Challenge 1
Hey, my lam, what’s up? You don’t know me, but I saw you in action during the playtest... nice moves!
Anyway, to introduce myself, I’m Cyberlam #49, of the Cryptography Flock, and I was wondering if you could lend me a hand with a little hac-... err, investigation.
There’s this juicy wallet I ran across, and the private key for it seems weak somehow, like it’s got a lot of small prime factors. If you help me crack it, I’ll deal you in for a slice of whatever’s inside...
Step one is figuring out who it belongs to, though. The big clues I’ve found are that they go by “The Alchemist”, are super-rich and like to throw their money around in auctions, and are kind of eccentric-slash-crazy, like chewing on the scenery while laughing maniacally, that kind of thing...
Argh, this is so close to a score for us, but I think I need just one more piece of info to nail this sucker down. Does anything ring a bell for you?
This is intended to be almost an immediate fill-in for people who have been participating in the Cyberlam auctions so far, since Alchemist 19 has been a well-known personality on the Lamina1 Discord almost from day one. However, the “so close to a score” wording also implies that we are looking for the prime number which is closest to 20, so you could get the answer of 19 that way too.
Challenge 2
Alchemist 19! Yeah, I think that’s our guy.
An eccentric billionaire, after he sold a couple of startups and then rode the crypto wave to the top a few years back. Holy Lam, can you imagine what this wallet might have in it?
And this totally explains the prime factors hidden in his private key, too. He seems to be obsessed with numbers, and I’ll bet he picked it by hand, starting with his own name. A classic security blunder, and that’s how we’re gonna crack it!
I found this old interview with him, where he says the world even runs on numbers:
“We all know of purely mathematical constants like pi, and physical constants like the speed of light. But did you know that there are things in-between? Dimensionless ratios, that don’t depend on how long your ruler is, but that are baked into the fine structure of the universe.
“One of the most mysterious of these constants controls the interactions between electrons and photons. It determines the fine structure of the atoms in our bodies, the very chemistry that keeps us alive. I’ve got a plaque with it on my wall so that I can keep pondering its value. Does it fall out inevitably from math, or was it chosen by a higher power to make the universe work? No one knows...”
If he’s making up numbers for his key, he must have picked this one! But I haven’t got a clue what number he might be referring to. Any ideas?
The main clue here is the repetition in the wording, as “constant” shows up three times in the quote, and “fine structure” shows up twice. There are other ways that you can also get to the fine-structure constant with web searches for things like important dimensionless numbers in electromagnetism, etc. Its value is pretty close to 1/137, so 137 is the answer we were looking for.
Note also that the mentions of pi and the speed of light, beyond being related to the fine-structure constant’s formula, are intended to foreshadow their use in Challenges 7 and 4.
Challenge 3
Eureka! Good work, my lam!
Meanwhile, I’ve been tracing some network connections associated with our pal 19, and I think I’ve found a data center that he owns. It’s registered to one of his companies, which sounds like a factory, so maybe it’s a secret project!
The network architecture is weird, though. The cluster seems to contain a little under 1000 servers, but the arrangement is something I haven’t seen before.
Normally you’d have one node connect to a few others, and each of those connect to a few others, etc. until you can get from any point A to point B in several hops. The number of nodes you can reach grows exponentially, so it’s nice and efficient.
But this data center seems to be scaling even faster than exponentially. Like one node connects to two others, each of which connects to three others, and so on. There was this one video clip where 19 bragged, “We’ve got the best network anywhere, exclamation point!” I mean, like, he said the punctuation in words. What a total wacko...
I can’t make sense of it. The whole scheme seems to be like 6 layers deep, and it’s hurting my brain to trace it all the way through. One node seems to be missing from the pattern, but I need to know how many I’m dealing with to be sure of that. Little help?
The puzzle is pointing you towards the factorial function, with several different clues. It “sounds like a factory”. It grows faster than exponentially. It is constructed from the sequence 2, 3, 4, etc. And its mathematical symbol is the exclamation point. The mention that the network is 6 layers deep means that we’re looking for 6! = 720, and the one missing node turns that into 719, a prime number which nicely ends in 19.
Challenge 4
Thanks! My hunch about the one node being missing was right, but now we’re going way deeper down the rabbit hole here.
I’ve been watching that extra node on the net, and its latency delays to the rest of the cluster are super-weird. Like they vary with a regular pattern, up and down, repeating every two hours or so.
... I think that it’s actually a satellite in orbit. So the ping varies between when it’s directly overhead, then farther away at the horizon, yada yada yada. Dammit Jim, I’m a hacker, not an astrophysicist!
But I think we could figure out which satellite it is if we knew its precise altitude in kilometers, since they’re all tracked by radar in a public database.
The minimum round-trip latency I saw was 13.2 milliseconds, which should be when it’s closest. Which isn’t bad, I could play Persona Collective with that kind of ping... sorry for the derail.
Anyway, do you have any guess how far a radio wave could get up to orbit and back in that time?
This is a relatively straightforward problem, just asking you to turn a radio-wave latency time into a distance. Radio travels at the speed of light, previously mentioned in challenge 2, which is 299,792,458 meters/sec. So in 13.2 milliseconds (0.0132 seconds), it would go 3,957,260 meters. Then divide by 2 to account for the distance both up to the satellite and back down, and you get 1,978,630 meters, or 1979 to the nearest kilometer, which is prime and also starts with 19.
Challenge 5
Score! There’s only one satellite at that exact altitude, and it’s in the database. A boring name, “R-Process”, but it seems to indeed be owned by one of 19’s companies.
Three things are strange about it, though. First, it’s supposedly part of a “data center in space” project, but when I spotted a photo of it getting prepped for the rocket launch, the solar panels were tiny. A bunch of CPUs should really be sucking down more juice.
Second, the launch manifest doesn’t add up. It was on a dedicated rocket, mostly empty, but the weight was way up there. Either they were flying something else secretly too, or that satellite is made of lead, heh heh.
And third, the orbit. Nobody else seems to operate at that altitude, maybe because of the radiation or something. I think this is the important part, and 19 probably picked the orbit for his own quirky reasons.
Looks like a reporter actually asked him about it during a press conference, and here’s what he said:
“Oh, indeed, it’s a special orbit. I wanted to use lucky #1000, but that was just too far away from the Earth, so I just flipped it around and ended up with this one instead. It’s actually a prime altitude for coverage of our ground stations, so the numbers cooperated with me once again.”
He’s not making any sense, though. 1000 km is actually closer than the real orbit, but he said that it was too far away, so I’m not sure what he’s talking about. Do you understand?
This puzzle is a bit more subtle, and most of the flavor text in the story is actually setting up the plot for Challenge 7 rather than being immediately relevant to this one. The primary clues from the quote are the reference to #1000, and the use of the word “prime”. We’re also looking for an answer which is related to the previous 1979 value but has been flipped around somehow.
Since it’s “#1000” of something instead of just “1000” as a value, the major question to ask is: what relevant things might we want to count? Hopefully “the primes” should spring to mind by now, so the answer is the 1000th prime number, which is 7919, just 1979 flipped around by its pairs of digits. As mentioned in the story, 79 is the atomic number of gold on the periodic table too, but there wasn’t enough information given to make use of that fact here.
Challenge 6
I think I’m learning from you, because I spotted something in the last two numbers: 79. Guess what that corresponds to on the periodic table, my lam? It’s shiny, shiny gold!
It’s all coming together now, why they call him the Alchemist. He’s super-talented with turning stuff into gold, obviously like startup companies! And he probably picked those numbers from the gold connection because of that, too.
I may have also found where the “19” came from, as it seems to be part of his origin story. Here’s a clip from an interview he did recently:
Reporter: “How do you feel about being labeled as an out-of-touch financial elite?”
Alchemist 19: “Elite? Ha-HA! Funny story about that. When I got started I actually had only a few hundred to my name, but I worked and learned and worked some more. 19 years later, I sold my first company for 62 million. That’s a 90% annual rate of return, so let that be a lesson for everyone: if you want to beat the stock market, invest in yourself! Although I must admit I found a lucky penny on the sidewalk that first day too...”
I think there could be something here. For someone like him who pays so much attention to numbers, he was sure dropping a lot of them in that interview.
But I don’t know what to do with them. Is there some way to back-calculate how much money he started with, like to the penny? Maybe that’s something he would have remembered and used in his key.
This is another relatively straightforward calculation, here using compound interest formulas. A 90% return means an annual growth factor of 1.90, and over 19 years it would be raised to the 19th power. This is a total growth of about 197,842. Dividing $62,000,000 by this value gives $313.38 as an initial value, and in pennies that would be 31,338.
But that value isn’t prime, so we have to pay attention to the mention that there was an extra penny found on the first day, so the real prime-number amount Alchemist 19 started with is actually one less instead: 31,337. This also spells “ELEET” in hacker jargon, which is why he was laughing at the repeated mention of the word “elite”.
Challenge 7
Trouble! We’ve got trouble, right here in Cyber City! With a capital “T”, and that rhymes with E-M-P!
I had it all wrong! The gold, the satellite, everything! It’s actually crazy, my lam, pure crazy-pills supervillain territory!
He’s not called the Alchemist because he turns companies into gold, he’s the Alchemist because he’s obsessed with turning real actual lead into gold, like ancient-wizard style. Specifically the lead that he has placed on that satellite, along with... oh... I don’t know... a farmers-be-damned genuine atomic bomb!!!
I looked up what the “r-process” actually is, and it’s some nuclear-physics thing for how elements can get transmuted into other elements – like gold! – in exploding stars, or in the closest, biggest explodey thing we can make that’s got lots and lots of neutrons! If it goes off, you can say bye-bye to anything with an antenna underneath it...
And worse, I can see what looks like a countdown timer on the satellite controller, with about 8 hours left on the clock. I think I can hack it, but I need to know exactly what duration it was originally set to, like to the millisecond! There’s this odd comment in the script:
# TODO: Pick a suitable timer value.
# Years, days, hours, seconds, milliseconds, let’s bake them up into a delicious round pie,
# and then sprinkle just the right little bit of spice on top.
What the heck does this mean? Help!!!
The final problem is intended to be the hardest, not because it involves difficult math or science but just because it’s a bit cryptic to figure out what to do. We’re trying to look for a duration in milliseconds which is somewhat larger than 8 hours, so that’s 28,800,000 as a starting point. But how to get the exact value?
One set of hints involves the different units of time, so we have values to play with like 365 days in a year, 24 hours in a day, 3600 seconds in an hour, and 1000 milliseconds in a second. We also have the clue referencing “a round pie”, so we’re probably looking for a value which is “around pi” in some sense.
If you multiply 365 * 24 * 3600, you get 31,536,000, and 3.1536 is pretty close to 3.1416. It’s a somewhat known approximation that the number of seconds in a year is about pi times 10 million. However, this answer isn’t prime, so we need to add a little bit more to it to satisfy that. The next larger primes are 31,536,017 and then 31,536,019, and obviously the “19” one would have appealed more to Alchemist 19 as the right little bit of spice, so that’s what he picked. Amusingly 31,536,049 is the next prime after that.
Conclusion
Nicely done! And surprise... there’s actually no bomb, whew! I kinda made it up, my bad...
Allow me to re-introduce myself. I’m Cyberlam #49, of the Alchemy Flock, also known as Alchemist 49. Oooh, what a plot twist, ha ha!
Alchemist 19 is actually a friend of mine, and he’s exponentially less crazy than I told you about. By which I mean maybe a factor of 1/e...
But you’re almost to the finish line! You’ve got 7 prime factors now, but they only combine to give you about 85 bits, and private keys are 256 bits worth of hexadecimal digits. What to do, what to do, what to do...
When you figure it out, you’ll be able to import the key into MetaMask and log into @cyberlam0049 on the Lamina1 Hub with address 0xFE2D442AbE15759D4F0Fa2cd1587cF3af25503F0 to claim your prize.
But you need to be quick about it, because only the early lam gets the hay! You’d better transfer the NFT to your own wallet before someone else with the same private key grabs it instead. And sorry if you were already too late...
I hope you enjoyed this little puzzle challenge, exploring some fun topics in math and science, whether or not you won. For getting this far, you’re certainly an honorary member of the Alchemy Flock now!
“Math is the language of the universe. So the more equations you know, the more you can converse with the cosmos.”
– Neil deGrasse Tyson
Getting from the 7 individual challenge answers to the final private key isn’t intended to be difficult for folks who’ve gotten this far, but it takes a couple of steps. Multiplying all the factors together gives:
19 * 137 * 719 * 1979 * 7919 * 31337 * 31536019 = 28985679235331911456247171
As mentioned, this is too small for a private key, but 85 bits is almost exactly 1/3 of the needed size, so if we just cube this value, using each prime factor 3 times, then we get:
24352886550139607650237938625586594941484847858819970592143821851628977581211
The triple repetition of “what to do” in the puzzle was intended to be a hint that you should do something three times like this. Converting the value to hexadecimal produces:
35d73ecdc9a73d1e68f4369eabf673d248c0a065f6332253b9de04cfd824ec9b
And that’s the private key we need to access the @cyberlam0049 account and claim the prize. I hope you enjoyed all of these problems as a wrap-up to Cryptoversal’s amazing Cyberlam Chapter 2!
– Alchemist 49, a.k.a. EdKeyes