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I am from Mathematics Background. I am getting an offer to a phd in cryptography but am also thinking of going in deep learning theory- which area has more scope in future in research?

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Both are pretty hot at the moment and will likely stay hot for a while, but the future remains unknown to us. Some truly key breakthrough in a field can quiet interest in it, or expand it.

But you will have an opportunity to re-examine the question over the course of a career and make judgements about whether to continue a line of research or move to another.

A decision made now isn't necessarily a "forever" decision.

"Prediction is hard. Especially about the future." (Source unknown)

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