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In order to convert 3-move id-schemes to digital signatures you have to replace the request (or the challenge) by a value of a secure hash function applied to the message. So in the case of the specific id-scheme you have to choose a random element from the set $\{0,1,2\}$ for your message. That is $H(m)\in \{0,1,2\}.$ For instance if you have $H(m)=0$ then ...


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We recently discussed this question with some colleagues and this is what we came up with (no guarantees): Grovers algorithm only outputs a correct answer if it is feed with a input set such that the target function evaluates as 1 for a single input and as 0 otherwise. The algorithm then iterates a subroutine $\sqrt{N}$ times (where $N$ is the size of the ...


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Found the answer: The $l$-torsion subgroup is isomorphic to a direct sum of two quotient groups: $E[l] \simeq \mathbb{Z}_n \oplus \mathbb{Z}_n$, hence the basis requires two points and the elements of $E[l]$ are represented by linear combinations of such a basis. [Reference: Elliptic Curves, Washington, section 3.1]


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In order to answer this question, we need to understand the basis behind all of modern cryptography, which is computational hardness. Today, we believe that we know how to construct block ciphers that are secure, except for brute force search (or almost that secure). However, we don't really know this. We also think that factoring is hard, and so on. All of ...


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Take a look at this chapter (Introduction to post-quantum cryptography by Daniel J. Bernstein). It gives an excellent overview of what is known. (Bottom line, Grover's is the best known today and so this can be thwarted by taking 256 bit keys.)



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