Argon2

Argon2 is a key derivation function that was selected as the winner of the Password Hashing Competition in July 2015. It was designed by Alex Biryukov, Daniel Dinu, and Dmitry Khovratovich from the University of Luxembourg. The reference implementation of Argon2 is released under a Creative Commons CC0 license (i.e. public domain) or the Apache License 2.0, and provides three related versions:
All three modes allow specification by three parameters that control:
Cryptanalysis.
While there is no public cryptanalysis applicable to Argon2d, there are two published attacks on the Argon2i function. The first attack is applicable only to the old version of Argon2i, while the second has been extended to the latest version (1.3)
The first attack shows that it is possible to compute a single-pass Argon2i function using between a quarter and a fifth of the desired space with no time penalty, and compute a multiple-pass Argon2i using only / < /2.71 space with no time penalty. According to the Argon2 authors, this attack vector was fixed in version 1.3.
The second attack shows that Argon2i can be computed by an algorithm which has complexity O(7/4 log()) for all choices of parameters (space cost), (time cost), and thread-count such that =∗. The Argon2 authors claim that this attack is not efficient if Argon2i is used with three or more passes. However, Joël Alwen and Jeremiah Blocki improved the attack and showed that in order for the attack to fail, Argon2i 1.3 needs more than 10 passes over memory.
Algorithm.
 Function Argon2
 Inputs:
 password (P): Bytes (0..232-1) Password (or message) to be hashed
 salt (S): Bytes (8..232-1) Salt (16 bytes recommended for password hashing)
 parallelism (p): Number (1..224-1) Degree of parallelism (i.e. number of threads)
 tagLength (T): Number (4..232-1) Desired number of returned bytes
 memorySizeKB (m): Number (8p..232-1) Amount of memory (in kibibytes) to use
 iterations (t): Number (1..232-1) Number of iterations to perform
 version (v): Number (0x13) The current version is 0x13 (19 decimal)
 key (K): Bytes (0..232-1) Optional key (Errata: PDF says 0..32 bytes, RFC says 0..232 bytes)
 associatedData (X): Bytes (0..232-1) Optional arbitrary extra data
 hashType (y): Number (0=Argon2d, 1=Argon2i, 2=Argon2id)
 Output:
 tag: Bytes (tagLength) The resulting generated bytes, tagLength bytes long
 "Generate initial 64-byte block H0."
 All the input parameters are concatenated and input as a source of additional entropy.
 Errata: RFC says H0 is 64-bits; PDF says H0 is 64-bytes.
 Errata: RFC says the Hash is H^, the PDF says it's ℋ (but doesn't document what ℋ is). It's actually Blake2b.
 Variable length items are prepended with their length as 32-bit little-endian integers.
 buffer ← parallelism ∥ tagLength ∥ memorySizeKB ∥ iterations ∥ version ∥ hashType
 ∥ Length(password) ∥ Password
 ∥ Length(salt) ∥ salt
 ∥ Length(key) ∥ key
 ∥ Length(associatedData) ∥ associatedData
 H0 ← Blake2b(buffer, 64) "//default hash size of Blake2b is 64-bytes"
 Calculate number of 1 KB blocks by rounding down memorySizeKB to the nearest multiple of 4*parallelism kibibytes
 blockCount ← Floor(memorySizeKB, 4*parallelism)
 Allocate two-dimensional array of 1 KiB blocks (parallelism rows x columnCount columns)
 columnCount ← blockCount / parallelism; //In the RFC, columnCount is referred to as q
 Compute the first and second block (i.e. column zero and one ) of each lane (i.e. row)
 for i ← 0 to parallelism-1 do for each row
 Bi[0] ← Hash(H0 ∥ 0 ∥ i, 1024) "//Generate a 1024-byte digest"
 Bi[1] ← Hash(H0 ∥ 1 ∥ i, 1024) "//Generate a 1024-byte digest"
 Compute remaining columns of each lane
 for i ← 0 to parallelism-1 do //for each row
 for j ← 2 to columnCount-1 do //for each subsequent column
 //i' and j' indexes depend if it's Argon2i, Argon2d, or Argon2id (See section 3.4)
 i′, j′ ← GetBlockIndexes(i, j) //the GetBlockIndexes function is not defined
 Bi[j] = G(Bi[j-1], Bi′[j′]) //the G hash function is not defined
 Further passes when iterations > 1
 for nIteration ← 2 to iterations do
 for i ← 0 to parallelism-1 do for each row
 for j ← 0 to columnCount-1 do //for each subsequent column
 //i' and j' indexes depend if it's Argon2i, Argon2d, or Argon2id (See section 3.4)
 i′, j′ ← GetBlockIndexes(i, j)
 if j == 0 then 
 Bi[0] = Bi[0] xor G(Bi[columnCount-1], Bi′[j′])
 else
 Bi[j] = Bi[j] xor G(Bi[j-1], Bi′[j′])
 Compute final block C as the XOR of the last column of each row
 C ← B0[columnCount-1]
 for i ← 1 to parallelism-1 do
 C ← C xor Bi[columnCount-1]
 Compute output tag
 return Hash(C, tagLength)
Variable-length hash function.
Argon2 makes use of a hash function capable of producing digests up to 232 bytes long. This hash function is internally built upon Blake2.
 Function Hash(message, digestSize)
 Inputs:
 message: Bytes (0..232-1) Message to be hashed
 digestSize: Integer (1..232) Desired number of bytes to be returned
 Output:
 digest: Bytes (digestSize) The resulting generated bytes, digestSize bytes long
 Hash is a variable-length hash function, built using Blake2b, capable of generating
 digests up to 232 bytes.
 If the requested digestSize is 64-bytes or lower, then we use Blake2b directly
 if (digestSize <= 64) then
 return Blake2b(digestSize ∥ message, digestSize) //concatenate 32-bit little endian digestSize with the message bytes
 For desired hashes over 64-bytes (e.g. 1024 bytes for Argon2 blocks),
 we use Blake2b to generate twice the number of needed 64-byte blocks,
 and then only use 32-bytes from each block
 Calculate the number of whole blocks (knowing we're only going to use 32-bytes from each)
 r ← Ceil(digestSize/32)-1;
 Generate r whole blocks.
 Initial block is generated from message
 V1 ← Blake2b(digestSize ∥ message, 64);
 Subsequent blocks are generated from previous blocks
 for i ← 2 to r do
 Vi ← Blake2b(Vi-1, 64)
 Generate the final (possibly partial) block
 partialBytesNeeded ← digestSize – 32*r;
 Vr+1 ← Blake2b(Vr, partialBytesNeeded)
 Concatenate the first 32-bytes of each block Vi
 (except the possibly partial last block, which we take the whole thing)
 Let Ai represent the lower 32-bytes of block Vi
 return A1 ∥ A2 ∥ ... ∥ Ar ∥ Vr+1