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Decryption error may happen: In our underlying HE scheme, the message space is the polynomial ring over modulus p. During the operations, δi,j and ηi,j increase and may become p which is congruent to 0 modulo p. Since we are interested in whether each value is zero or not, incorrect results may be derived if the values become congruent to 0 modulo p by accident. 2 Overview of Improvement For the number m of pointer variables and the maximal pointer level n, the followings are our solutions. – Level-by-level Analysis: We analyze pointers of the same level together from the highest to lowest in order to decrease the depth of the arithmetic circuit described in the Sect.

Because performing the step 1 entails m homomorphic multiplications on each δ¯i,j , and repeating the 2 two steps m2 times performs about mm homomorphic multiplications on each δ¯i,j . Output Determination. Decsk (δ¯i,j ) = 0 and 1 ≤ i, j ≤ m} Why Do We Not Represent the Algorithm by a Boolean Circuit? One may wonder why we represent the pointer analysis algorithm by an arithmetic circuit rather than a Boolean circuit. As an example of applying the Trans rule, ηi,k ∧ δk,j . However, this representation we might update δi,j by δi,j ← 1≤k≤m causes more multiplicative depth than our current approach.

Efficient fully homomorphic encryption from (standard) LWE. In: FOCS (2011) 6. : Accelerating sorting of fully homomorphic encrypted data. , Vaudenay, S. ) INDOCRYPT 2013. LNCS, vol. 8250, pp. 262–273. Springer, Heidelberg (2013) 7. : Batch fully homomorphic encryption over the integers. Q. ) EUROCRYPT 2013. LNCS, vol. 7881, pp. 315–335. Springer, Heidelberg (2013) 8. : Homomorphic computation of edit distance. In: IACR Cryptology ePrint Archive, 2015:132. WAHC (2015) (to appear) 9. : Fully homomorphic encryption over the integers.

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