Secure data transformations

A method of providing a system for secure processing of data by performing a secure sequence of n transformations Ti, wherein i=1, . . . ,n, to a data value, using encrypted representations of the data value to obtain a secure outcome, the method comprising acts of:

• providing a processor;
• configuring the processor to apply a first transformation to the data value w0 to obtain an obfuscated representation (X0, Y0) of w0, wherein the obfuscated representation (X0, Y0) contains a redundancy that depends on an input variable r;
• configuring the processor to, for each of i=1, . . . ,n−1, apply a second transformation ui, to compute (Xi, Yi) from (X(i−1), Y(i−1)), such that (Xi, Yi)=ui(X(i−1), Y(i−1));
• and
• configuring the processor to apply a third transformation G that depends on X(n−1), Y(n−1), and r, by computing wn=G(X(i−1), Y(i−),r), to obtain the secure outcome, wherein wn=Tn∘ . . . ∘T1 (w0), the third transformation being performed by applying a fourth transformation un such that Xn=un(X(n−1), Y(n−1)), and a fifth transformation F that depends on Xn and r, by computing wn=F(Xn,r), to obtain the value of wn;
• and outputting of the secure outcome,
• wherein (Xi, Yi)=Ψi(wi, σi), for i=0,1, . . . ,n, wherein Ψ_i is a predefined obfuscation function that defines a one-to-one relation between (Xi, Yi) and (wi, σi), wi being processed data and σi being redundant state variable for obfuscating wi, and wherein Ψi satisfies a condition that there is a one-to-one mapping that maps any value of (Xi, σi) to a value of (wi, Yi) in such a way that (Xi, Yi)=Ψi(wi, σi) such that a change of Xi or Yi causes a change of the redundant state variable σi and makes unpredictable the secure outcome of the sequence of transformations wn, wherein σ0 depends on r, wherein wi=Ti(w(i−1) and σi=gi (σ(i−1)) for i=1, . . . ,n for predetermined functions Ti and gi, and
• wherein the first transformation means, the second transformation means, and the third transformation means are configured to obfuscate the values of w1, . . . ,w(n−1) and σ0, . . . σn.

A method of secure processing of data by performing a secure sequence of n transformations Ti, wherein i=1, . . . ,n, to a data value to obtain a secure outcome, using encrypted representations of the data value, comprising acts of:

• applying a first transformation to the data value w0 to obtain an obfuscated representation(X0, Y0) of w0, wherein the obfuscated representation contains a redundancy that depends on an input variable r;
• for each of i=1, . . . ,n−1, applying a second transformation ui to compute (Xi, Yi) from (X(i−1), Y(i−1)), such that (Xi, Yi)=ui(X(i−1), Y(i−1));
• and
• applying a third transformation G that depends on X(n−1), Y(n−1), and r, by computing wn=G(X(i−1), Y(i−1), r), to obtain the secure outcome of the sequence of transformations, wherein wn=Tn∘ . . . ∘T1(w0), the third transformation being performed by applying a fourth transformation un such that Xn=un(X(n−1), Y(n−1)), and a fifth transformation F that depends on Xn and r, by computing wn=F(Xn, r), to obtain the value of wn;
• and outputting of the secure outcome,
• wherein (Xi, Yi)=Ψi(wi, σi), for i=0,1, . . . ,n, wherein Ψi is a predefined obfuscation function that defines a one-to-one relation between (Xi, Yi) and (wi,σi), wi being processed data and σi being redundant state variable for obfuscating wi, and wherein Ψi satisfies a condition that there is a one-to-one mapping that maps any value of (Xi, σi) to a value of (wi, Yi) in such a way that (Xi, Yi)=Ψi(wi, σi) such that a change of Xi or Yi causes a change of the redundant state variable σi and makes unpredictable the secure outcome,
• wherein σ0 depends on r, and wherein wi=Ti(w(i−1)) and σi=gi(σ(i−1)) for i=1, . . . ,n for predetermined functions Ti and gi, wherein w1, . . . , w(n−1) and σ0, . . . σn are obfuscated in the acts of applying the first, second and third transformations.