By Tibor Jager

Generic staff algorithms resolve computational difficulties outlined over algebraic teams with no exploiting homes of a selected illustration of crew parts. this can be modeled by means of treating the crowd as a black-box. the truth that a computational challenge can't be solved by means of a fairly constrained category of algorithms could be obvious as help in the direction of the conjecture that the matter can be not easy within the classical Turing computer version. in addition, a decrease complexity sure for convinced algorithms is a useful perception for the hunt for cryptanalytic algorithms.

Tibor Jager addresses a number of basic questions relating algebraic black-box versions of computation: Are the ordinary workforce version and its versions an affordable abstraction? What are the constraints of those types? do we sit back those versions to convey them toward the reality?

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This game corresponds to the generic DCR experiment described above, with b = 0. That is, the algorithm interacts with an oracle O0 whose initial list con$ tents is L1 = (1, x), where x ← Z∗N is a random element of Z∗N . We have Pr[A O0 (N, , e) = 1] = Pr[A O (N, , e) = 1 | b = 0]. 54 5 The Generic Composite Residuosity Problem Game 1. We change the way the challenge x is sampled. Instead of choosing $ $ x ← Z∗N , O1 samples x ← ZN . We assume that O1 does so by choosing two $ $ integers x0 ← ZN and x1 ← ZN −1 and setting x = x1 N + x0 .

Note that for x ∈ Z∗N we have (x | N) ∈ {1, −1}. Let JN := {x ∈ Z∗N : (x | N) = 1} be the set of elements of ZN having Jacobi symbol 1. Thus, we can perceive the problem of computing the Jacobi symbol as a subset membership problem (C , V ) over ZN with C = Z∗N and V = JN . 2. The cardinality |JN | of the set of elements having Jacobi symbol 1 depends on whether N is a square in N. 6]. This is an immediate consequence of the deﬁnition of the Jacobi symbol. Now we are ready to apply our main theorem to show that there is no efﬁcient generic ring algorithm computing the Jacobi symbol efﬁciently, unless factoring N is easy.

2] and independently in [AM09, Lemma 7] to generic ring algorithms that may also compute inverses, still for the case C = ZN with N the product of two primes. To prove our theorem we have to generalize this to the general case where algorithms may compute inverses, N = ∏i=1 pei i is the product of at least two different primes, and where C ⊆ ZN may be a subset of ZN . 1 Sequence of Games Let Succi (A ) denote the event that A solves the given instance of the subset membership problem successfully in Game i.