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We observe that we have y j · s j ? [?X,X] n ,X = 2 Y ?, for all j ? HS. We also observe that, if A makes all encryption queries QEncrypt(j, x 0,j , x 1,j , t) for all indexes j ? HS for a given tag t, so does B for the same tag t. By construction, B only makes a functional decryption query QDKeygen(f y ) for a function f y when A makes a partial functional key query QDKeygen(j, f y ) for the last honest sender's index j ? HS. In the Finalize step of Definition 2.14, Condition 3 requires that, for any function f y involved in queries QDKeygen(i, f y ) for all honest senders' indexes i ? HS, the condition f y (X 0 ) = f y (X 1 ) be satisfied for any pair of messages X 0 =, Asymptotically compact adaptively secure lattice IBEs and verifiable random functions via generalized partitioning techniques, 2017. ,
, From (38), it follows that, if secret bit of B's challenger is b = 0, B is playing Game 1 with A. Similarly
Game 3 and Game 2 are computationally indistinguishable as long as that the MCFE scheme of Section 3 provides IND-sec security. Under the LWE q,m,n1,?1 assumption, we have | Pr ,