Cryptography

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By Jürgen Müller

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The Tangled Web: A Guide to Securing Modern Web Applications

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Ransomware. Defending Against Digital Extortion

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Decryption is given as follows: q q q We have ρy = ρ i=1 xi ai = i=1 (ρai )xi = i=1 (ρ + αi )xi ∈ Fqd . Let µρ ∈ Fq [X] be the minimum polynomial of ρ; hence µρ is irreducible of degree d and Fq [X]/µρ Fq [X] ∼ = Fqd as rings via X → ρ. Let f ∈ Fq [X] such that deg(f ) < d and X y ≡ f (mod µρ ). Then f + µρ ∈ Fq [X] is monic of degree d such that q f + µρ ≡ X y (mod µρ ). Since i=1 (X + αi )xi ∈ Fq [X] has the same properties q we conclude f + µρ = i=1 (X + αi )xi ∈ Fq [X]. Hence f + µρ splits into linear factors in Fq [X], and [x1 , .

Given s ∈ N0 , as well as n ∈ N and a knapsack sequence [a1 , . . , an ] ∈ Nn0 , the subset sum problem is to find a n sequence [x1 , . . , xn ] ∈ {0, 1}n such that i=1 xi ai = s, or to show that such a sequence does not exist; in general a solution sequence is not necessarily unique. The subset sum problem is solved by the following recursive algorithm S, returning S(s; a1 , . . , an ), which is a solution sequence if such a sequence exists, or fail otherwise: if n = 1 then if s = 0 then return [0] if s = a1 then return [1] return fail T := S(s; a1 , .

Any such discrete logarithm problem yields a generalised ElGamal cryptosystem, which is secure only if solving the underlying discrete logarithm problem is computationally difficult. There are discrete logarithm problems which can be solved in polynomial time; e. g. for n ∈ N the additive group Z/nZ is cyclic with generator a ∈ (Z/nZ)∗ , and for x ∈ Z/nZ we have loga (x) = xa−1 ∈ Z/nZ, which hence can be computed in polynomial time by the Euclidean algorithm. Still, there are finite commutative groups possessing cyclic subgroups having conjecturally difficult discrete logarithm problems: a) (Z/pZ)∗ where p ∈ N is a prime, and (Z/nZ)∗ where n ∈ N composite.

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