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An algorithm is fundamentally defined as a step-by-step method of solving a problem. It represents a finite sequence of well-defined instructions or procedures designed to perform a task or to find a solution to a particular issue. The crucial aspect of an algorithm is that it provides a clear and systematic approach to problem-solving, which can apply to various fields, including computer science, mathematics, and data processing.

The other options, while relevant to certain domains, do not capture the essential nature of what an algorithm is. A logical structure for data storage pertains more to data organization and design rather than the procedural aspect of problem-solving that algorithms encapsulate. A method for verifying identities involves authentication techniques, which are not inherently algorithms themselves but can implement algorithms as part of their processes. Lastly, a technique for data compression involves specific algorithms but focuses solely on one application of algorithms rather than defining what they are overall. Thus, the definition that highlights the step-by-step approach of solving problems is the most accurate representation of an algorithm.