Download e-book for kindle: Black-Box Models of Computation in Cryptology by Tibor Jager

By Tibor Jager

ISBN-10: 3834819905

ISBN-13: 9783834819901

Generic staff algorithms remedy computational difficulties outlined over algebraic teams with out exploiting homes of a selected illustration of team parts. this is often modeled by way of treating the crowd as a black-box. the truth that a computational challenge can't be solved by way of a fairly limited classification of algorithms could be noticeable as help in the direction of the conjecture that the matter can be demanding within the classical Turing desktop version. in addition, a reduce complexity sure for yes algorithms is a worthy perception for the hunt for cryptanalytic algorithms.

Tibor Jager addresses a number of basic questions relating algebraic black-box versions of computation: Are the regularly occurring team version and its editions a cheap abstraction? What are the restrictions of those versions? will we sit back those types to carry them in the direction of the reality?

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Example text

In particular, B computes gcd(Pk (y), N) for y ← U [C ] and the straight line program Pk P satisfying Pr Pk (x) ∈ ZN \ Z∗N and Pk (x ) ∈ Z∗N : x, x ← C $ = max Pr Pk (x) ∈ ZN \ Z∗N and Pk (x ) ∈ Z∗N : x, x ← C $ . 3 algorithm B finds a factor in this step with probability at least γ1 . Moreover, B evaluates any pair Pi , Pj of straight line programs in P with a uni$ formly random element y ← U [C ] and computes gcd(Pi (y) − Pj (y), N). So in par$ ticular B computes gcd(Pi (y) − Pj (y), N) with y ← U [C ] for the pair of straight line programs Pi , Pj P satisfying $ Pr Pi (x) ≡N Pj (x) and Pi (x ) ≡N Pj (x ) : x, x ← C = max −1≤i< j≤t $ Pr Pi (x) ≡N Pj (x) and Pi (x ) ≡N Pj (x ) : x, x ← C .

T. t. t. j = i and Δ(y) ≡ 0 mod pei i and Δ(y) ≡ 0 mod p j j ≤ Pr [gcd(N, Δ(y)) ∈ {1, N}] . 4 Let C ⊆ ZN and V ⊆ C with |V | > 1. The subset membership problem defined by (C , V ) $ is: given x ← C , decide whether x ∈ V . In this chapter we will consider only subset membership problems such that |C | = 2 · |V |, this is sufficient for all our applications. We formalize the notion of subset membership problems in the generic ring model as a game between an algorithm A and a generic ring oracle O.

We replace the original oracle O with an equivalent oracle O1 . Oracle O1 proceeds exactly like O, except for the following. Instead of list L, O1 maintains a sequence P storing the sequence of computations performed by A . P is initialized to the empty sequence. 1. Due to our construction of the generic ring oracle (newly computed ring elements are always appended to L, see Chapter 2) we have Pk−2 (x) = Lk 38 4 Analysis of Cryptographic Assumptions in the Generic Ring Model for all k ∈ {1, . .

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Black-Box Models of Computation in Cryptology by Tibor Jager


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