We propose efficient strategies for calculating point tripling on Hessian (8M+5S), Jacobi-intersection (7M+5S), Edwards (8M+5S) and Huff (10M + 5S) curves, together with a fast quintupling formula on Edwards curves. M is the cost of a field multiplication and S is the cost of a field squaring. To get the best speeds for single-scalar multiplication without regarding perstored points, computational cost between different double-base representation algorithms with various forms of curves is analyzed. Generally speaking, tree-based approach achieves best timings on inverted Edwards curves; yet under exceptional environment, near optimal controlled approach also worths being considered.
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