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I have been reading about differential and linear cryptanalysis. They were mainly introduced by Adi Shamir and Biham to show weakness of DES. However, many of articles state that they have been considered as a technique to evaluate all other block ciphers. My question is, are they beneficial for public key cryptography? or are they limited to private key schemes ?

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Differential Cryptanalysis is attributed to Shamir&Biham, but Linear Cryptanalysis is generally attributed to Matsui (although The Block Cipher Companion says it was used earlier by Tardy-Corfdir & Gilbert). Coppersmith notes they were known by the NSA prior to this –  figlesquidge Dec 11 '13 at 10:18
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They are generally relevant only to symmetric-key cryptography (e.g., block ciphers, hash functions, message authentication codes). There's no deep reason why -- it's just that differential and linear cryptanalysis tend to be effective against the sort of structure that are commonly used in block ciphers, but not very effective against the sort of designs that are commonly used for public-key crypto.

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Probably there are much better methods in every case to break or heavily decrease the bit strength of asymmetric procedures than differential or linear cryptanalysis. Those are useful when there is no alternative, but in asymmetric procedures there is alway some better method. Think also that symmetric encryption is smaller in structure (so differential and linear cryptanalysis is more feasible) and so faster than asymmetric encryption, as in symmetric encryption speed matters much more than in asymmetric, where the function performed is the important thing.

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