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4

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 $\beta {y_0}^2 = {x_0}^3 ... 0 No, the DNA computing method in the article can not solve cryptographically interesting instances of the DLP problem. That's not ruling out that DNA computing could. The main issue is that the method in the article is hopelessly inefficient: sure it has a number of steps linear with the key size$n$, but even in theory it uses exponentially much material:$\...

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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 ...

1

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 ...

2

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 ...

0

Its depend on the protocol used. The last and more efficient is Groth16 that use only 3 curve points in its proofs. You can see the size of the keys and the the proofs in the table 2 of Groth's paper. The computation complexity of the pairing depends of whats curve is used. In general ZoKrates/Ethereum and ZCash use the bn256 curve paramters and the pairing ...

0

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 >&...

3

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 $\mathbb{Z}/11\mathbb{Z}$). ...

0

Suppose you want to compute $100 \times 3$, but you only know how to do addition. You can do it with 99 additions: $$3+3=6$$ $$6+3=9$$ $$9+3=2$$ $$...$$ $$294+3=297$$ $$297+3=300$$ But this is slow. How can we speed this up? First we make bundles of 2, 4, 8, 16, 32 and 64 hops. $$2 \times 3 = 3+3=6$$ $$4 \times 3 = 6+6=12$$ $$8 \times 3 = 12+12=24$$ 16 \...

2

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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primitive methods defined here curve domain parameters available here check whether public_key_xy = scalar_mult(private_key,curve.g) matches expected target

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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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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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I am implementing a ECDH using Curve25519 to communicate two system. One system have library that use for weierstrass curve only, ... Curve25519 is the underlaying curve designed for X25519 Diffie-Hellman function (details see RFC 7748), it's designed to be more easily to implement correctly than Weierstrass curves. Curve25519 and secp256r1 (or any other ...

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