Strong white-box cryptography
A method is provided for generating an output from an input according to a secret using a white-box implementation of a cryptographic function having a first operation, a second operation, and a third operation. The method applies the input to a first operation to generate a first intermediate result, applies the first intermediate result to a second operation to generate a second intermediate result, and applies the second intermediate result to a third operation to generate the output, wherein at least two of the first operation, the second operation, and the third operation is implemented by a plurality of interconnected logic elements, the interconnection of the plurality of logic elements being comprised of one of a non-algebraic interconnection of logic elements and an algebraic interconnection of logic elements having obfuscated boundaries between the at least one of the first operation, the second operation and the third operation.
1. A method for providing digital signal security comprising:
generating at least one output from at least one input and at least one secret that uses a white-box implementation of a randomized cryptographic function, the randomized cryptographic function obtained by at least one random operation performed on the whole of a nonrandomized function;
providing the randomized cryptographic function as a plurality of interconnected logic elements, the interconnection of the plurality of logic elements being comprised of non-algebraic interconnection of logic elements and algebraic interconnection of logic elements having obfuscated boundaries between the at least one input and output;
wherein the non-algebraic interconnected logic elements comprises a cyclic connection of at least a portion of the plurality of interconnected logic elements that produces an equivalent input-output relationship of a non-cyclic connection of logic elements;
wherein the logic elements are Boolean circuit gates; and
wherein the randomized cryptographic function has fewer Boolean circuit gates than the whole of the nonrandomized function.
2. The method of claim 1 wherein the logic elements are implemented by associated lookup tables.
3. The method of claim 1 , wherein the randomized cryptographic function is implemented by a subset (S) of the plurality of interconnected logic elements.
4. The method of claim 3 , wherein the subset (S) of the plurality of interconnected logic elements is defined by:
slicing a graph of the plurality of interconnected logic elements into the subset (S) of the plurality of interconnected logic elements; and
converting the subset (S) into a logic element having a fan-in equal to a number of inputs to the slice.
5. The method of claim 4 , wherein each of the plurality of interconnected logic elements are Boolean hardware circuit gates.
6. The method of claim 4 , wherein the each of the plurality of interconnected logic elements are implemented by an associated lookup table.
7. The method of claim 1 , wherein the plurality of interconnected logic elements is randomized by performing a plurality of Mal'tsev superposition operations on a non-randomized interconnection of plurality of logic elements for computing the randomized cryptographic function.
8. The method of claim 1 , wherein the plurality of interconnected logic elements is a randomized plurality of interconnected logic elements, randomized by:
generating a non-randomized plurality of interconnected logic elements C1 for computing the randomized cryptographic function;
performing at least one of the following on the non-randomized plurality of interconnected logic elements—C1 to generate the randomized plurality of interconnected logic elements:
introduction of a variable to the non-randomized plurality of interconnected logic elements C1;
permutation of a variable of the non-randomized plurality of interconnected logic elements C1;
deletion of a logic element of the non-randomized plurality of interconnected logic elements C1, the deleted logic element being an input gate having all outgoing edges removed and assigned to another input gate of the plurality of interconnected logic elements C1; and
substitution of a logic gate of the plurality of interconnected logic elements C1 by a second interconnection of logic elements C2.
9. An apparatus for providing digital signal security comprising:
a processor;
a memory, communicatively coupled to the processor, the memory storing instructions comprising instructions for:
generating at least one output from at least one input according to a secret that uses a white-box implementation of a randomized cryptographic function, the randomized cryptographic function obtained by at least one random operation performed on the whole of a nonrandomized function;
providing the randomized cryptographic function as a plurality of interconnected logic elements, the interconnection of the plurality of logic elements being comprised of anon-algebraic interconnection of logic elements and an algebraic interconnection of logic elements having obfuscated boundaries between the at least one input and output;
wherein the non-algebraic plurality of interconnected logic elements comprises a cyclic connection of at least a portion of the plurality of interconnected logic elements that produces an equivalent input-output relationship of a non-cyclic connection of logic elements,
wherein the logic elements are Boolean circuit gates; and
wherein the randomized cryptographic function has fewer Boolean circuit gates than the whole of the nonrandomized function.
10. The apparatus of claim 9 , wherein the randomized cryptographic function is implemented by a subset (S) of the plurality of interconnected logic elements.
11. The apparatus of claim 10 , wherein the subset (S) of the plurality of interconnected logic elements is defined by:
slicing a graph of the plurality of interconnected logic elements into the subset (S) of the plurality of interconnected logic elements; and
converting the subset (S) into a logic element having a fan-in equal to a number of inputs to the slice.
12. The apparatus of claim 11 , wherein the each of the plurality of interconnected logic elements are implemented by an associated lookup table.
13. A non-transitory computer readable medium containing program instructions for causing a computer to perform the method of:
generating at least one output from at least one input according to a secret that uses a white-box implementation of a randomized cryptographic function, the randomized cryptographic function obtained by at least one random operation performed on the whole of a nonrandomized function;
providing the randomized cryptographic function as a plurality of interconnected logic elements, the interconnection of the plurality of logic elements being comprised of non-algebraic interconnection of logic elements and algebraic interconnection of logic elements having obfuscated boundaries between at the least one input and output;
wherein the non-algebraic interconnected logic elements comprises a cyclic connection of at least a portion of the plurality of interconnected logic elements,
wherein the logic elements are Boolean circuit gates; and
wherein the randomized cryptographic function has fewer Boolean circuit gates than the whole of the nonrandomized function.