Searching for a key in 2^256 and searching for a key in 2^70 are not the same problem on different scales. They are different problems. The first is physically impossible, as the intermediate trail measured. The second is a project: expensive, time-consuming, with existing hardware and people who have already executed it. Everything in this module resides in that difference.
A search range is exactly what the name implies: a beginning and an end. Saying that a key is in the range of seventy bits means it is greater than or equal to 2^69 and less than 2^70 — meaning there are 2^69 candidates, and none outside of it. The range does not make the math weaker. It reduces the field where one searches.
The correct measure of this field is the number of bits, and it is what gives the correct proportions. Each additional bit doubles the work: a 71-bit range costs twice as much as a 70-bit one, and an 80-bit one costs a thousand times more. Talking in bits instead of numbers with twenty digits prevents intuition from getting lost — the difference between 70 and 80 seems small written like this, yet it is the difference between weeks and centuries.
Now the distinction that decides everything in this module, and that many people discover late. There are two completely different situations, depending on what is known about the target.
In the first, the public key is known. This happens whenever the address has spent something: to spend, it is necessary to present the public key, and from that moment it is published on the blockchain forever. With the public key in hand, the problem becomes a discrete logarithm in a small range, and the square root algorithms in the upcoming lessons attack 2^69 candidates with something around 2^35 steps. Thirty-five bits is an afternoon's work.
In the second, only the address is known. An address is the hash of the public key, and a hash cannot be undone — Lesson 1 of Module 4 emphasized this. Without the public key, there is no curve to attack, no square root shortcut, and only the long path remains: draw a number, derive the public key, calculate the hash, compare with the address, repeat. It takes 2^69 complete rounds, not 2^35.
Keep this asymmetry in mind, because it explains almost the entire story of the challenges that the last module will recount: those that fell first were generally the ones whose public key was exposed, and those that resist are the ones that remain untouched, protected not by the curve, but by the hash function in front of it.
It is also worth mentioning where the small ranges come from, because they do not appear on their own. Some are deliberate: in 2015, someone created a wallet where the key for each address was drawn within a range of increasing size, one bit at a time, and funded all 256. Others are accidental and much sadder: a phrase chosen by a person may have perhaps thirty bits of unpredictability, a faulty generator can reduce 256 bits to just a few dozen, and a test program that became a real wallet usually draws from a tiny range.
A correctly drawn key, with 256 bits of true entropy, is not in any range — or rather, it is in the only range that no one sweeps. Nothing in this module threatens it, and it is important that this is stated before the six upcoming lessons.
What comes next is the honest account of each method, starting with the simplest of all. In the next lesson, how much it costs to try one number at a time — in keys per second, in electricity bills, and the exact point where the math stops adding up.