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Versions: (draft-josefsson-argon2) 00 01 02 03 04 05 06 07 08

Network Working Group                                        A. Biryukov
Internet-Draft                                                   D. Dinu
Intended status: Informational                  University of Luxembourg
Expires: April 12, 2020                                  D. Khovratovich
                                                         ABDK Consulting
                                                            S. Josefsson
                                                                  SJD AB
                                                        October 10, 2019


    The memory-hard Argon2 password hash and proof-of-work function
                       draft-irtf-cfrg-argon2-07

Abstract

   This document describes the Argon2 memory-hard function for password
   hashing and proof-of-work applications.  We provide an implementer-
   oriented description with test vectors.  The purpose is to simplify
   adoption of Argon2 for Internet protocols.  This document is a
   product of the Crypto Forum Research Group (CFRG) in the IRTF.

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
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   Drafts is at https://datatracker.ietf.org/drafts/current/.

   Internet-Drafts are draft documents valid for a maximum of six months
   and may be updated, replaced, or obsoleted by other documents at any
   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on April 12, 2020.

Copyright Notice

   Copyright (c) 2019 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (https://trustee.ietf.org/license-info) in effect on the date of
   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect



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   to this document.  Code Components extracted from this document must
   include Simplified BSD License text as described in Section 4.e of
   the Trust Legal Provisions and are provided without warranty as
   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'  . . . . . . . . . . . .   7
     3.4.  Indexing  . . . . . . . . . . . . . . . . . . . . . . . .   7
       3.4.1.  Getting the 32-bit values J_1 and J_2 . . . . . . . .   8
       3.4.2.  Mapping J_1 and J_2 to reference block index  . . . .   8
     3.5.  Compression function G  . . . . . . . . . . . . . . . . .   9
     3.6.  Permutation P . . . . . . . . . . . . . . . . . . . . . .  11
   4.  Parameter Choice  . . . . . . . . . . . . . . . . . . . . . .  12
   5.  Test Vectors  . . . . . . . . . . . . . . . . . . . . . . . .  13
     5.1.  Argon2d Test Vectors  . . . . . . . . . . . . . . . . . .  13
     5.2.  Argon2i Test Vectors  . . . . . . . . . . . . . . . . . .  14
     5.3.  Argon2id Test Vectors . . . . . . . . . . . . . . . . . .  16
   6.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  17
   7.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  17
   8.  Security Considerations . . . . . . . . . . . . . . . . . . .  17
     8.1.  Security as hash function and KDF . . . . . . . . . . . .  17
     8.2.  Security against time-space tradeoff attacks  . . . . . .  17
     8.3.  Security for time-bounded defenders . . . . . . . . . . .  18
     8.4.  Recommendations . . . . . . . . . . . . . . . . . . . . .  18
   9.  References  . . . . . . . . . . . . . . . . . . . . . . . . .  18
     9.1.  Normative References  . . . . . . . . . . . . . . . . . .  18
     9.2.  Informative References  . . . . . . . . . . . . . . . . .  19
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  19

1.  Introduction

   This document describes the Argon2 [ARGON2ESP] memory-hard function
   for password hashing and proof-of-work applications.  We provide an
   implementer oriented description with 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 [BLAKE2].

   For further background and discussion, see the Argon2 paper [ARGON2].

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
   document are to be interpreted as described in RFC 2119 [RFC2119].

   This document represents the consensus of the Crypto Forum Research
   Group (CFRG).

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





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   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

   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.  MUST have length from 0 to 2^(32) - 1 bytes.

   o  Nonce S, which is a salt for password hashing applications.  MUST
      have length not greater than 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 MUST
      be an integer value from 1 to 2^(24)-1.



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   o  Tag length T MUST be an integer number of bytes from 4 to 2^(32)-
      1.

   o  Memory size m MUST be an 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.

   o  Number of iterations t (used to tune the running time
      independently of the memory size) MUST be an integer number from 1
      to 2^(32)-1.

   o  Version number v MUST be one byte 0x13.

   o  Secret value K is OPTIONAL.  If used, it MUST have length not
      greater than 2^(32)-1 bytes.

   o  Associated data X is OPTIONAL.  If used, it MUST have length not
      greater than 2^(32)-1 bytes.

   o  Type y of Argon2: MUST be 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





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   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.

             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:






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             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:

           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 can be
   computed in parallel and do 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



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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]:

                   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.



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   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].

   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:






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     ( 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

   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.



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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}):

            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.





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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
   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.






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   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.

   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.  Test Vectors

   This section contains test vectors for Argon2.

5.1.  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



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   Block 0000 [  3]: 3ea8efd4d55ac0d1
   ...
   Block 0031 [124]: 28d17914aea9734c
   Block 0031 [125]: 6a4622176522e398
   Block 0031 [126]: 951aa08aeecb2c05
   Block 0031 [127]: 6a6c49d2cb75d5b6

    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

5.2.  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



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   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
                       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





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5.3.  Argon2id Test Vectors

   =======================================
   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



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   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

6.  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.

7.  IANA Considerations

   None.

8.  Security Considerations

8.1.  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.

8.2.  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



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   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
   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.

8.3.  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.

8.4.  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.

9.  References

9.1.  Normative References

   [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>.





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   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", RFC 2119, March 1997,
              <http://www.rfc-editor.org/info/rfc2119>.

9.2.  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>.

   [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>.

   [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







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   Daniel Dinu
   University of Luxembourg

   Email: dumitru-daniel.dinu@uni.lu


   Dmitry Khovratovich
   ABDK Consulting

   Email: khovratovich@gmail.com


   Simon Josefsson
   SJD AB

   Email: simon@josefsson.org
   URI:   http://josefsson.org/


































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