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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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Curve 25519 (X25519, Ed25519) Convert coordinates between Montgomery curve and twisted Edwards curve
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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