If you knew the entire equation of an elliptic curve $$y^2 = x^3 + ax + b~( mod ~~m)$$ given $a, b$ and $m$ which are all more than 70 digits in length, as well as a point $P$ and the point $t P$, with different orders, would it be possible to calculate $t$? What would be the fastest way to do this?
(I'm aware that $tP$ and $P$ not sharing the same order makes this less secure but I'm wondering how exactly to exploit it).
You mention your point has order $$93556643250795678718734474880013829509\\196181230338248789325711173791286325820$$ which factorizes as
$$2^2\cdot 3\cdot 5\cdot 7\cdot137\cdot593\cdot 24337\cdot 25589\cdot3637793\cdot5733569\cdot106831998530025000830453\cdot1975901744727669147699767.$$
Without much computing power, we can apply the Pohlig-Hellman algorithm to obtain $$t\bmod4, t\bmod3,t\bmod5,\ldots,t\bmod5733569.$$
Using the Chinese Remainder Theorem, we can combine all these results to get
$$t\bmod\left(4\cdot3\cdot5\cdots5733569\right),$$ i.e. $$t\bmod443208349730265573969192476820.$$
As @Ruggero remarks, the other primes are only about 80 bits. So with $\approx 40$ bits of computing power we could also break that discrete logarithm. It takes a little more effort than the others though, so if $t < 400\cdots 0$, then you may as well only do the easy ones.
Hint: this allows one to find a factor of $t$; find out how.
As an example, consider the multiplicative group modulo a prime, with a generator $g$. If $g^t$ is a quadratic residue, what does that say about $t$?
