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This is more an answer to why the uniqueness of the sums effects the size so much that the case for $52485332$ becomes trivial (its way to long for a comment). When all sums must be unique, then they must result in different integers. Because there are $500^{1000}$ possible sums, there are also $500^{1000}$ different integer results for that. the lowest case ...

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So, the only way must be an Exhaustive Search Algorithm As I mentioned in my comment, there are practical methods for nonhuge values of $m$ (and $m=52485332$ is not huge). Here is the outline of one such method (which assumes all the sets consist of nonnegative integers): We have an array $A_{n, m+1}$; each element of the array $A_{a, b}$ will note how we ...

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A mono-alphabetic substitution cipher simply replaces each symbol with another symbol, in a 1:1 fashion. So indeed you have 26 symbols or letters in the ciphertext. Now say you write down the ABC: ABCDEFGHIJKLMNOPQRSTUVWXYZ Each of these letters will need to be substituted by another one to go from plaintext to ciphertext. Lets use the same same symbols ...

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"Prime factorization" is not worth interest, for primes are their own factorization. Factorization into primes is not used for key generation. I conclude the question asks: When generating primes during generation of public/private key pairs for crytosystems based on hardness of factorization (RSA, Rabin, Pailler…), is there a limit on the size of ...

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If it does matter, what is the current state of the art elliptic curve and how does it compare with popular elliptic curves such as Curve25519 or secp256k1? Well, if you have an elliptic curve with a large subgroup of size $q$ (which is prime), then we know how to compute a discrete log within that subgroup in $O(\sqrt{q})$ time, and this applies to all ...

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In short, AES-256 beats AES-128. Use AES-256 which is the golden standard; Cryptanalysis The attacks on AES-256 doesn't make it insecure practically, even after 20 years the best attack has the complexity of $2^{254.3}$ for AES-256 and $2^{126.0}$ for AES-128 2015 Improving the Biclique Cryptanalysis of AES, Biaoshuai Tao and Hongjun Wu. The related key ...

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The problem is that you are trying to put a generic security comparison on AES-128 and AES-256. I would argue that if you're going to dig deeper - as you currently do - then this idea that there is a "generic security level" as I would coin it, is flawed. As you said, there is a related key attack that is only valid for AES-256, mainly related to ...

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The larger the key size the slower the operational performance. Is it true? Yes, but it depends on the algorithm how much difference it makes. And if the algorithm is slow in the first place then the speed difference may make more of an impact. For instance, AES-128 has 10 rounds while AES-256 has 14 rounds. So choosing AES-256 is generally only slightly ...

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