accuracy of the algorithms. These algorithms apply uniformly from electrically small to electrically large conducting screens. With only slight modifications the present analysis can be also used to obtain exponentially converging Galerkin solutions.