Network Working Group A. Biryukov
Internet-Draft D. Dinu
Intended status: Informational D. Khovratovich
Expires: March 26, 2017 University of Luxembourg
S. Josefsson
SJD AB
September 22, 2016

The memory-hard Argon2 password hash and proof-of-work function
draft-irtf-cfrg-argon2-01

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.

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Table of Contents

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. 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 the cache and memory organization of the recent Intel and AMD processors. Argon2 has two variants: Argon2d and Argon2i. Argon2d is faster and uses data-depending 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. Argon2i is slower as it makes more passes over the memory to protect from tradeoff attacks.

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

2. Notation and Conventions

x**y --- x multiplied by itself y times

a*b --- multiplication of a and b

c-d --- substraction of c with d

E_f --- variable E with subscript index f

g / h --- g divided by h

I(j) --- function I evaluated on parameter j

K || L --- string K concatenated with string L

a ^ b --- bitwise exclusive-or between a and b

a mod b --- remainder of a modulo b, always in range [0, b-1]

a >>> n --- rotation of a to the right by n bits

trunc(a) --- the 64-bit value a truncated to the 32 least significant bits

extract(a, i) --- the i-th set of 32-bits from a

|A| --- the number of elements in set A

3. Argon2 Algorithm

3.1. Argon2 Inputs and Outputs

Argon2 has the following input parameters:

The Argon2 output is a T-length string.

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. Here H is the BLAKE2b [I-D.saarinen-blake2] hash function, and the compression function G is based on its internal permutation. A variable-length hash function H' built upon H is also used. G and H' are described in later section.

          H_0 = H(p, T, m, t, v, y, length(P), P, length(S), S,
                  length(K), K, length(X), X)
        
          m' = 4 * p * floor (m / 4p)
        
          B[i][0] = H'(H_0, 0, i)
        
          B[i][1] = H'(H_0, 1, i)
        
          B[i][j] = G(B[i][j-1], B[i'][j'])
        
          B[i][0] = G(B[i][q-1], B[i'][j'])
          B[i][j] = G(B[i][j-1], B[i'][j'])
        
          C = B[0][q-1] XOR B[1][q-1] XOR ... XOR B[p-1][q-1]
        

The Argon2 operation is as follows.

  1. Establish H_0 as the 64-bit value as shown in the figure below. H is BLAKE2b and the non-strings p, T, m, t, v, y, length(P), length(S), length(K), and length(X) are treated as a 32-bit little-endian encoding of the integer.
  2. Allocate the memory as m' 1024-byte blocks where m' is derived as:
  3. Compute B[i][0] for all i ranging from (and including) 0 to (not including) p.
  4. Compute B[i][1] for all i ranging from (and including) 0 to (not including) p.
  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 i' and j' are determined differently for Argon2d and Argon2i.
  6. If the number of iterations t is larger than 1, we repeat the steps however replacing the computations with the following expression:
  7. After t steps have been iterated, the final block C is computed as the XOR of the last column:
  8. The output tag is computed as H'(C).

3.3. Variable-length hash function H'

        if T <= 64
            H'(X) = H_T(T||X)
        else
            r = ceil(T/32)-2
            V_1 = H_64(T||X)
            V_2 = H_64(V_1)
            ...
            V_r = H_64(V_{r-1})
            V_{r+1} = H_{T-32*r}(V_{r}) 
            H'(X) = A_1 || A_2 || ... || A_r || V_{r+1}
      

Let H_x be a hash function with x-byte output (in our case H_x is BLAKE2b, which supports x between 1 and 64 inclusive). Let V_i be a 64-byte block, and A_i be its first 32 bytes, and T < 2**32 be the tag length in bytes, encoded in little-endian as 32-bit integer. Then we define:

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 = extract(B[i][j-1], 1)
                J_2 = extract(B[i][j-1], 2)
              

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]:

3.4.1.2. Argon2i

    ( r || l || s || m' || t || x || i || 0 ), 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
    x  -- the Argon2 type (0 for Argon2d and 1 for Argon2i)
    i  -- the counter (starts from 1 in each segment)
   

Each application of the 2-round compression function G in the counter mode gives 128 64-bit values J_1 || J_2. The first input is the all zero block and the second input is constructed as follows:

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 R contains the indices that can be referenced according to the following rules:

  1. If l is the currnt lane, then R includes the indices of all blocks computed in this lane that are not overwritten yet, excluding B[i][j-1]
  2. If l is not the current lane, then R 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 R is excluded.

                J_1 in [0, |R|) -> |R|(1 - J_1**2 / 2**64)
              

We are going to take a block from R with a non-uniform distribution over [0, |R|):

                x = J_1**2 / 2**32
                y = (|R| * x) / 2**32
                z = |R| - 1 - y
              

To avoid floating poit computation, the following approximation is used:

The value of z gives the reference block index in R.

3.5. Compression function G

        P(A_0, A_1, ... ,A_7) = (B_0, B_1, ... ,B_7)
      

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:

  ( 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)
      

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 rowwise, and then columnwise to get Z:

        G: (X, Y) -> R = X XOR Y -P-> Q -P-> Z -P-> Z XOR R
      
                         +---+       +---+
                         | X |       | Y |
                         +---+       +---+
                           |           |
                           ---->XOR<----
                         --------|
                         |      \ /
                         |     +---+
                         |     | R |
                         |     +---+
                         |       |
                         |      \ /
                         |   P rowwise
                         |       |
                         |      \ /
                         |     +---+
                         |     | Q |
                         |     +---+
                         |       |
                         |      \ /
                         |  P columnwise
                         |       |
                         |      \ /
                         |     +---+
                         |     | Z |
                         |     +---+
                         |       |
                         |      \ /
                         ------>XOR
                                 |
                                \ /
      

Argon2 compression function G.

Finally, G outputs Z XOR R:

3.6. Permutation P

         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
      
        G(v_0, v_4,  v_8, v_12)
        G(v_1, v_5,  v_9, v_13)
        G(v_2, v_6, v_10, v_14)
        G(v_3, v_7, v_11, v_15)

        G(v_0, v_5, v_10, v_15)
        G(v_1, v_6, v_11, v_12)
        G(v_2, v_7,  v_8, v_13)
        G(v_3, v_4,  v_9, v_14)
      
        a <- (a + b + 2 * trunc(a) * trunc(b)) mod 2**64
        d <- (d ^ a) >>> 32
        c <- (c + d + 2 * trunc(c) * trunc(d)) mod 2**64
        b <- (b ^ c) >>> 24

        a <- (a + b + 2 * trunc(a) * trunc(b)) mod 2**64
        d <- (d ^ a) >>> 16
        c <- (c + d + 2 * trunc(c) * trunc(d)) mod 2**64
        b <- (b ^ c) >>> 63
      

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}):

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:

Argon2i is optimized for more realistic settings, where the adversary possibly can access the same machine, use its CPU or mount cold-boot attacks. We use three passes to get rid entirely of the password in the memory. We suggest the following settings:

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 Argon2i.
  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 is a viable threat, 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. Example Code

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);
}
  
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);
}
  
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];
  }
}
  
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) {
    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 {
       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);
}
  
uint32_t index_alpha(const argon2_instance_t *instance,
                     const argon2_position_t *position, 
                     uint32_t pseudo_rand,
                     int same_lane) {
  /*
   * Pass 0:
   *  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 -
                      (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;
}
  
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) {
        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) {
    free(thr_data);
  }

  return ARGON2_OK;
}
  

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

 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
                    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
      

7. Acknowledgements

TBA

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 is 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 Argon2i is the ranking tradeoff attack[AB15], which reduces the time-area product by the factor of 3. The best attack on 4 passes and more of Argon2i is [AB16], but its reduction factor does not exceed 2 up to 32 GiB of memory. To completely prevent time-space tradeoffs from [AB16], number t of passes must exceed binary logarithm of memory minus 26.

The best attack on t-pass Argon2d is the ranking tradeoff attack, which reduces the time-area product by the factor of 1.33.

10. References

10.1. Normative References

[I-D.saarinen-blake2] Saarinen, M. and J. Aumasson, "The BLAKE2 Cryptographic Hash and MAC", Internet-Draft draft-saarinen-blake2-06, August 2015.

10.2. Informative References

[ARGON2] Biryukov, A., Dinu, D. and D. Khovratovich, "Argon2: the memory-hard function for password hashing and other applications", WWW <https://www.cryptolux.org/images/0/0d/Argon2.pdf>, October 2015.
[CBS16] Corrigan-Gibbs, H., Boneh, D. and S. Schechter, "Balloon Hashing: Provably Space-Hard Hash Functions with Data-Independent Access Patterns", WWW <https://eprint.iacr.org/2016/027.pdf>, January 2016.
[AB16] Alwen, J. and J. Blocki, "Efficiently Computing Data-Independent Memory-Hard Functions", WWW <https://eprint.iacr.org/2016/115.pdf>, December 2015.
[AB15] Biryukov, A. and D. Khovratovich, "Tradeoff Cryptanalysis of Memory-Hard Functions", Asiacrypt'15 <https://eprint.iacr.org/2015/227.pdf>, December 2015.

Authors' Addresses

Alex Biryukov University of Luxembourg EMail: alex.biryukov@uni.lu
Daniel Dinu University of Luxembourg EMail: dumitru-daniel.dinu@uni.lu
Dmitry Khovratovich University of Luxembourg EMail: dmitry.khovratovich@uni.lu
Simon Josefsson SJD AB EMail: simon@josefsson.org URI: http://josefsson.org/