scrypt: A new key derivation function
Doing our best to thwart TLAs armed with ASICs
Colin Percival
Tarsnap
cperciva@tarsnap.com

May 9, 2009

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

scrypt: A new key derivation function
Making bcrypt obsolete
Colin Percival
Tarsnap
cperciva@tarsnap.com

May 9, 2009

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

scrypt: A new key derivation function
Are you sure your SSH keys are safe?
Colin Percival
Tarsnap
cperciva@tarsnap.com

May 9, 2009

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

What are key derivation functions?
You have a password.
You want a generate a derived key from that password.
Verifying passwords for user authentication.
Encrypting or signing files.

In most situations where passwords are used, they are passed
to a key derivation function first.
In most situations where key derivation functions aren’t used,
they should be!

Examples of key derivation functions:
DES CRYPT [R. Morris, 1979]
MD5 CRYPT [P. H. Kamp, 1994]
bcrypt [N. Provos and D. Mazières, 1999]
PBKDF2 [B. Kaliski, 2000]
MD5 (not really a key derivation function!)
Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

Security of key derivation functions
Attack model: Assume that the attacker can mount an offline
attack.
Attacker has access to /etc/master.passwd and wants to
find the users’ passwords.
Attacker has an encrypted file and wants to decrypt it.

For all reasonable key derivation functions, the only feasible
attack is to repeatedly try passwords until you find the right
one.
This is called a “brute force” attack.

If it takes twice as long to check if a password is correct, it
will take twice as long to find the right password.
. . . as long as the attacker is using the same software as you.

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

Hardware-based brute force attacks

CREDIT: Randall Munroe / xkcd.com

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

Hardware-based brute force attacks
Some organizations have the resources to design and fabricate
custom password-cracking integrated circuits.
The US National Security Agency
The UK Government Communications Headquarters?
The Communications Security Establishment of Canada?
The Chinese government?
Organized crime?
The Electronic Frontier Foundation?

Using ASICs, it is possible to pack many copies of a
cryptographic circuit onto a single piece of silicon.
Moore’s law: Every 18–24 months, a new generation of
semiconductor manufacturing processes makes CPUs faster.
. . . password-cracking ASICs get faster AND can fit more
copies of a password-cracking circuit.

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

Hardware brute-force attack cost
The cost of a hardware brute-force attack is meaured in
dollar-seconds.
Password cracking is embarrassingly parallel, so if you use
twice as much hardware you can crack the key in half the time.

Cost of ASICs ≍ size of ASICs.
A strong key derivation function is one which can only be
computed by using a large circuit for a long time.

J. Kelsey, B. Schneier, C. Hall and D. Wagner, 1998: Use
“32-bit arithmetic and moderately large amounts of RAM”.
An example of a “moderately large amount of RAM”: 1 kB.

If we use a ridiculously large amount of RAM, hardware
attacks will be even more expensive.

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

Memory-hard algorithms
Definition
A memory-hard algorithm on a Random Access Machine is an
algorithm which uses

 S(n) space and T (n) operations, where
S(n) ∈ Ω T (n)1−ǫ .
Conceptually, a memory-hard algorithm is one which comes
close to using the largest amount of storage possible for an
algorithm with the same running time.
. . . and consequently the largest circuit area possible.

The HEKS key derivation algorithm [A.G. Reinhold, 1999] is
memory-hard, but it isn’t very secure, since it can be
effectively parallelized.
Secure key derivation functions require a large die area and a
lot of time to compute.
Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

Sequential memory-hard functions
Definition
A sequential memory-hard function is a function which
(a) can be computed by a memory-hard algorithm on a Random
Access Machine in T (n) operations; and
(b) cannot be computed on a Parallel Random Access Machine
with S ∗ (n) processors and S ∗ (n) space in expected time T ∗ (n)
where S ∗ (n)T ∗ (n) = O(T (n)2−x ) for any x > 0.
Not only do memory-hard functions require lots of storage,
but they also cannot be parallelized efficiently.
If we can find a key derivation function which is sequential
memory-hard, it should be very secure against hardware
attack.

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

ROMix
Algorithm (ROMix)
Given a hash function H, an input B, and an integer parameter N,
compute
Vi = H i (B)
0≤i <N
and X = H N (B), then iterate
j ← Integerify(X ) mod N
X ← H(X ⊕ Vj )
N times; and output X .
The function Integerify can be any bijection from {0, 1}k to
{0 . . . 2k − 1}.
ROMix fills V with pseudorandom values, then accesses them
in a pseudorandom order.
Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

ROMix
Theorem
Under the random oracle model, the class of functions ROMix are
sequential memory-hard.
The random oracle model is a very standard assumption in
proofs relating to hash functions.
The proof is roughly 2 pages long.
I could probably spend my entire talk explaining the proof.
. . . but I won’t.
If you’re interested, go read the paper I wrote about this.

It’s much easier to prove that an algorithm runs in specified
time and space than to prove a minimum bound on the time
and space used by any algorithm which computes a function.

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

scrypt

Use PBKDF2 to convert a password into a bitstream.
Feed this bitstream to ROMix.
Feed the output of ROMix back to PBKDF2 to generate the
derived key.
Cryptographic primitives used:
HMAC-SHA256
Salsa20/8 core

The Salsa20/8 core outputs lots of bits very fast, which
means that scrypt can use lots of memory.
Approximately 1 byte of RAM per 10 clock cycles on a Core 2
processor.
We can quickly require a large semiconductor area.

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

Estimating hardware brute force attack costs

It’s hard to get accurate information about how much it costs
to build password-cracking machines.
Oddly enough, the NSA doesn’t publish this data.

The best we can do for most KDFs is to count cryptographic
operations and assume that they are responsible for most of
the time and die area.
This is probably a fairly accurate approximation, since key
derivation functions only have a very small amount of
non-cryptographic computations.

For scrypt we also need to look at the die area required for
storage.

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

Estimating hardware brute force attack costs
Approximate circuit complexity and performance for
cryptographic primitives, based on a 130 nm semiconductor
process:
A DES circuit with ≈ 4000 gates of logic can encrypt data at
2000 Mbps.
An MD5 circuit with ≈ 12000 gates of logic can hash data at
2500 Mbps.
A SHA256 circuit with ≈ 20000 gates of logic can hash data
at 2500 Mbps.
A Blowfish circuit with ≈ 22000 gates of logic and 4 kiB of
SRAM can encrypt data at 1000 Mbps.
A Salsa20/8 circuit with ≈ 24000 gates of logic can output a
keystream at 2000 Mbps.

I’m using 130 nm as a reference point because this is what I
could get the most data for.

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

Estimating hardware brute force attack costs
Very approximate estimates of VLSI area and cost:
Each
Each
Each
VLSI

gate of random logic requires ≈ 5 µm2 of VLSI area.
bit of SRAM requires ≈ 2.5 µm2 of VLSI area.
bit of DRAM requires ≈ 0.1 µm2 of VLSI area.
circuits cost ≈ 0.1$/mm2 .

These values are based on a 130 nm process circa 2002.
These values have a very wide error margin.
Non-cryptographic parts of ASICs (e.g., I/O), chip packaging,
boards, power supplies, and operating costs could increase
password-cracking costs by a factor of 10.
Improvements in semiconductor technology since 2002 could
reduce password-cracking costs by a factor of 10.
Improved cryptographic circuits could reduce costs by a factor
of 10.

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

Key derivation functions
Non-parameterized KDFs:
DES CRYPT
MD5 CRYPT
MD5

KDFs tuned for interactive logins (t ≤ 100 ms):
PBKDF2-HMAC-SHA256, c = 86000
bcrypt, cost = 11
scrypt, N = 214 , r = 8, p = 1

KDFs tuned for file encryption (t ≤ 5 s):
PBKDF2-HMAC-SHA256, c = 4300000
bcrypt, cost = 16
scrypt, N = 220 , r = 8, p = 1

Running time based on a 2.5 GHz Core 2 (aka. my laptop).

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

Passwords

6 lower-case letters; e.g., “sfgroy”.
8 lower-case letters; e.g., “ksuvnwyf”.
8 characters selected from the 95 printable 7-bit ASCII
characters; e.g., “6,uh3y[a”.
10 characters selected from the 95 printable 7-bit ASCII
characters; e.g., “H.*W8Jz&r3”.
A 40-character string of text; e.g., “This is a
40-character string of English”.
Entropy estimated according to formula from NIST: 1st
character has 4 bits of entropy; 2nd–8th characters have 2 bits
of entropy each; 9th–20th characters have 1.5 bits of entropy
each; 21st and later characters have 1 bit of entropy each.

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

Estimated brute force attack costs

Estimated cost of hardware to crack a password in 1 year.
KDF
DES CRYPT
MD5
MD5 CRYPT
PBKDF2 (100 ms)
bcrypt (95 ms)
scrypt (64 ms)
PBKDF2 (5.0 s)
bcrypt (3.0 s)
scrypt (3.8 s)

6 letters

8 letters

8 chars

10 chars

40-char text

< $1
< $1
< $1
< $1
< $1
< $1
< $1
< $1
$900

< $1
< $1
< $1
< $1
$4
$150
$29
$130
$610k

< $1
< $1
$130
$18k
$130k
$4.8M
$920k
$4.3M
$19B

< $1
$1.1k
$1.1M
$160M
$1.2B
$43B
$8.3B
$39B
$175T

< $1
$1
$1.4k
$200k
$1.5M
$52M
$10M
$47M
$210B

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

KDF brute force attack costs

When used for interactive logins, scrypt is . . .
≈ 25 times more expensive to attack than bcrypt,
≈ 28 times more expensive to attack than PBKDF2,
and ≈ 215 times more expensive to attack than MD5 CRYPT.

When used for file encryption, scrypt is . . .
≈ 212 times more expensive to attack than bcrypt,
≈ 215 times more expensive to attack than PBKDF2,
and ≈ 237 times more expensive to attack than MD5.

openssl enc uses MD5 as a key derivation function.
OpenSSH uses MD5 as a key derivation function for
passphrases on key files.
Are you sure that your SSH keys are safe?

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function

Availability

More details at http://www.tarsnap.com/scrypt/.
Source code for scrypt.
A simple file encryption/decryption utility.
A 16-page paper

Questions?

Colin Percival Tarsnap cperciva@tarsnap.com

scrypt: A new key derivation function