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Does there exist any sorts of algebraic relationship between Original and Sub Key of DES? I understand that by solving K-Maps of 64-bit Input/48-bit Output, I can get 16 algebraic relationships relating each sub key with the original key. Has someone done it already? Can this be of any use in exploiting the vulnerabilities of DES?

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Laurie Brown and Jeniffer Seberry (see paper here performed a comprehensive analysis of the key schedule of DES, published in AUSCRYPT90 (conference series which later on turned into ASIACRYPT), and suggested some changes to strengthen the key schedule.

This is a good starting point to start hunting citations, and see what else has been done since then.

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