July 2020

# Cryptography is not Magic

Don’t roll your own crypto.

Broken crypto often looks like it works.

The best way to build crypto is to learn how to break it.

Use existing, vetted, well reviewed libraries.

Leave it to the experts.

*Infosec people and their followers.*

The running theme seems to be that cryptography is a kind of Dark Magic, best left to anointed High Priests. Us mere Mortals cannot hope to wield it safely without first becoming one of those vaunted Experts — a futile endeavour for those of us who know their place.

While being a good first order approximation, that kind of gate keeping is problematic on a number of levels. First, some people back in the real world have needs that current crypto systems don’t always fill. Insulting them with “you don’t know what you are doing” does not help. Second, it has a couple perverse effects:

- It stops many reasonable people from studying the subject.
- It causes less reasonable people to charge ahead anyway.
- If cryptography is an Art, we could make things up as we go along. And invent “unbreakable” ciphers that are anything but.

I would like to advocate a different approach. Cryptography has
rules, which are both simpler and more fundamental than we might think.
We could tell people *what* they don’t know, and give them some
pointers: introductory books or
courses,
or specific words and concepts. Here, I will outline what you need to
look for before you roll your own crypto system. I’ve identified 3 broad
categories: implementing crypto, designing protocols, and designing
primitives.

*Note: this is an opinion piece, so let me clarify where I came
from. I program professionally since 2007, and started to teach myself
cryptography 4 years ago. I wrote a crypto library, whose recent audit
uncovered no major flaw.*

## Implementing crypto

Perhaps surprisingly, implementing cryptographic primitives &
protocols requires little cryptographic knowledge. *Choosing*
what to implement is more delicate (you need to know the state of the
art), but once you’ve made your choice, your only worries are side
channels and correctness.

Side channels have one rule: don’t let any data flow from secrets to that channel. Interesting side channels include timings, energy consumption, electromagnetic emissions… Most can be ignored most of the time (energy consumption for instance requires physical access), except timings. Never ignore timings, they’re part of most threat models.

On most CPUs, the timing side channel can be eliminated by removing all secret dependent branches and all secret dependent indices. Some CPUs also have variable time arithmetic operations. Watch out for multiplications and shifts by variable amounts in particular.

Also be careful around compilers and interpreters. No mainstream language specifies how to make constant time code, so your tools could insert secret dependent branches where the source code had none. High level languages in particular tend to have variable time arithmetic. In practice, low level languages like C are fairly reasonable, but I’ve seen exceptions.

Note that some primitives are more amenable to constant time software implementations than others: Chacha20 is naturally immune to timing attacks on most platforms, while AES requires special care if you don’t have hardware support.

Then we have correctness. The slightest error may throw cryptographic
guarantees out the window, so we cannot tolerate errors. Your code must
be bug free, period. It’s not easy, but it *is* simple: it’s all
about *tests* and *proofs*.

### Correctness of ciphers and hashes

Ciphers and hashes are fairly easy to test. At their core, they are about taking an input, and mangle it so thoroughly that the slightest change in the input will completely garble the output. In practice, the slightest programming error tends to change the end result completely.

All you have to do is compare the output of your primitive with a
reference implementation or test vectors. Ideally, do that for all
possible input & output lengths. *Including zero (empty
inputs)*. Practically, you can stop at a couple iterations of your
biggest internal loop, to be sure you hit all data paths.

If you have an init-update-final interface, try all possible cut-off points in the updates to make sure you get the same results. I’ve caught several bugs that way.

### Correctness of modular arithmetic

Modular arithmetic (polynomial hashes, elliptic curves…) is more
delicate. Now we’re adding or multiplying big numbers (130 bits, 255
bits…) that do not fit in a single machine register. Many libraries
don’t run in constant time, and custom code is often faster. So you end
up dividing those big numbers in several *limbs*, which can fit
in a single machine register.

*(You can think of a limb as a huge digit: instead of working in
base 10, we can work in base 2³², and store the corresponding “digits”
in 32-bit registers.)*

Once you’ve applied the precautions used for ciphers and hashes, the
biggest trap is *limb overflow*. Not a problem with naive
implementations, but there is one dangerous trick that often drastically
improves performance: *delayed carry propagation*. Make your
limbs smaller than the range of the registers they are stored into (26
bits limbs with 32 bits registers for instance), so that they can
temporarily exceed their normal range.

With this trick, some overflow bugs happen rarely enough that random tests will not catch them. There are two ways to address the problem: either don’t delay carry propagation to begin with (it’s safest), or write a mathematical proof that shows your algorithm never triggers an overflow. Ideally you’d have a machine verify that proof.

### Correctness of elliptic curves

Elliptic curves have their own traps. If you have explicit formulas
telling you what arithmetic operation to perform in what order, no
problem (beyond comparing to a reference implementation and limb
overflow). If you want to be clever, make sure you know *exactly*
what you are doing. If you see some mathematical term you haven’t seen
before (like “birational equivalence”), don’t guess,
*look it up*. I broke that rule once, and it wasn’t pretty.

Also note that Elliptic curves often require conditional selection, swapping, or even lookups. There are techniques to do that in constant time. Use them. Failing to do so has enabled actual exploits.

### Correctness of constructions & Protocols

Constructions and protocols are generally much simpler to implement than their underlying primitives. The same precautions apply, with a couple differences:

- Don’t forget the checks. If the protocol says you must abort upon a
certain condition, you
*absolutely must*. Failing to check something almost certainly destroys the security properties of the protocol or construction you are implementing. - Try the protocol (or construction) with corrupted inputs. Corrupt
every single byte, perhaps every single
*bit*, one by one, and test whether this causes the protocol to abort. If it doesn’t, you may have forgotten a check. - That’s a corollary from “no conditional branches”, but it bears repeating: when you compare buffers, use constant time code. Crypto libraries generally provide the comparison routines you need.

## Designing protocols

That’s where things become interesting. Where implementing crypto was
mostly tedious, designing protocols is *delicate*. One is not
necessarily harder than the other, but they’re very different.

This is no longer about faithfully implementing an unambiguous
specification. This is about addressing a *threat model*. This
means being aware of what you’re up against, addressing that threat, and
proving that you have done so. On top of that, the protocol should
allow, even encourage, good APIs that are hard to misuse.

### Threat model

We need to realistically and *precisely* define the
capabilities of the adversary. A typical threat model is the untrusted
network, where the adversary can basically do anything: watch messages,
intercept messages, forge messages, replay messages…

Some adversaries can also steal your keys. Maybe the police comes
with a warrant. Maybe your computer is being hacked. You can limit the
damage with forward secrecy (messages sent before the key was
stolen can’t be decrypted), and *key compromise impersonation
resistance* (where stealing your keys don’t allow the attacker to
impersonate *others* when they are talking *to you*).

Yet another threat is meta data analysis. Maybe you want to hide your
identity to outsiders. Some protocols achieve some measure of
*anonymity*, where it is hard for the attacker to determine who
is talking to whom.

Or maybe you fear *traffic analysis*, where the size and
timings of the messages reveal too much about the content. You could
mitigate that by padding your messages to some standard size, or even
send a constant stream of data, and fill the bandwidth you don’t use
with noise.

### Proofs

Once you know your threat model, you need to demonstrate that your protocol addresses it. For instance, when the adversary is an untrusted network, you need IND-CCA2 (indistinguishability under adaptive chosen ciphertext attack). It’s a formalisation of what you need to combat active adversaries (man in the middle). It’s defined thus:

- We allow the adversary to ask an
*Oracle*to encrypt or decrypt anything they please. - At any point, the adversary can issue a challenge, where they
provide
*two*plaintexts, and the oracle responds with only*one*ciphertext. - IND-CCA2 is achieved if the adversary cannot tell which plaintext the ciphertext is from, without asking for a decryption of that particular ciphertext.

Let’s try an example with *passive* adversaries, for which we
only need IND-CPA: independence under chosen *plaintext* attack.
It’s the same as IND-CCA2, except we don’t have the decryption
oracle.

Alice needs to send messages to Bob, using a shared secret key
`K0`

, and a stream cipher `Stream`

that takes a
key as input, and outputs an arbitrarily long key stream. (Real stream
ciphers also have a nonce, but we’ll make do without one for the sake
of the exercise.) To send her messages, Alice will use *key
erasure*, or *ratchet*:

```
K1, S1 = Stream(K0)
msg1 = plaintext XOR S1
K2, S2 = Stream(K1)
msg2 = plaintext XOR S2
K3, S3 = Stream(K2)
msg3 = plaintext XOR S3
...
```

To send a message, we first split the key stream in two parts: a new
encryption key, and a data stream. Note that `K1`

is the
first bytes of `Stream(K0)`

, and `S1`

is the rest,
so they’re independent from each other. Same for `K2`

and
`S2`

, etc. We can’t break that one with the encryption oracle
alone (IND-CPA). Here’s why:

- For any
`n`

, if`K(n)`

is an independent random string, then so are`K(n+1)`

and`S(n+1)`

, which come from`Stream(K(n))`

`K0`

is an independent random string.- Corollary:
`S1`

,`S2`

,`S3`

… are all independent random strings. As a whole, they act like a one time pad. - Conclusion: there is no relation whatsoever between different ciphertexts. To the attacker, they all look like independent random strings. When the attacker submits its challenge, the answer will be just as random, and there will be no way to tell which plaintext it came from.

Not the most rigorous proof. That will do for this simple construction under this simple threat model, but a more complex protocol may not tolerate as much hand waving. And if you want a machine to check your proof (a good thing to strive for), it’ll have to be perfect.

Another limitation is my choice of the *symbolic model*: I am
assuming the stream cipher is unbreakable, and will always be
indistinguishable from a true random number generator to anyone who
doesn’t know the key. Works well enough for most purposes, but keep in
mind that real cryptographic primitives aren’t perfect: they can’t
generate a genuinely infinite stream without repeating themselves, and a
finite amount of computing work is enough to break them. Those limits
are high enough to be ignored in many cases, but depending on the
primitive you use, and what you use them for, they may not be.

The biggest limitation of all of course is that we only proved
IND-CPA here. This construction is only immune to *passive*
attackers, that only listen and never talk. What we really want is to
protect against *active* attackers, that could perform a full Man
in the Middle attack. A chosen ciphertext attack easily destroys our
construction:

- Send the challenge with the plaintexts
`p1`

and`p2`

. Get back the ciphertext`c`

. - Generate some non-null string
`r`

. - Ask the oracle to decrypt
`c XOR r`

, get the plaintext`p`

back. - Compute
`p XOR r`

. It will be equal to`p1`

or`p2`

, depending on which it came from.

We basically tricked the oracle into decrypting the challenge
ciphertext, without breaking the rules: we didn’t *directly*
asked to decrypt `c`

, we only took advantage of ciphertext malleability to get around the
restriction. To avoid this, we need authentication, which I’ll leave as an exercise to
the reader for brevity’s sake.

### API

The proper way to achieve IND-CCA2 is to first encrypt the plaintext,
*then* compute an authentication tag over the ciphertext.
Encrypt-then-mac. We could instead authenticate the *plaintext*,
and only then encrypt the whole thing (including the authentication
tag). Mac-then-encrypt. It can be done correctly, and we can even prove
IND-CCA2. It is also a bad idea.

Ideally, we want the recipient to verify first. This has some advantages:

- No time lost decrypting corrupted messages.
- No memory allocated or overwritten for corrupted messages.
- No temptation to provide a detached API in two steps, where the users could forget the authentication step.

What your users need is *authenticated encryption*, where
decryption either successfully decrypts the message, or does nothing at
all. That’s the only way you can be sure no information will leak
because of corrupted messages.

In fact, “success or nothing” is a rule I try generalise to all my
APIs: check the inputs, *then* act on them. If the inputs are
wrong, don’t do anything, just return an error to the caller.

## Designing primitives

That’s the tough one. There’s generally no way to prove that a given
primitive is indeed as secure as it claims to be. There’s an obvious
tension between performance and security, and designing something as
secure *and* as fast as the state of the art is not trivial to
say the least.

I have also no experience in the domain. I can only give a few pointers, that will hopefully make you realise what it means to stare into the abyss.

### Symmetric crypto

All ciphers and hashes I know are designed around a core
*permutation*: mangling the hell out of a block of data to
produce a pseudo-random result:

**Hashes**use that permutation to build a*compression function*, whose output is shorter than the sum of its inputs, then repeatedly use it to compress arbitrarily long messages into short digests.**Block ciphers**are like a compression function (with the key and a piece of data as input), except it’s reversible if you have the key.**Stream ciphers**are something of a blend between hashes and block ciphers: they mix the key and some input (often a nonce and counter) to output a random piece of data, but they don’t need to be reversible.**Password hashes**are their own beast. They’re designed to require huge amounts of resources to make it harder to brute force passwords, so the underlying permutations tend to be very different.

Some permutations (Keccak, Xoodoo, Gimli…) are designed to rule them all, but they don’t have to. The most popular ciphers and hashes have their own permutation, tailored just for them. Chacha20’s permutation for instance produces zero when the input is all zero. Not very random, but we don’t care, because the input is never zero: part of it is a constant, that aims to “break symmetry”.

Speaking of which: before you make a serious attempt at designing your hash, cipher, or permutation, there are some terms and concepts you probably should be deeply familiar with:

**Indistinguishability.**Basically what we ask of most ciphers and hashes: the output must be indistinguishable from random under some conditions (like not knowing the keys). The primitive is broken if we find a “distinguisher”.**Memory hardness.**Good password hashes are optimised to require lots of resources. Our machines have lots of RAM, we want to use it so brute-force guesses are more expensive. The idea is to make honest verifications fast enough to be tolerable, and dishonest guesses as expensive as possible (ideally as expensive as a honest verification, even if the attacker can use custom hardware).**Cache hardness.**Our machines have fast cache hierarchies. We could as well take advantage of them, make brute-force guesses even more expensive without slowing down honest attempts.**Rounds.**Permutations are generally about repeating a core permutation, the “round”, several time to achieve security. Chacha20 for instance repeats its core loop 20 times. Each round is strictly speaking a permutation, but we don’t call them that.**Security margin.**Cryptographic literature shows that increasing the number of rounds in a primitive makes it harder to break. So when we study a primitive, we try and break weaker versions, with fewer rounds. The security margin of a primitive is the difference between the minimum number of rounds required to thwart all known attacks, and the actual number of rounds. Chacha20 for instance has a margin of 12 rounds: as of 2020, 8 rounds stop all known attacks, and we have 20.**Confusion.**It’s how many bits of input you need to define a bit of output. If I wave my hands a lot, the higher the confusion, the more effective a single round is at contributing to security.**Diffusion.**It’s the ability of a permutation round to propagate a change from the input to the output. Like confusion, higher diffusion increases the security contribution of a single round (if I wave my hands a lot).**Symmetry.**I’m not familiar with the term myself. I believe it refers to algebraic properties of a permutation, which can help cryptanalysis in some conditions. It must be “broken”, or otherwise be rendered inexploitable to get a secure result. Or so I gathered.**Cryptanalysis.**There are lots of ways to try and break symmetric primitives. Here’s a small (not exhaustive) list of attacks one must absolutely consider when designing a block cipher (courtesy of /u/Quibbler2412 on Reddit):*linear cryptanaysis (single, multiple, zero-correlation), differential cryptanalysis (single, truncated, impossible), differential-linear cryptanalysis, boomerang attacks, integral attacks, division properties, meet-in-the-middle, invariant subspace attacks, slide attacks, related-key attacks, algebraic cryptanalysis, cube attacks.*

Those are an incomplete list of terms I stumbled upon. My hope is by
the time you get familiar with those, you’ll have a better idea of what
*else* you should study.

### Polynomial hashes

The primary use case of of polynomial hashes is message authentication codes. They’re not as easy to use as regular hashes, but they’re very fast, and their security properties are provable. If you’re going to invent one, you need to produce a proper security reduction (a mathematical proof of how secure it is, given some assumptions).

Also note that Poly1305 and GHASH already cover a lot of ground. Poly1305 is designed to allow simple and efficient software implementations, while GHASH is very hardware friendly. One of them may already offer what you need.

### Elliptic curves

The modern way to use the difficulty of discrete logarithm. Unlike
other primitives, I feel like elliptic curves are *found* more
than they are invented. Granted, they’re not trivial. The maths involved
are definitely intimidating, and a single mistake could destroy
security.

One way to find a curve is to target a security level, choose a prime that matches that security level (generally twice as big), and follow the SafeCurves recommendations. The first curve you find is almost certainly the one you want.

If you want to play with extension fields instead of prime fields, I can only point out that some extension fields are more efficient on tiny embedded targets, and the security literature around binary extension fields is less stable than it is for prime fields.

I also strongly advise sticking to Montgomery or (twisted) Edwards
curves. They are relatively simple, and they’re *fast*. They
don’t have a prime order, but that can be worked around, or even solved entirely. Avoid short
Weierstraß curves. They’re not broken, but they are slower and harder to
implement securely.

### Post quantum cryptography

I know even less than Jon Snow. Can’t help you, sorry.

## Conclusion

I won’t sugar coat it, rolling your own crypto is not easy. Mistakes are easy to make, and the stakes are often high — getting it wrong can even get people killed. Don’t rush it, and if you can, seek guidance and feedback. Some communities are very welcoming.

Still, we *can* teach ourselves. The rules may not be easy to
follow, but they are simple:

- Make it bug free. Test, test, test. Prove what you can. Be extra-rigorous around constructions and protocols.
- Know your side channels. Make it at least constant time.
- If you’re inventing something, know the state of the art, including cryptanalysis. Even if what you want does not exist, what does may help you design it.

*Discuss on r/crypto,
r/programming,
Lobsters,
Hacker
News.*