Ethereum's Justin Drake: AI May Break Crypto Keys Before Quantum Era
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https://nile1.com/ethereums-justin-drake-ai-may-break-crypto-keys-before-quantum-era/
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#cryptocurrencyWallets #ecdsa #ellipticCurve #freshAddresses #justinDrake #openai #privateKeys #quantumComp

The Cofactor can be directly computed from the curve:

E = EllipticCurve(GF(2^256-2^32-2^9-2^8-2^7-2^6-2^4-1), [0,7])
G=E.lift_x(55066263022277343669578718895168534326250603453777594175500187360389116729240)
n=E.cardinality()
order=G.order()
assert n == order
print(n, order)

Because this is a _modular_ curve, its points correspond to (isomorphism classes of) elliptic curves with a bit of extra structure. The four non-cusp points correspond to the curves listed here:

https://www.lmfdb.org/EllipticCurve/Q/50/a/

They have short Weierstrass equations:

\\[ […]

📄 Skyscraper: Fast Hashing on Big Primes: doi.org/10.46586/tch...

A Selmer‑inspired framework creates elliptic curves via a Las Vegas algorithm that retries until a curve passes cryptographic checks. The curves run constant‑time. Read more: https://getnews.me/selmer-inspired-method-produces-transparent-elliptic-curve-parameters/ #ellipticcurve #selmer

EllipticCurve.btc
每一笔比特币交易的安全，
都来自这条曲线。
y² = x³ + 7
没有椭圆曲线，就没有比特币。
数学即信任。
#Bitcoin #EllipticCurve #secp256k1

So EllipticCurve(GF(191),[4,128]) and E2=EllipticCurve(GF(137),[4,128]) have |E|=168 and |E2|=142. E3=EllipticCurve(Zmod(137*191),[4,128]). Counting using lift_x on E3 gives 168*142. One point is double counted, but I think we also leave out the O point so it works out.

Consider

E=EllipticCurve(Zmod(17*19),[3,7])

where this is true. One question I have, which I suspect is true, is that as long as your decomposition for your multiple doesn't ever involve 14, you won't ever compute a different point.

Alright; so technically this "works":

```
def elliptic_curve_factorization(N):
E = EllipticCurve(Zmod(N), [3,7])
p = random_point_on_curve(E)
acc = p
while True:
try:
acc += acc
except:
break
return gcd(N,acc[0]-p[0])
```

4/ Affine Coordinates `(x,y)` is the standard, intuitive way to represent a point on a 2D plane. You have an x-value and a y-value. Addition and Doubling ops in ECC require modular inversion(clock math).

#math #cryptography #affine #ellipticcurve