# Secure and Privacy-Preserving Data Aggregation Protocols for by Sen, Jaydip

By Sen, Jaydip

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Parity, circuits, and the polynomial-time hierarchy, Mathematical Systems Theory 17: 13–27. Goldreich, O. (2001). Foundations of Cryptography. Basic Tools, Cambridge University Press. Goldreich, O. (2004). Foundations of Cryptography II. Basic Applications, Cambridge University Press. Goldwasser, S. & Bellare, M. (2001). Lecture Notes on Cryptography, Summer course on cryptography at MIT. , Hirsch, E. A. & Pervyshev, K. (2009). A complete public-key cryptosystem, Groups, Complexity, and Cryptology 1: 1–12.

Y n − 1 ) y n ⊕ y1 ⊕ y n − 1 ⊕ y n − 2 , (18) ··· x3 = ( y1 ⊕ . . ⊕ y n − 1 ) y n ⊕ y1 ⊕ y n − 1 ⊕ . . ⊕ y4 , x1 = ( y1 ⊕ . . ⊕ y n − 1 ) y n ⊕ y1 ⊕ y n − 1 ⊕ . . ⊕ y3 . Lemma 17. The family of functions { f n }∞ n=1 is feebly one-way of order 2. Proof. It is easy to see that f n can be computed in n + 1 gates. Each component function of f n−1 , except for the last one, depends non-trivially of all n variables, and all component functions are different. Therefore, to compute f n−1 we need at least (n − 1) + (n − 2) = 2n − 3 gates (since f n is invertible, Proposition 6 is applicable to f n and f n−1 ).

Let us now consider the following problem: can we deﬁne a set of (simple) tests to know on which processor we are? As a practical example: is it possible to know whether we are on a mobile phone or on a computer? Malicious Cryptology and Mathematics Malicious Cryptology and Mathematics 37 15 The Intel Assembly Language instruction CPUID can be used both on Intel and AMD processors, but it has at least two severe drawbacks: • it is easy to “ﬁnd” it whenever scanning the ﬁle (malware detection issue); • some other processors cannot recognize and process this instruction.