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The short answer is that LLL (or more generally, lattice reduction methods) is useful when you can convert your problem into finding a small linear combination of known vectors. Let's take your example and make it concrete. Let $m = 2^{64}$, $a = 7244019458077122845$, $b=1$, and the we have a generator that outputs the upper 16 bits of the state at each ...


3

what is the purpose of this line of code? while (bits - val + (n - 1) < 0); From a syntactic perspective, it ends the do opened above. From a functional perspective, it makes the pseudo-random number generated uniform on the specified interval. I'll focus on the later aspect. Assume n was $3\cdot2^{29}\,$ (that is 3<<29) rather than $6$. After the ...


3

The Falcon digital signature algorithm is as "meaty" as one can be. It has an NTRU-like key generation procedure and public key lattice structure; its security is based on conjectured difficulty of lattice reduction in classical and quantum computers; its efficiency is based on number-theoretic transformation and fast discrete normal ...


2

The RSA keys generated by the (Java) code in the question are 1024-bit, thus are not considered secure (even though there has not yet been any successful attack of a 1024-bit RSA key where that size was the only issue, see this). Security authorities no longer recommend anything below 2048-bit, and none that I know condones 2048-bit after 2030. Other common ...


1

Definition: $$NS_a(\alpha,\beta) = \# \{x| 0 \leq x < 64, (\oplus_{s=0}^{5}(x[s] \bullet \alpha[s])) = (\oplus_{t=0}^{3}(S_{a}(x)[t] \bullet \beta[t]))\}$$ Here, $x$ is your input to the S-box, and $S_a(x)$ is the output of the corresponding S-box. $\alpha$ and $\beta$ are the masks. Then, $NS_a(\alpha,\beta)$ is the number of coincidences that exor of ...


1

It is natural to wonder if Arora-Ge can break LPN, but as you suspect, it does not work. The essential problem is that, because the modulus is $q=2$, the method does not find the unique solution $s$, nor does it even narrow down the set of possible solutions. The reason is that the first step of the algorithm converts each LPN sample into a quadratic (in $s$)...


1

Almost all block cipher performs some types of substitution and permutation in every round function. For AES, ShiftRow, MixColumn, AddRoundKey are linear operations; only S-box operation is a nonlinear operation. If all operations in a round function are linear then there is no advantage of having multiple(10,12,14) iterative encryption rounds to produce the ...


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