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Let $E$ be an elliptic curve over $\mathbf F_p$ given by the equation $y^2 = x^3 + Ax + B$. Then a quadratic twist $E'$ of $E$ is given by the equation $\beta y^2 = x^3 + Ax + B$ where $\beta$ is not a square. Now you have a point $(x_0, y_0)$ you suppose is not on $E'$. Then to make it on a twist, you need to find $\beta$ such that $\beta {y_0}^2 = {x_0}^3 ...


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TL:DR; The algorithm is working in parallel, however not practical due to the number of DNA strands requirements for safe-curves. Generic Discrete logarithm algorithms can easily beat this, [see here]. The algorithm The algorithms work in theory. The authors use $Q =lP$ let change it $Q = [x]P$. Their method, actually, is highly parallelized like any DNA ...


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meet-in-the-middle attacks rely on storing the results in a map Only the naive version of MitM does. The impractical memory requirement (size and accesses) of naive MitM can be greatly lowered with a relatively modest increase in other computation costs, and the calculation distributed to independent devices. The reference paper is Paul C. van Oorschot and ...


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In elliptic curve algebra, operations are computed over a field. In your case, the field is $\mathbb{F}_{11}$. $\mathbb{F}_{11}$ is a finite field containing 11 elements ($\{0,1,2,...,10\}$), and like every field there is and addition and multiplicative law. Theses two laws work other the ring of integers modulo 11 (often noted $\mathbb{Z}/11\mathbb{Z}$). ...


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Diffie-Hellman on Curve25519 and Curve448 is specified on RFC 7748 and the algorithm clearly mentions that the scalar must be a multiple of the cofactor (by setting the 2 or 3 least significant bits to zero depending of the curve). If you choose a key with least significant bits not all zeros, then it depends of the implementation. An correct implementation ...


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With Elliptic Curves, we can compute the order of any point (and, in particular, the point $nP$; this is especially easy on the curves we actually use for ECC, because those curves typically have an order $hq$, for a small $h$ and a large prime $q$ (and the order of any point is a divisor of $hq$). So, if $q$ is the order of the point $nP$, and if $n$ is ...


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No, DNA computing doesn't harness quantum phenomenon to achieve parallelisation/quantum speedup, DNA computing still falls under the classical classification. Landauer's principle asserts that there is a minimum possible amount of energy required to erase one bit of information, known as the Landauer limit Taking into account the Landauer limit on ...


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The comparison can be made according to know attacks and their timings. Your source doesn't provide a reference and date backs to 2015, so that is not a good site as keylength.com. The current values of this answer is taken for 2019. ECC For ECC 128 bit security means we need 256-bit curves due to the generic discrete log attacks that have $\mathcal{O}(\...


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Note you're doing direct division instead of modular inverse. Since there is no division in elliptic curve arithmetic, you need to do mod inverse. pow function could be used as a helper here; it takes third argument as modulus: >>> P = 2 ** 255 - 19 >>> X = 15112221349535400772501151409588531511454012693041857206046113283949847762202 >&...


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