Network Working Group A. Biryukov
Internet-Draft D. Dinu
Intended status: Informational University of Luxembourg
Expires: May 26, 2019 D. Khovratovich
ABDK Consulting
S. Josefsson
SJD AB
November 22, 2018
The memory-hard Argon2 password hash and proof-of-work function
draft-irtf-cfrg-argon2-04
Abstract
This document describes the Argon2 memory-hard function for password
hashing and proof-of-work applications. We provide an implementer-
oriented description together with sample code and test vectors. The
purpose is to simplify adoption of Argon2 for Internet protocols.
Status of This Memo
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This Internet-Draft will expire on May 26, 2019.
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document authors. All rights reserved.
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described in the Simplified BSD License.
Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
2. Notation and Conventions . . . . . . . . . . . . . . . . . . 3
3. Argon2 Algorithm . . . . . . . . . . . . . . . . . . . . . . 4
3.1. Argon2 Inputs and Outputs . . . . . . . . . . . . . . . . 4
3.2. Argon2 Operation . . . . . . . . . . . . . . . . . . . . 5
3.3. Variable-length hash function H' . . . . . . . . . . . . 6
3.4. Indexing . . . . . . . . . . . . . . . . . . . . . . . . 7
3.4.1. Getting the 32-bit values J_1 and J_2 . . . . . . . . 7
3.4.2. Mapping J_1 and J_2 to reference block index . . . . 8
3.5. Compression function G . . . . . . . . . . . . . . . . . 9
3.6. Permutation P . . . . . . . . . . . . . . . . . . . . . . 10
4. Parameter Choice . . . . . . . . . . . . . . . . . . . . . . 11
5. Example Code . . . . . . . . . . . . . . . . . . . . . . . . 13
6. Test Vectors . . . . . . . . . . . . . . . . . . . . . . . . 22
6.1. Argon2d Test Vectors . . . . . . . . . . . . . . . . . . 22
6.2. Argon2i Test Vectors . . . . . . . . . . . . . . . . . . 23
6.3. Argon2id Test Vectors . . . . . . . . . . . . . . . . . . 24
7. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 26
8. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 26
9. Security Considerations . . . . . . . . . . . . . . . . . . . 26
9.1. Security as hash function and KDF . . . . . . . . . . . . 26
9.2. Security against time-space tradeoff attacks . . . . . . 26
9.3. Security for time-bounded defenders . . . . . . . . . . . 27
9.4. Recommendations . . . . . . . . . . . . . . . . . . . . . 27
10. References . . . . . . . . . . . . . . . . . . . . . . . . . 27
10.1. Normative References . . . . . . . . . . . . . . . . . . 27
10.2. Informative References . . . . . . . . . . . . . . . . . 27
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 28
1. Introduction
This document describes the Argon2 memory-hard function for password
hashing and proof-of-work applications. We provide an implementer
oriented description together with sample code and test vectors. The
purpose is to simplify adoption of Argon2 for Internet protocols.
This document corresponds to version 1.3 of the Argon2 hash function.
Argon2 summarizes the state of the art in the design of memory-hard
functions [HARD]. It is a streamlined and simple design. It aims at
the highest memory filling rate and effective use of multiple
computing units, while still providing defense against tradeoff
attacks. Argon2 is optimized for the x86 architecture and exploits
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the cache and memory organization of the recent Intel and AMD
processors. Argon2 has one primary variant: Argon2id, and two
supplementary variants: Argon2d and Argon2i. Argon2d uses data-
dependent memory access, which makes it suitable for cryptocurrencies
and proof-of-work applications with no threats from side-channel
timing attacks. Argon2i uses data-independent memory access, which
is preferred for password hashing and password-based key derivation.
Argon2id works as Argon2i for the first half of the first iteration
over the memory, and as Argon2d for the rest, thus providing both
side-channel attack protection and brute-force cost savings due to
time-memory tradeoffs. Argon2i makes more passes over the memory to
protect from tradeoff attacks [AB15].
Argon2 can be viewed as a mode of operation over a fixed-input-length
compression function G and a variable-input-length hash function H.
Even though Argon2 can be potentially used with arbitrary function H,
as long as it provides outputs up to 64 bytes, in this document it
MUST be BLAKE2b.
For further background and discussion, see the Argon2 paper [ARGON2].
2. Notation and Conventions
x^y --- integer x multiplied by itself integer y times
a*b --- multiplication of integer a and integer b
c-d --- substraction of integer c with integer d
E_f --- variable E with subscript index f
g / h --- integer g divided by integer h. The result is rational
number
I(j) --- function I evaluated on integer parameter j
K || L --- string K concatenated with string L
a XOR b --- bitwise exclusive-or between bitstrings a and b
a mod b --- remainder of integer a modulo integer b, always in range
[0, b-1]
a >>> n --- rotation of 64-bit string a to the right by n bits
trunc(a) --- the 64-bit value, truncated to the 32 least significant
bits
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floor(a) --- the largest integer not bigger than a
ceil(a) --- the smallest integer not smaller than a
extract(a, i) --- the i-th set of 32-bits from bitstring a, starting
from 0-th
|A| --- the number of elements in set A
LE32(a) --- 32-bit integer a converted to bytestring in little
endian. Example: 123456 (decimal) is 40 E2 01 00.
LE64(a) --- 64-bit integer a converted to bytestring in little
endian. Example: 123456 (decimal) is 40 E2 01 00 00 00 00 00.
int32(s) --- 32-bit string s is converted to non-negative integer in
little endian.
int64(s) --- 64-bit string s is converted to non-negative integer in
little endian.
length(P) --- the bytelength of string P expressed as 32-bit integer
3. Argon2 Algorithm
3.1. Argon2 Inputs and Outputs
Argon2 has the following input parameters:
o Message string P, which is a password for password hashing
applications. May have any length from 0 to 2^(32) - 1 bytes.
o Nonce S, which is a salt for password hashing applications. May
have any length from 8 to 2^(32)-1 bytes. 16 bytes is recommended
for password hashing. Salt SHOULD be unique for each password.
o Degree of parallelism p determines how many independent (but
synchronizing) computational chains (lanes) can be run. It may
take any integer value from 1 to 2^(24)-1.
o Tag length T may be any integer number of bytes from 4 to 2^(32)-
1.
o Memory size m can be any integer number of kibibytes from 8*p to
2^(32)-1. The actual number of blocks is m', which is m rounded
down to the nearest multiple of 4*p.
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o Number of iterations t (used to tune the running time
independently of the memory size) can be any integer number from 1
to 2^(32)-1.
o Version number v is one byte 0x13.
o Secret value K (serves as key if necessary, but we do not assume
any key use by default) may have any length from 0 to 2^(32)-1
bytes.
o Associated data X may have any length from 0 to 2^(32)-1 bytes.
o Type y of Argon2: 0 for Argon2d, 1 for Argon2i, 2 for Argon2id.
The Argon2 output, or "tag" is a string T bytes long.
3.2. Argon2 Operation
Argon2 uses an internal compression function G with two 1024-byte
inputs and a 1024-byte output, and an internal hash function H^x()
with x being its output length in bytes. Here H^x() applied to
string A is the BLAKE2b [BLAKE2] function, which takes
(d,|dd|,kk=0,nn=x) as parameters where d is A padded to a multiple of
128 bytes and partitioned into 128-byte blocks. The compression
function G is based on its internal permutation. A variable-length
hash function H' built upon H is also used. G is described in
Section Section 3.5 and H' is described in Section Section 3.3.
The Argon2 operation is as follows.
1. Establish H_0 as the 64-byte value as shown below.
H_0 = H^(64)(LE32(p) || LE32(T) || LE32(m) || LE32(t) || LE32(v) || LE32(y) || LE32(length(P)) || P || LE32(length(S)) || S || LE32(length(K)) || K || LE32(length(X)) || X)
H_0 generation
2. Allocate the memory as m' 1024-byte blocks where m' is derived
as:
m' = 4 * p * floor (m / 4p)
Memory allocation
For p lanes, the memory is organized in a matrix B[i][j] of
blocks with p rows (lanes) and q = m' / p columns.
3. Compute B[i][0] for all i ranging from (and including) 0 to (not
including) p.
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B[i][0] = H'^(128)(H_0 || LE32(0) || LE32(i))
Lane starting blocks
4. Compute B[i][1] for all i ranging from (and including) 0 to (not
including) p.
B[i][1] = H'^(128)(H_0 || LE32(1) || LE32(i))
Second lane blocks
5. Compute B[i][j] for all i ranging from (and including) 0 to (not
including) p, and for all j ranging from (and including) 2) to
(not including) q. The block indices l and z are determined for
each i, j differently for Argon2d, Argon2i, and Argon2id
(Section Section 3.4).
B[i][j] = G(B[i][j-1], B[l][z])
Further block generation
6. If the number of iterations t is larger than 1, we repeat the
steps however replacing the computations with the following
expression:
B[i][0] = G(B[i][q-1], B[l][z])
B[i][j] = G(B[i][j-1], B[l][z])
Further passes
7. After t steps have been iterated, the final block C is computed
as the XOR of the last column:
C = B[0][q-1] XOR B[1][q-1] XOR ... XOR B[p-1][q-1]
Final block
8. The output tag is computed as H'^T(C).
3.3. Variable-length hash function H'
Let V_i be a 64-byte block, and W_i be its first 32 bytes. Then we
define:
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if T <= 64
H'^T(A) = H^T(LE32(T)||A)
else
r = ceil(T/32)-2
V_1 = H^(64)(LE32(T)||A)
V_2 = H^(64)(V_1)
...
V_r = H^(64)(V_{r-1})
V_{r+1} = H^(T-32*r)(V_{r})
H'^T(X) = W_1 || W_2 || ... || W_r || V_{r+1}
Tag computation
3.4. Indexing
To enable parallel block computation, we further partition the memory
matrix into S = 4 vertical slices. The intersection of a slice and a
lane is a segment of length q/S. Segments of the same slice are
computed in parallel and may not reference blocks from each other.
All other blocks can be referenced.
slice 0 slice 1 slice 2 slice 3
___/\___ ___/\___ ___/\___ ___/\___
/ \ / \ / \ / \
+----------+----------+----------+----------+
| | | | | > lane 0
+----------+----------+----------+----------+
| | | | | > lane 1
+----------+----------+----------+----------+
| | | | | > lane 2
+----------+----------+----------+----------+
| ... ... ... | ...
+----------+----------+----------+----------+
| | | | | > lane p - 1
+----------+----------+----------+----------+
Single-pass Argon2 with p lanes and 4 slices
3.4.1. Getting the 32-bit values J_1 and J_2
3.4.1.1. Argon2d
J_1 is given by the first 32 bits of block B[i][j-1], while J_2 is
given by the next 32-bits of block B[i][j-1]:
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J_1 = int32(extract(B[i][j-1], 1))
J_2 = int32(extract(B[i][j-1], 2))
Deriving J1,J2 in Argon2d
3.4.1.2. Argon2i
Each application of the 2-round compression function G in the counter
mode gives 128 64-bit values X, which are viewed as X1||X2 and
converted to J_1=int32(X1) and J_2=int32(X2). The first input to G
is the all zero block and the second input to G is constructed as
follows:
( LE64(r) || LE64(l) || LE64(s) || LE64(m') || LE64(t) || LE64(y) || LE64(i) || ZERO ), where
r -- the pass number
l -- the lane number
s -- the slice number
m' -- the total number of memory blocks
t -- the total number of passes
y -- the Argon2 type (0 for Argon2d, 1 for Argon2i, 2 for Argon2id)
i -- the counter (starts from 1 in each segment)
ZERO -- the 968-byte zero string.
Input to compute J1,J2 in Argon2i
The values r, l, s, m', t, x, i are represented as 8 bytes in little-
endian.
3.4.1.3. Argon2id
If the pass number is 0 and the slice number is 0 or 1, then compute
J_1 and J_2 as for Argon2i, else compute J_1 and J_2 as for Argon2d.
3.4.2. Mapping J_1 and J_2 to reference block index
The value of l = J_2 mod p gives the index of the lane from which the
block will be taken. For the firt pass (r=0) and the first slice
(s=0) the block is taken from the current lane.
The set W contains the indices that can be referenced according to
the following rules:
1. If l is the current lane, then W includes the indices of all
blocks in the last S - 1 = 3 segments computed and finished, as
well as the blocks computed in the current segment in the current
pass excluding B[i][j-1].
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2. If l is not the current lane, then W includes the indices of all
blocks in the last S - 1 = 3 segments computed and finished in
lane l. If B[i][j] is the first block of a segment, then the
very last index from W is excluded.
We are going to take a block from W with a non-uniform distribution
over [0, |W|) using the mapping
J_1 -> |W|(1 - J_1^2 / 2^(64))
Computing J1
To avoid floating point computation, the following approximation is
used:
x = J_1^2 / 2^(32)
y = (|W| * x) / 2^(32)
z = |W| - 1 - y
Computing J1, part 2
The value of z gives the reference block index in W.
3.5. Compression function G
Compression function G is built upon the BLAKE2b round function P. P
operates on the 128-byte input, which can be viewed as 8 16-byte
registers:
P(A_0, A_1, ... ,A_7) = (B_0, B_1, ... ,B_7)
Blake round function P
Compression function G(X, Y) operates on two 1024-byte blocks X and
Y. It first computes R = X XOR Y. Then R is viewed as a 8x8 matrix
of 16-byte registers R_0, R_1, ... , R_63. Then P is first applied
to each row, and then to each column to get Z:
( Q_0, Q_1, Q_2, ... , Q_7) <- P( R_0, R_1, R_2, ... , R_7)
( Q_8, Q_9, Q_10, ... , Q_15) <- P( R_8, R_9, R_10, ... , R_15)
...
(Q_56, Q_57, Q_58, ... , Q_63) <- P(R_56, R_57, R_58, ... , R_63)
( Z_0, Z_8, Z_16, ... , Z_56) <- P( Q_0, Q_8, Q_16, ... , Q_56)
( Z_1, Z_9, Z_17, ... , Z_57) <- P( Q_1, Q_9, Q_17, ... , Q_57)
...
( Z_7, Z_15, Z 23, ... , Z_63) <- P( Q_7, Q_15, Q_23, ... , Q_63)
Core of compression function G
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Finally, G outputs Z XOR R:
G: (X, Y) -> R -> Q -> Z -> Z XOR R
+---+ +---+
| X | | Y |
+---+ +---+
| |
---->XOR<----
--------|
| \ /
| +---+
| | R |
| +---+
| |
| \ /
| P rowwise
| |
| \ /
| +---+
| | Q |
| +---+
| |
| \ /
| P columnwise
| |
| \ /
| +---+
| | Z |
| +---+
| |
| \ /
------>XOR
|
\ /
Argon2 compression function G.
3.6. Permutation P
Permutation P is based on the round function of BLAKE2b. The 8
16-byte inputs S_0, S_1, ... , S_7 are viewed as a 4x4 matrix of
64-bit words, where S_i = (v_{2*i+1} || v_{2*i}):
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v_0 v_1 v_2 v_3
v_4 v_5 v_6 v_7
v_8 v_9 v_10 v_11
v_12 v_13 v_14 v_15
Matrix element labeling
It works as follows:
GB(v_0, v_4, v_8, v_12)
GB(v_1, v_5, v_9, v_13)
GB(v_2, v_6, v_10, v_14)
GB(v_3, v_7, v_11, v_15)
GB(v_0, v_5, v_10, v_15)
GB(v_1, v_6, v_11, v_12)
GB(v_2, v_7, v_8, v_13)
GB(v_3, v_4, v_9, v_14)
Feeding matrix elements to GB
GB(a, b, c, d) is defined as follows:
a = (a + b + 2 * trunc(a) * trunc(b)) mod 2^(64)
d = (d XOR a) >>> 32
c = (c + d + 2 * trunc(c) * trunc(d)) mod 2^(64)
b = (b XOR c) >>> 24
a = (a + b + 2 * trunc(a) * trunc(b)) mod 2^(64)
d = (d XOR a) >>> 16
c = (c + d + 2 * trunc(c) * trunc(d)) mod 2^(64)
b = (b XOR c) >>> 63
Details of GB
The modular additions in GB are combined with 64-bit multiplications.
Multiplications are the only difference to the original BLAKE2b
design. This choice is done to increase the circuit depth and thus
the running time of ASIC implementations, while having roughly the
same running time on CPUs thanks to parallelism and pipelining.
4. Parameter Choice
Argon2d is optimized for settings where the adversary does not get
regular access to system memory or CPU, i.e. he can not run side-
channel attacks based on the timing information, nor he can recover
the password much faster using garbage collection. These settings
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are more typical for backend servers and cryptocurrency minings. For
practice we suggest the following settings:
o Cryptocurrency mining, that takes 0.1 seconds on a 2 Ghz CPU using
1 core -- Argon2d with 2 lanes and 250 MB of RAM.
Argon2id is optimized for more realistic settings, where the
adversary possibly can access the same machine, use its CPU or mount
cold-boot attacks. We suggest the following settings:
o Backend server authentication, that takes 0.5 seconds on a 2 GHz
CPU using 4 cores -- Argon2id with 8 lanes and 4 GiB of RAM.
o Key derivation for hard-drive encryption, that takes 3 seconds on
a 2 GHz CPU using 2 cores - Argon2id with 4 lanes and 6 GiB of
RAM.
o Frontend server authentication, that takes 0.5 seconds on a 2 GHz
CPU using 2 cores - Argon2id with 4 lanes and 1 GiB of RAM.
We recommend the following procedure to select the type and the
parameters for practical use of Argon2.
1. Select the type y. If you do not know the difference between
them or you consider side-channel attacks as viable threat,
choose Argon2id.
2. Figure out the maximum number h of threads that can be initiated
by each call to Argon2.
3. Figure out the maximum amount m of memory that each call can
afford.
4. Figure out the maximum amount x of time (in seconds) that each
call can afford.
5. Select the salt length. 128 bits is sufficient for all
applications, but can be reduced to 64 bits in the case of space
constraints.
6. Select the tag length. 128 bits is sufficient for most
applications, including key derivation. If longer keys are
needed, select longer tags.
7. If side-channel attacks are a viable threat, or if you're
uncertain, enable the memory wiping option in the library call.
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8. Run the scheme of type y, memory m and h lanes and threads, using
different number of passes t. Figure out the maximum t such that
the running time does not exceed x. If it exceeds x even for t =
1, reduce m accordingly.
9. Hash all the passwords with the just determined values m, h, and
t.
5. Example Code
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void fill_block(const block *prev_block,
const block *ref_block,
block *next_block) {
block blockR, block_tmp;
unsigned i;
copy_block(&blockR, ref_block);
xor_block(&blockR, prev_block);
copy_block(&block_tmp, &blockR);
/* Now blockR = ref_block + prev_block and bloc_tmp = ref_block +
prev_block */
/* Apply Blake2 on columns of 64-bit words: (0,1,...,15),
then (16,17,..31)... finally (112,113,...127) */
for (i = 0; i < 8; ++i) {
BLAKE2_ROUND_NOMSG(
blockR.v[16 * i], blockR.v[16 * i + 1],
blockR.v[16 * i + 2], blockR.v[16 * i + 3],
blockR.v[16 * i + 4], blockR.v[16 * i + 5],
blockR.v[16 * i + 6], blockR.v[16 * i + 7],
blockR.v[16 * i + 8], blockR.v[16 * i + 9],
blockR.v[16 * i + 10], blockR.v[16 * i + 11],
blockR.v[16 * i + 12], blockR.v[16 * i + 13],
blockR.v[16 * i + 14], blockR.v[16 * i + 15]);
}
/* Apply Blake2 on rows of 64-bit words: (0,1,16,17,...112,113),
then (2,3,18,19,...,114,115), ... and finally
(14,15,30,31,...,126,127) */
for (i = 0; i < 8; i++) {
BLAKE2_ROUND_NOMSG(
blockR.v[2 * i], blockR.v[2 * i + 1],
blockR.v[2 * i + 16], blockR.v[2 * i + 17],
blockR.v[2 * i + 32], blockR.v[2 * i + 33],
blockR.v[2 * i + 48], blockR.v[2 * i + 49],
blockR.v[2 * i + 64], blockR.v[2 * i + 65],
blockR.v[2 * i + 80], blockR.v[2 * i + 81],
blockR.v[2 * i + 96], blockR.v[2 * i + 97],
blockR.v[2 * i + 112], blockR.v[2 * i + 113]);
}
copy_block(next_block, &block_tmp);
xor_block(next_block, &blockR);
}
Example code
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void fill_block_with_xor(const block *prev_block,
const block *ref_block,
block *next_block) {
block blockR, block_tmp;
unsigned i;
copy_block(&blockR, ref_block);
xor_block(&blockR, prev_block);
copy_block(&block_tmp, &blockR);
/* Saving the next block contents for XOR over */
xor_block(&block_tmp, next_block);
/* Now blockR = ref_block + prev_block and bloc_tmp = ref_block +
prev_block + next_block*/
/* Apply Blake2 on columns of 64-bit words: (0,1,...,15) , then
(16,17,..31),... and finally (112,113,...127) */
for (i = 0; i < 8; ++i) {
BLAKE2_ROUND_NOMSG(
blockR.v[16 * i], blockR.v[16 * i + 1],
blockR.v[16 * i + 2], blockR.v[16 * i + 3],
blockR.v[16 * i + 4], blockR.v[16 * i + 5],
blockR.v[16 * i + 6], blockR.v[16 * i + 7],
blockR.v[16 * i + 8], blockR.v[16 * i + 9],
blockR.v[16 * i + 10], blockR.v[16 * i + 11],
blockR.v[16 * i + 12], blockR.v[16 * i + 13],
blockR.v[16 * i + 14], blockR.v[16 * i + 15]);
}
/* Apply Blake2 on rows of 64-bit words:
(0,1,16,17,...112,113), then
(2,3,18,19,...,114,115), ... and finally
(14,15,30,31,...,126,127) */
for (i = 0; i < 8; i++) {
BLAKE2_ROUND_NOMSG(
blockR.v[2 * i], blockR.v[2 * i + 1],
blockR.v[2 * i + 16], blockR.v[2 * i + 17],
blockR.v[2 * i + 32], blockR.v[2 * i + 33],
blockR.v[2 * i + 48], blockR.v[2 * i + 49],
blockR.v[2 * i + 64], blockR.v[2 * i + 65],
blockR.v[2 * i + 80], blockR.v[2 * i + 81],
blockR.v[2 * i + 96], blockR.v[2 * i + 97],
blockR.v[2 * i + 112], blockR.v[2 * i + 113]);
}
copy_block(next_block, &block_tmp);
xor_block(next_block, &blockR);
}
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Example code page 2
void generate_addresses(const argon2_instance_t *instance,
const argon2_position_t *position,
uint64_t *pseudo_rands) {
block zero_block, input_block, address_block,tmp_block;
uint32_t i;
init_block_value(&zero_block, 0);
init_block_value(&input_block, 0);
if (instance != NULL && position != NULL) {
input_block.v[0] = position->pass;
input_block.v[1] = position->lane;
input_block.v[2] = position->slice;
input_block.v[3] = instance->memory_blocks;
input_block.v[4] = instance->passes;
input_block.v[5] = instance->type;
for (i = 0; i < instance->segment_length; ++i) {
if (i % ARGON2_ADDRESSES_IN_BLOCK == 0) {
input_block.v[6]++;
init_block_value(&tmp_block, 0);
init_block_value(&address_block, 0);
fill_block_with_xor(&zero_block, &input_block, &tmp_block);
fill_block_with_xor(&zero_block, &tmp_block, &address_block);
}
pseudo_rands[i] = address_block.v[i % ARGON2_ADDRESSES_IN_BLOCK];
}
}
Example code page 3
void fill_segment(const argon2_instance_t *instance,
argon2_position_t position) {
block *ref_block = NULL, *curr_block = NULL;
uint64_t pseudo_rand, ref_index, ref_lane;
uint32_t prev_offset, curr_offset;
uint32_t starting_index;
uint32_t i;
int data_independent_addressing;
/* Pseudo-random values that determine the reference block
position */
uint64_t *pseudo_rands = NULL;
if (instance == NULL) {
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return;
}
data_independent_addressing = (instance->type == Argon2_i);
pseudo_rands = (uint64_t *)malloc(sizeof(uint64_t) *
(instance->segment_length));
if (pseudo_rands == NULL) {
return;
}
if (data_independent_addressing) {
generate_addresses(instance, &position, pseudo_rands);
}
starting_index = 0;
if ((0 == position.pass) && (0 == position.slice)) {
/* we have already generated the first two blocks */
starting_index = 2;
}
/* Offset of the current block */
curr_offset = position.lane * instance->lane_length +
position.slice * instance->segment_length +
starting_index;
if (0 == curr_offset % instance->lane_length) {
/* Last block in this lane */
prev_offset = curr_offset + instance->lane_length - 1;
} else {
/* Previous block */
prev_offset = curr_offset - 1;
}
for (i = starting_index; i < instance->segment_length;
++i, ++curr_offset, ++prev_offset) {
/*1.1 Rotating prev_offset if needed */
if (curr_offset % instance->lane_length == 1) {
prev_offset = curr_offset - 1;
}
/* 1.2 Computing the index of the reference block */
/* 1.2.1 Taking pseudo-random value from the previous block */
if (data_independent_addressing) {
pseudo_rand = pseudo_rands[i];
} else {
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pseudo_rand = instance->memory[prev_offset].v[0];
}
/* 1.2.2 Computing the lane of the reference block */
ref_lane = ((pseudo_rand >> 32)) % instance->lanes;
if ((position.pass == 0) && (position.slice == 0)) {
/* Can not reference other lanes yet */
ref_lane = position.lane;
}
/* 1.2.3 Computing the number of possible reference block
within the lane. */
position.index = i;
ref_index = index_alpha(instance, &position,
pseudo_rand & 0xFFFFFFFF,
ref_lane == position.lane);
/* 2 Creating a new block */
ref_block = instance->memory +
instance->lane_length * ref_lane + ref_index;
curr_block = instance->memory + curr_offset;
if (instance->version == ARGON2_OLD_VERSION_NUMBER) {
/* version 1.2.1 and earlier: overwrite, not XOR */
fill_block(instance->memory + prev_offset, ref_block,
curr_block);
} else {
if(0 == position.pass) {
fill_block(instance->memory + prev_offset, ref_block,
curr_block);
} else {
fill_block_with_xor(instance->memory + prev_offset,
ref_block, curr_block);
}
}
}
free(pseudo_rands);
}
Example code page 4
uint32_t index_alpha(const argon2_instance_t *instance,
const argon2_position_t *position,
uint32_t pseudo_rand,
int same_lane) {
/*
* Pass 0:
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* This lane : all already finished segments plus already
* constructed blocks in this segment
* Other lanes : all already finished segments
* Pass 1+:
* This lane : (SYNC_POINTS - 1) last segments plus
* already constructed blocks in this segment
* Other lanes : (SYNC_POINTS - 1) last segments
*/
uint32_t reference_area_size;
uint64_t relative_position;
uint32_t start_position, absolute_position;
if (0 == position->pass) {
/* First pass */
if (0 == position->slice) {
/* First slice */
reference_area_size =
position->index - 1; /* all but the previous */
} else {
if (same_lane) {
/* The same lane => add current segment */
reference_area_size = position->slice *
instance->segment_length +
position->index - 1;
} else {
reference_area_size = position->slice *
instance->segment_length +
((position->index == 0) ? (-1) : 0);
}
}
} else {
/* Second pass */
if (same_lane) {
reference_area_size = instance->lane_length -
instance->segment_length +
position->index - 1;
} else {
reference_area_size = instance->lane_length -
instance->segment_length +
((position->index == 0) ? (-1) : 0);
}
}
/* 1.2.4. Mapping pseudo_rand to 0..<reference_area_size-1>
and produce relative position */
relative_position = pseudo_rand;
relative_position = relative_position * relative_position >> 32;
relative_position = reference_area_size - 1 -
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(reference_area_size * relative_position >> 32);
/* 1.2.5 Computing starting position */
start_position = 0;
if (0 != position->pass) {
start_position = (position->slice == ARGON2_SYNC_POINTS - 1)
? 0
: (position->slice + 1) *
instance->segment_length;
}
/* 1.2.6. Computing absolute position */
absolute_position = (start_position + relative_position) %
instance->lane_length; /* absolute position */
return absolute_position;
}
Example code page 5
int fill_memory_blocks(argon2_instance_t *instance) {
uint32_t r, s;
argon2_thread_handle_t *thread = NULL;
argon2_thread_data *thr_data = NULL;
if (instance == NULL || instance->lanes == 0) {
return ARGON2_THREAD_FAIL;
}
/* 1. Allocating space for threads */
thread = calloc(instance->lanes, sizeof(argon2_thread_handle_t));
if (thread == NULL) {
return ARGON2_MEMORY_ALLOCATION_ERROR;
}
thr_data = calloc(instance->lanes, sizeof(argon2_thread_data));
if (thr_data == NULL) {
free(thread);
return ARGON2_MEMORY_ALLOCATION_ERROR;
}
for (r = 0; r < instance->passes; ++r) {
for (s = 0; s < ARGON2_SYNC_POINTS; ++s) {
int rc;
uint32_t l;
/* 2. Calling threads */
for (l = 0; l < instance->lanes; ++l) {
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argon2_position_t position;
/* 2.1 Join a thread if limit is exceeded */
if (l >= instance->threads) {
rc = argon2_thread_join(thread[l - instance->threads]);
if (rc) {
free(thr_data);
free(thread);
return ARGON2_THREAD_FAIL;
}
}
/* 2.2 Create thread */
position.pass = r;
position.lane = l;
position.slice = (uint8_t)s;
position.index = 0;
/* preparing the thread input */
thr_data[l].instance_ptr = instance;
memcpy(&(thr_data[l].pos), &position,
sizeof(argon2_position_t));
rc = argon2_thread_create(&thread[l], &fill_segment_thr,
(void *)&thr_data[l]);
if (rc) {
free(thr_data);
free(thread);
return ARGON2_THREAD_FAIL;
}
/* fill_segment(instance, position); */
/*Non-thread equivalent of the lines above */
}
/* 3. Joining remaining threads */
for (l = instance->lanes - instance->threads; l < instance->lanes;
++l) {
rc = argon2_thread_join(thread[l]);
if (rc) {
return ARGON2_THREAD_FAIL;
}
}
}
}
if (thread != NULL) {
free(thread);
}
if (thr_data != NULL) {
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free(thr_data);
}
return ARGON2_OK;
}
Example code page 6
6. Test Vectors
This section contains test vectors for Argon2.
6.1. Argon2d Test Vectors
=======================================
Argon2d version number 19
=======================================
Memory: 32 KiB
Iterations: 3
Parallelism: 4 lanes
Tag length: 32 bytes
Password[32]: 01 01 01 01 01 01 01 01
01 01 01 01 01 01 01 01
01 01 01 01 01 01 01 01
01 01 01 01 01 01 01 01
Salt[16]: 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02
Secret[8]: 03 03 03 03 03 03 03 03
Associated data[12]: 04 04 04 04 04 04 04 04 04 04 04 04
Pre-hashing digest: b8 81 97 91 a0 35 96 60
bb 77 09 c8 5f a4 8f 04
d5 d8 2c 05 c5 f2 15 cc
db 88 54 91 71 7c f7 57
08 2c 28 b9 51 be 38 14
10 b5 fc 2e b7 27 40 33
b9 fd c7 ae 67 2b ca ac
5d 17 90 97 a4 af 31 09
After pass 0:
Block 0000 [ 0]: db2fea6b2c6f5c8a
Block 0000 [ 1]: 719413be00f82634
Block 0000 [ 2]: a1e3f6dd42aa25cc
Block 0000 [ 3]: 3ea8efd4d55ac0d1
...
Block 0031 [124]: 28d17914aea9734c
Block 0031 [125]: 6a4622176522e398
Block 0031 [126]: 951aa08aeecb2c05
Block 0031 [127]: 6a6c49d2cb75d5b6
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After pass 1:
Block 0000 [ 0]: d3801200410f8c0d
Block 0000 [ 1]: 0bf9e8a6e442ba6d
Block 0000 [ 2]: e2ca92fe9c541fcc
Block 0000 [ 3]: 6269fe6db177a388
...
Block 0031 [124]: 9eacfcfbdb3ce0fc
Block 0031 [125]: 07dedaeb0aee71ac
Block 0031 [126]: 074435fad91548f4
Block 0031 [127]: 2dbfff23f31b5883
After pass 2:
Block 0000 [ 0]: 5f047b575c5ff4d2
Block 0000 [ 1]: f06985dbf11c91a8
Block 0000 [ 2]: 89efb2759f9a8964
Block 0000 [ 3]: 7486a73f62f9b142
...
Block 0031 [124]: 57cfb9d20479da49
Block 0031 [125]: 4099654bc6607f69
Block 0031 [126]: f142a1126075a5c8
Block 0031 [127]: c341b3ca45c10da5
Tag: 51 2b 39 1b 6f 11 62 97
53 71 d3 09 19 73 42 94
f8 68 e3 be 39 84 f3 c1
a1 3a 4d b9 fa be 4a cb
6.2. Argon2i Test Vectors
=======================================
Argon2i version number 19
=======================================
Memory: 32 KiB
Iterations: 3
Parallelism: 4 lanes
Tag length: 32 bytes
Password[32]: 01 01 01 01 01 01 01 01
01 01 01 01 01 01 01 01
01 01 01 01 01 01 01 01
01 01 01 01 01 01 01 01
Salt[16]: 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02
Secret[8]: 03 03 03 03 03 03 03 03
Associated data[12]: 04 04 04 04 04 04 04 04 04 04 04 04
Pre-hashing digest: c4 60 65 81 52 76 a0 b3
e7 31 73 1c 90 2f 1f d8
0c f7 76 90 7f bb 7b 6a
5c a7 2e 7b 56 01 1f ee
ca 44 6c 86 dd 75 b9 46
9a 5e 68 79 de c4 b7 2d
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08 63 fb 93 9b 98 2e 5f
39 7c c7 d1 64 fd da a9
After pass 0:
Block 0000 [ 0]: f8f9e84545db08f6
Block 0000 [ 1]: 9b073a5c87aa2d97
Block 0000 [ 2]: d1e868d75ca8d8e4
Block 0000 [ 3]: 349634174e1aebcc
...
Block 0031 [124]: 975f596583745e30
Block 0031 [125]: e349bdd7edeb3092
Block 0031 [126]: b751a689b7a83659
Block 0031 [127]: c570f2ab2a86cf00
After pass 1:
Block 0000 [ 0]: b2e4ddfcf76dc85a
Block 0000 [ 1]: 4ffd0626c89a2327
Block 0000 [ 2]: 4af1440fff212980
Block 0000 [ 3]: 1e77299c7408505b
...
Block 0031 [124]: e4274fd675d1e1d6
Block 0031 [125]: 903fffb7c4a14c98
Block 0031 [126]: 7e5db55def471966
Block 0031 [127]: 421b3c6e9555b79d
After pass 2:
Block 0000 [ 0]: af2a8bd8482c2f11
Block 0000 [ 1]: 785442294fa55e6d
Block 0000 [ 2]: 9256a768529a7f96
Block 0000 [ 3]: 25a1c1f5bb953766
...
Block 0031 [124]: 68cf72fccc7112b9
Block 0031 [125]: 91e8c6f8bb0ad70d
Block 0031 [126]: 4f59c8bd65cbb765
Block 0031 [127]: 71e436f035f30ed0
Tag: c8 14 d9 d1 dc 7f 37 aa
13 f0 d7 7f 24 94 bd a1
c8 de 6b 01 6d d3 88 d2
99 52 a4 c4 67 2b 6c e8
6.3. Argon2id Test Vectors
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=======================================
Argon2id version number 19
=======================================
Memory: 32 KiB, Iterations: 3, Parallelism: 4 lanes, Tag length: 32 bytes
Password[32]: 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01
01 01 01 01 01 01 01 01 01 01 01 01 01 01 01 01
Salt[16]: 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02 02
Secret[8]: 03 03 03 03 03 03 03 03
Associated data[12]: 04 04 04 04 04 04 04 04 04 04 04 04
Pre-hashing digest: 28 89 de 48 7e b4 2a e5 00 c0 00 7e d9 25 2f
10 69 ea de c4 0d 57 65 b4 85 de 6d c2 43 7a 67 b8 54 6a 2f 0a
cc 1a 08 82 db 8f cf 74 71 4b 47 2e 94 df 42 1a 5d a1 11 2f fa
11 43 43 70 a1 e9 97
After pass 0:
Block 0000 [ 0]: 6b2e09f10671bd43
Block 0000 [ 1]: f69f5c27918a21be
Block 0000 [ 2]: dea7810ea41290e1
Block 0000 [ 3]: 6787f7171870f893
...
Block 0031 [124]: 377fa81666dc7f2b
Block 0031 [125]: 50e586398a9c39c8
Block 0031 [126]: 6f732732a550924a
Block 0031 [127]: 81f88b28683ea8e5
After pass 1:
Block 0000 [ 0]: 3653ec9d01583df9
Block 0000 [ 1]: 69ef53a72d1e1fd3
Block 0000 [ 2]: 35635631744ab54f
Block 0000 [ 3]: 599512e96a37ab6e
...
Block 0031 [124]: 4d4b435cea35caa6
Block 0031 [125]: c582210d99ad1359
Block 0031 [126]: d087971b36fd6d77
Block 0031 [127]: a55222a93754c692
After pass 2:
Block 0000 [ 0]: 942363968ce597a4
Block 0000 [ 1]: a22448c0bdad5760
Block 0000 [ 2]: a5f80662b6fa8748
Block 0000 [ 3]: a0f9b9ce392f719f
...
Block 0031 [124]: d723359b485f509b
Block 0031 [125]: cb78824f42375111
Block 0031 [126]: 35bc8cc6e83b1875
Block 0031 [127]: 0b012846a40f346a
Tag: 0d 64 0d f5 8d 78 76 6c 08 c0 37 a3 4a 8b 53 c9 d0
1e f0 45 2d 75 b6 5e b5 25 20 e9 6b 01 e6 59
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7. Acknowledgements
We thank greatly the following authors who helped a lot in preparing
and reviewing this document: Jean-Philippe Aumasson, Samuel Neves,
Joel Alwen, Jeremiah Blocki, Bill Cox, Arnold Reinhold, Solar
Designer, Russ Housley, Stanislav Smyshlyaev, Kenny Paterson, Alexey
Melnikov.
8. IANA Considerations
None.
9. Security Considerations
9.1. Security as hash function and KDF
The collision and preimage resistance levels of Argon2 are equivalent
to those of the underlying BLAKE2b hash function. To produce a
collision, 2^(256) inputs are needed. To find a preimage, 2^(512)
inputs must be tried.
The KDF security is determined by the key length and the size of the
internal state of hash function H'. To distinguish the output of
keyed Argon2 from random, minimum of (2^(128),2^length(K)) calls to
BLAKE2b are needed.
9.2. Security against time-space tradeoff attacks
Time-space tradeoffs allow computing a memory-hard function storing
fewer memory blocks at the cost of more calls to the internal
comression function. The advantage of tradeoff attacks is measured
in the reduction factor to the time-area product, where memory and
extra compression function cores contribute to the area, and time is
increased to accomodate the recomputation of missed blocks. A high
reduction factor may potentially speed up preimage search.
The best attacks on the 1-pass and 2-pass Argon2i is the low-storage
attack described in [CBS16], which reduces the time-area product
(using the peak memory value) by the factor of 5. The best attack on
3-pass and more Argon2i is [AB16] with reduction factor being a
function of memory size and the number of passes. For 1 gibibyte of
memory: 3 for 3 passes, 2.5 for 4 passes, 2 for 6 passes. The
reduction factor grows by about 0.5 with every doubling the memory
size. To completely prevent time-space tradeoffs from [AB16], the
number of passes must exceed binary logarithm of memory minus 26.
Asymptotically, the best attack on 1-pass Argon2i is given in [BZ17]
with maximal advantage of the adversary upper bounded by O(m^(0.233))
where m is the number of blocks. This attack is also asymptotically
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optimal as [BZ17] also prove the upper bound on any attack of
O(m^(0.25)).
The best tradeoff attack on t-pass Argon2d is the ranking tradeoff
attack, which reduces the time-area product by the factor of 1.33.
The best attack on Argon2id can be obtained by complementing the best
attack on the 1-pass Argon2i with the best attack on a multi-pass
Argon2d. Thus the best tradeoff attack on 1-pass Argon2id is the
combined low-storage attack (for the first half of the memory) and
the ranking attack (for the second half), which bring together the
factor of about 2.1. The best tradeoff attack on t-pass Argon2id is
the ranking tradeoff attack, which reduces the time-area product by
the factor of 1.33.
9.3. Security for time-bounded defenders
A bottleneck in a system employing the password-hashing function is
often the function latency rather than memory costs. A rational
defender would then maximize the bruteforce costs for the attacker
equipped with a list of hashes, salts, and timing information, for
fixed computing time on the defender's machine. The attack cost
estimates from [AB16] imply that for Argon2i, 3 passes is almost
optimal for the most of reasonable memory sizes, and that for Argon2d
and Argon2id, 1 pass maximizes the attack costs for the constant
defender time.
9.4. Recommendations
The Argon2id variant with t=1 and maximum available memory is
recommended as a default setting for all environments. This setting
is secure against side-channel attacks and maximizes adversarial
costs on dedicated bruteforce hardware.
10. References
10.1. Normative References
[RFC7693] Saarinen, M-J., Ed. and J-P. Aumasson, "The BLAKE2
Cryptographic Hash and Message Authentication Code (MAC)",
RFC 7693, DOI 10.17487/RFC7693, November 2015,
<https://www.rfc-editor.org/info/rfc7693>.
10.2. Informative References
[AB15] Biryukov, A. and D. Khovratovich, "Tradeoff Cryptanalysis
of Memory-Hard Functions", Asiacrypt 2015, December 2015,
<https://eprint.iacr.org/2015/227.pdf>.
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[AB16] Alwen, J. and J. Blocki, "Efficiently Computing Data-
Independent Memory-Hard Functions", Crypto 2016, December
2015, <https://eprint.iacr.org/2016/115.pdf>.
[ARGON2] Biryukov, A., Dinu, D., and D. Khovratovich, "Argon2: the
memory-hard function for password hashing and other
applications", WWW www.cryptolux.org, October 2015,
<https://www.cryptolux.org/images/0/0d/Argon2.pdf>.
[ARGON2ESP]
Biryukov, A., Dinu, D., and D. Khovratovich, "Argon2: New
Generation of Memory-Hard Functions for Password Hashing
and Other Applications", Euro SnP 2016, March 2016,
<https://www.cryptolux.org/images/0/0d/Argon2ESP.pdf>.
[BLAKE2] Saarinen, M-J., Ed. and J-P. Aumasson, "The BLAKE2
Cryptographic Hash and Message Authentication Code (MAC)",
RFC 7693, November 2015,
<https://www.rfc-editor.org/info/rfc7693>.
[BZ17] Blocki, J. and S. Zhou, "On the Depth-Robustness and
Cumulative Pebbling Cost of Argon2i", TCC 2017, May 2017,
<https://eprint.iacr.org/2017/442.pdf>.
[CBS16] Corrigan-Gibbs, H., Boneh, D., and S. Schechter, "Balloon
Hashing: Provably Space-Hard Hash Functions with Data-
Independent Access Patterns", Asiacrypt 2016, January
2016, <https://eprint.iacr.org/2016/027.pdf>.
[HARD] Alwen, J. and V. Serbinenko, "High Parallel Complexity
Graphs and Memory-Hard Functions", STOC 2015, 2014,
<https://eprint.iacr.org/2014/238.pdf>.
Authors' Addresses
Alex Biryukov
University of Luxembourg
Email: alex.biryukov@uni.lu
Daniel Dinu
University of Luxembourg
Email: dumitru-daniel.dinu@uni.lu
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Dmitry Khovratovich
ABDK Consulting
Email: khovratovich@gmail.com
Simon Josefsson
SJD AB
Email: simon@josefsson.org
URI: http://josefsson.org/
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