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Though, no doubt, data saving at client end, itself has many limitations. But still what is the best approach. - What would be the best medium - And what will be the procedure.
HTML5 is the best answer, but if you want browser compatibility , then secure cookies I'd say ? make sure use the latest technologies to encrypt and encode.
if you are using pure asp.net there is no way to do this, by mixing new features of HTML5 in memory stroage you can temporary do that, its not clear from your question what data you need to stroe, you can check the
https://www.html5rocks.com/en/features/storage
there are plenty, Xml, csv depending on your knowledge