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**Argon2** is a key derivation function that was selected as the winner of the Password Hashing Competition in July 2015.^{[1]}^{[2]} It was designed by Alex Biryukov, Daniel Dinu, and Dmitry Khovratovich from the University of Luxembourg.^{[3]} 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:

- Argon2d maximizes resistance to GPU cracking attacks. It accesses the memory array in a password dependent order, which reduces the possibility of time–memory trade-off (TMTO) attacks, but introduces possible side-channel attacks.
- Argon2i is optimized to resist side-channel attacks. It accesses the memory array in a password independent order.
- Argon2id is a hybrid version. It follows the Argon2i approach for the first pass over memory and the Argon2d approach for subsequent passes. The Internet draft
^{[4]}recommends using Argon2id except when there are reasons to prefer one of the other two modes.

All three modes allow specification by three parameters that control:

- execution time
- memory required
- degree of parallelism

While there is no public cryptanalysis applicable to Argon2d, there are two published attacks on the Argon2i function.

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 N/e < N/2.71 space with no time penalty.^{[5]} According to the Argon2 authors, this attack vector was fixed in version 1.3.^{[6]}

The second attack shows that Argon2i can be computed by an algorithm which has complexity O(n^{7/4} log(n)) for all choices of parameters σ (space cost), τ (time cost), and thread-count such that n=σ∗τ.^{[7]} The Argon2 authors claim that this attack is not efficient if Argon2i is used with three or more passes.^{[6]} 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.^{[8]}

FunctionArgon2Inputs:password (P): Bytes (0..2^{32}-1) Password (or message) to be hashed salt (S): Bytes (8..2^{32}-1) Salt (16 bytes recommended for password hashing) parallelism (p): Number (1..2^{24}-1) Degree of parallelism (i.e. number of threads) tagLength (T): Number (4..2^{32}-1) Desired number of returned bytes memorySizeKB (m): Number (8p..2^{32}-1) Amount of memory (in kibibytes) to use iterations (t): Number (1..2^{32}-1) Number of iterations to perform version (v): Number (0x13) The current version is 0x13 (19 decimal) key (K): Bytes (0..2^{32}-1) Optional key (Errata: PDF says 0..32 bytes, RFC says 0..2^{32}bytes) associatedData (X): Bytes (0..2^{32}-1) Optional arbitrary extra data hashType (y): Number (0=Argon2d, 1=Argon2i, 2=Argon2id)Output:tag: Bytes (tagLength) The resulting generated bytes, tagLength bytes longGenerate initial 64-byte block HAll the input parameters are concatenated and input as a source of additional entropy. Errata: RFC says H_{0}._{0}is 64-bits; PDF says H_{0}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 H_{0}← Blake2b(buffer, 64)//default hash size of Blake2b is 64-bytesCalculate 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 asqCompute the first and second block (i.e. column zero and one ) of each lane (i.e. row)fori ← 0toparallelism-1dofor each row B_{i}[0] ← Hash(H_{0}∥ 0 ∥ i, 1024)//Generate a 1024-byte digestB_{i}[1] ← Hash(H_{0}∥ 1 ∥ i, 1024)//Generate a 1024-byte digestCompute remaining columns of each lanefori ← 0toparallelism-1do//for each rowforj ← 2tocolumnCount-1do//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 B_{i}[j] = G(B_{i}[j-1], B_{i′}[j′]) //the G hash function is not defined Further passes when iterations > 1fornIteration ← 2toiterationsdofori ← 0toparallelism-1dofor each rowforj ← 2tocolumnCount-1do//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) B_{i}[0] = G(B_{i}[columnCount-1], B_{i′}[j′]) B_{i}[j] = G(B_{i}[j-1], B_{i′}[j′]) Compute final blockCas the XOR of the last column of each row C ← B_{0}[columnCount-1]fori ← 1toparallelism-1doC ← CxorB_{i}[columnCount-1] Compute output tagreturnHash(C, tagLength)

Argon2 makes use of a hash function capable of producing digests up to 2^{32} bytes long. This hash function is internally built upon Blake2.

FunctionHash(message, digestSize)Inputs:message: Bytes (0..2^{32}-1) Message to be hashed digestSize: Integer (1..2^{32}) Desired number of bytes to be returned Output: digest: Bytes (digestSize) The resulting generated bytes, digestSize bytes longHashis a variable-length hash function, built using Blake2b, capable of generating digests up to 2^{32}bytes. If the requested digestSize is 64-bytes or lower, then we use Blake2b directlyif(digestSize <= 64)thenreturnBlake2b(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 V_{1}← Blake2b(digestSize ∥ message, 64); Subsequent blocks are generated from previous blocksfori ← 2tordoV_{i}← Blake2b(V_{i-1}, 64) Generate the final (possibly partial) block partialBytesNeeded ← digestSize – 32*r; V_{r+1}← Blake2b(V_{r}, partialBytesNeeded) Concatenate the first 32-bytes of each block V_{i}(except the possibly partial last block, which we take the whole thing) Let A_{i}represent the lower 32-bytes of block V_{i}returnA_{1}∥ A_{2}∥ ... ∥ A_{r}∥ V_{r+1}

**^**"Password Hashing Competition"**^**Jos Wetzels (2016-02-08). "Open Sesame: The Password Hashing Competition and Argon2" (PDF).**^**Argon2: the memory-hard function for password hashing and other applications, Alex Biryukov, et al, October 1, 2015**^**[datatracker.ietf.org] The memory-hard Argon2 password hash and proof-of-work function, draft-irtf-cfrg-argon2-03, accessed August 16, 2017**^**Henry Corrigan-Gibbs, Dan Boneh, Stuart Schechter (2016-01-14). "Balloon Hashing: Provably Space-Hard Hash Functions with Data-Independent Access Patterns" (PDF).- ^
^{a}^{b}"[Cfrg] Argon2 v.1.3".*www.ietf.org*. Retrieved 2016-10-30. **^**Joel Alwen, Jeremiah Blocki (2016-02-19). "Efficiently Computing Data-Independent Memory-Hard Functions" (PDF).**^**Joël Alwen, Jeremiah Blocki (2016-08-05). "Towards Practical Attacks on Argon2i and Balloon Hashing" (PDF).