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Searchable Symmetric Encryption aims at making possible searching over an encrypted database stored on an untrusted server while keeping privacy of both the queries and the data, by allowing some small controlled leakage to the server. Recent work shows that dynamic schemes -- in which the data is efficiently updatable -- leaking some information on updated keywords are subject to devastating adaptative attacks breaking the privacy of the queries. The only way to thwart this attack is to design forward private schemes whose update procedure does not leak if a newly inserted element matches previous search queries.