LSH (hash function)

LSH is a cryptographic hash function designed in 2014 by South Korea to provide integrity in general-purpose software environments such as PCs and smart devices.^[1] LSH is one of the cryptographic algorithms approved by the Korean Cryptographic Module Validation Program (KCMVP). And it is the national standard of South Korea (KS X 3262).

Specification

The overall structure of the hash function LSH is shown in the following figure.

 

Overall structure of LSH

The hash function LSH has the wide-pipe Merkle-Damgård structure with one-zeros padding. The message hashing process of LSH consists of the following three stages.

1. Initialization:
   • One-zeros padding of a given bit string message.
   • Conversion to 32-word array message blocks from the padded bit string message.
   • Initialization of a chaining variable with the initialization vector.
2. Compression:
   • Updating of chaining variables by iteration of a compression function with message blocks.
3. Finalization:
   • Generation of an ${\displaystyle n}$ -bit hash value from the final chaining variable.

function Hash function LSH input: Bit string message ${\displaystyle m}$  output: Hash value ${\displaystyle h\in \{0,1\}^{n}}$  procedure ${\displaystyle \qquad }$ One-zeros padding of ${\displaystyle m}$  ${\displaystyle \qquad }$ Generation of ${\displaystyle t}$  message blocks ${\displaystyle \{{\textsf {M}}^{(i)}\}_{i=0}^{t-1}}$ , where ${\displaystyle t={\Big \lceil }{\frac {|m|+1}{32w}}{\Big \rceil }}$  from the padded bit string ${\displaystyle \qquad }$ ${\displaystyle {\textsf {CV}}^{(0)}\leftarrow {\textsf {IV}}}$  ${\displaystyle \qquad }$ for ${\displaystyle i=0}$  to ${\displaystyle (t-1)}$  do ${\displaystyle \qquad }$ ${\displaystyle \qquad }$ ${\displaystyle {\textsf {CV}}^{(i+1)}\leftarrow {\textrm {CF}}({\textsf {CV}}^{(i)},{\textsf {M}}^{(i)})}$  ${\displaystyle \qquad }$ end for ${\displaystyle \qquad }$ ${\displaystyle h\leftarrow {\textrm {FIN}}_{n}({\textsf {CV}}^{(t)})}$  ${\displaystyle \qquad }$ return ${\displaystyle h}$

The specifications of the hash function LSH are as follows.

Hash function LSH specifications

Algorithm	Digest size in bits (${\displaystyle n}$ )	Number of step functions (${\displaystyle N_{s}}$ )	Chaining variable size in bits	Message block size in bits	Word size in bits (${\displaystyle w}$ )
LSH-256-224	224	26	512	1024	32
LSH-256-256	256
LSH-512-224	224	28	1024	2048	64
LSH-512-256	256
LSH-512-384	384
LSH-512-512	512

Initialization

Let ${\displaystyle m}$  be a given bit string message. The given ${\displaystyle m}$  is padded by one-zeros, i.e., the bit ‘1’ is appended to the end of ${\displaystyle m}$ , and the bit ‘0’s are appended until a bit length of a padded message is ${\displaystyle 32wt}$ , where ${\displaystyle t=\lceil (|m|+1)/32w\rceil }$  and ${\displaystyle \lceil x\rceil }$  is the smallest integer not less than ${\displaystyle x}$ .

Let ${\displaystyle m_{p}=m_{0}\|m_{1}\|\ldots \|m_{(32wt-1)}}$  be the one-zeros-padded ${\displaystyle 32wt}$ -bit string of ${\displaystyle m}$ . Then ${\displaystyle m_{p}}$  is considered as a ${\displaystyle 4wt}$ -byte array ${\displaystyle m_{a}=(m[0],\ldots ,m[4wt-1])}$ , where ${\displaystyle m[k]=m_{8k}\|m_{(8k+1)}\|\ldots \|m_{(8k+7)}}$  for all ${\displaystyle 0\leq k\leq (4wt-1)}$ . The ${\displaystyle 4wt}$ -byte array ${\displaystyle m_{a}}$  converts into a ${\displaystyle 32t}$ -word array ${\displaystyle {\textsf {M}}=(M[0],\ldots ,M[32t-1])}$  as follows.

${\displaystyle M[s]\leftarrow m[ws/8+(w/8-1)]\|\ldots \|m[ws/8+1]\|m[ws/8]}$  ${\displaystyle (0\leq s\leq (32t-1))}$ 

From the word array ${\displaystyle {\textsf {M}}}$ , we define the ${\displaystyle t}$  32-word array message blocks ${\displaystyle \{{\textsf {M}}^{(i)}\}_{i=0}^{t-1}}$  as follows.

${\displaystyle {\textsf {M}}^{(i)}\leftarrow (M[32i],M[32i+1],\ldots ,M[32i+31])}$  ${\displaystyle (0\leq i\leq (t-1))}$ 

The 16-word array chaining variable ${\displaystyle {\textsf {CV}}^{(0)}}$  is initialized to the initialization vector ${\displaystyle {\textsf {IV}}}$ .

${\displaystyle {\textsf {CV}}^{(0)}[l]\leftarrow {\textsf {IV}}[l]}$  ${\displaystyle (0\leq l\leq 15)}$ 

The initialization vector ${\displaystyle {\textsf {IV}}}$  is as follows. In the following tables, all values are expressed in hexadecimal form.

LSH-256-224 initialization vector

${\displaystyle {\textsf {IV}}[0]}$	${\displaystyle {\textsf {IV}}[1]}$	${\displaystyle {\textsf {IV}}[2]}$	${\displaystyle {\textsf {IV}}[3]}$	${\displaystyle {\textsf {IV}}[4]}$	${\displaystyle {\textsf {IV}}[5]}$	${\displaystyle {\textsf {IV}}[6]}$	${\displaystyle {\textsf {IV}}[7]}$
068608D3	62D8F7A7	D76652AB	4C600A43	BDC40AA8	1ECA0B68	DA1A89BE	3147D354
${\displaystyle {\textsf {IV}}[8]}$	${\displaystyle {\textsf {IV}}[9]}$	${\displaystyle {\textsf {IV}}[10]}$	${\displaystyle {\textsf {IV}}[11]}$	${\displaystyle {\textsf {IV}}[12]}$	${\displaystyle {\textsf {IV}}[13]}$	${\displaystyle {\textsf {IV}}[14]}$	${\displaystyle {\textsf {IV}}[15]}$
707EB4F9	F65B3862	6B0B2ABE	56B8EC0A	CF237286	EE0D1727	33636595	8BB8D05F

LSH-256-256 initialization vector

${\displaystyle {\textsf {IV}}[0]}$	${\displaystyle {\textsf {IV}}[1]}$	${\displaystyle {\textsf {IV}}[2]}$	${\displaystyle {\textsf {IV}}[3]}$	${\displaystyle {\textsf {IV}}[4]}$	${\displaystyle {\textsf {IV}}[5]}$	${\displaystyle {\textsf {IV}}[6]}$	${\displaystyle {\textsf {IV}}[7]}$
46A10F1F	FDDCE486	B41443A8	198E6B9D	3304388D	B0F5A3C7	B36061C4	7ADBD553
${\displaystyle {\textsf {IV}}[8]}$	${\displaystyle {\textsf {IV}}[9]}$	${\displaystyle {\textsf {IV}}[10]}$	${\displaystyle {\textsf {IV}}[11]}$	${\displaystyle {\textsf {IV}}[12]}$	${\displaystyle {\textsf {IV}}[13]}$	${\displaystyle {\textsf {IV}}[14]}$	${\displaystyle {\textsf {IV}}[15]}$
105D5378	2F74DE54	5C2F2D95	F2553FBE	8051357A	138668C8	47AA4484	E01AFB41

LSH-512-224 initialization vector

${\displaystyle {\textsf {IV}}[0]}$	${\displaystyle {\textsf {IV}}[1]}$	${\displaystyle {\textsf {IV}}[2]}$	${\displaystyle {\textsf {IV}}[3]}$
0C401E9FE8813A55	4A5F446268FD3D35	FF13E452334F612A	F8227661037E354A
${\displaystyle {\textsf {IV}}[4]}$	${\displaystyle {\textsf {IV}}[5]}$	${\displaystyle {\textsf {IV}}[6]}$	${\displaystyle {\textsf {IV}}[7]}$
A5F223723C9CA29D	95D965A11AED3979	01E23835B9AB02CC	52D49CBAD5B30616
${\displaystyle {\textsf {IV}}[8]}$	${\displaystyle {\textsf {IV}}[9]}$	${\displaystyle {\textsf {IV}}[10]}$	${\displaystyle {\textsf {IV}}[11]}$
9E5C2027773F4ED3	66A5C8801925B701	22BBC85B4C6779D9	C13171A42C559C23
${\displaystyle {\textsf {IV}}[12]}$	${\displaystyle {\textsf {IV}}[13]}$	${\displaystyle {\textsf {IV}}[14]}$	${\displaystyle {\textsf {IV}}[15]}$
31E2B67D25BE3813	D522C4DEED8E4D83	A79F5509B43FBAFE	E00D2CD88B4B6C6A

LSH-512-256 initialization vector

${\displaystyle {\textsf {IV}}[0]}$	${\displaystyle {\textsf {IV}}[1]}$	${\displaystyle {\textsf {IV}}[2]}$	${\displaystyle {\textsf {IV}}[3]}$
6DC57C33DF989423	D8EA7F6E8342C199	76DF8356F8603AC4	40F1B44DE838223A
${\displaystyle {\textsf {IV}}[4]}$	${\displaystyle {\textsf {IV}}[5]}$	${\displaystyle {\textsf {IV}}[6]}$	${\displaystyle {\textsf {IV}}[7]}$
39FFE7CFC31484CD	39C4326CC5281548	8A2FF85A346045D8	FF202AA46DBDD61E
${\displaystyle {\textsf {IV}}[8]}$	${\displaystyle {\textsf {IV}}[9]}$	${\displaystyle {\textsf {IV}}[10]}$	${\displaystyle {\textsf {IV}}[11]}$
CF785B3CD5FCDB8B	1F0323B64A8150BF	FF75D972F29EA355	2E567F30BF1CA9E1
${\displaystyle {\textsf {IV}}[12]}$	${\displaystyle {\textsf {IV}}[13]}$	${\displaystyle {\textsf {IV}}[14]}$	${\displaystyle {\textsf {IV}}[15]}$
B596875BF8FF6DBA	FCCA39B089EF4615	ECFF4017D020B4B6	7E77384C772ED802

LSH-512-384 initialization vector

${\displaystyle {\textsf {IV}}[0]}$	${\displaystyle {\textsf {IV}}[1]}$	${\displaystyle {\textsf {IV}}[2]}$	${\displaystyle {\textsf {IV}}[3]}$
53156A66292808F6	B2C4F362B204C2BC	B84B7213BFA05C4E	976CEB7C1B299F73
${\displaystyle {\textsf {IV}}[4]}$	${\displaystyle {\textsf {IV}}[5]}$	${\displaystyle {\textsf {IV}}[6]}$	${\displaystyle {\textsf {IV}}[7]}$
DF0CC63C0570AE97	DA4441BAA486CE3F	6559F5D9B5F2ACC2	22DACF19B4B52A16
${\displaystyle {\textsf {IV}}[8]}$	${\displaystyle {\textsf {IV}}[9]}$	${\displaystyle {\textsf {IV}}[10]}$	${\displaystyle {\textsf {IV}}[11]}$
BBCDACEFDE80953A	C9891A2879725B3E	7C9FE6330237E440	A30BA550553F7431
${\displaystyle {\textsf {IV}}[12]}$	${\displaystyle {\textsf {IV}}[13]}$	${\displaystyle {\textsf {IV}}[14]}$	${\displaystyle {\textsf {IV}}[15]}$
BB08043FB34E3E30	A0DEC48D54618EAD	150317267464BC57	32D1501FDE63DC93

LSH-512-512 initialization vector

${\displaystyle {\textsf {IV}}[0]}$	${\displaystyle {\textsf {IV}}[1]}$	${\displaystyle {\textsf {IV}}[2]}$	${\displaystyle {\textsf {IV}}[3]}$
ADD50F3C7F07094E	E3F3CEE8F9418A4F	B527ECDE5B3D0AE9	2EF6DEC68076F501
${\displaystyle {\textsf {IV}}[4]}$	${\displaystyle {\textsf {IV}}[5]}$	${\displaystyle {\textsf {IV}}[6]}$	${\displaystyle {\textsf {IV}}[7]}$
8CB994CAE5ACA216	FBB9EAE4BBA48CC7	650A526174725FEA	1F9A61A73F8D8085
${\displaystyle {\textsf {IV}}[8]}$	${\displaystyle {\textsf {IV}}[9]}$	${\displaystyle {\textsf {IV}}[10]}$	${\displaystyle {\textsf {IV}}[11]}$
B6607378173B539B	1BC99853B0C0B9ED	DF727FC19B182D47	DBEF360CF893A457
${\displaystyle {\textsf {IV}}[12]}$	${\displaystyle {\textsf {IV}}[13]}$	${\displaystyle {\textsf {IV}}[14]}$	${\displaystyle {\textsf {IV}}[15]}$
4981F5E570147E80	D00C4490CA7D3E30	5D73940C0E4AE1EC	894085E2EDB2D819

Compression

In this stage, the ${\displaystyle t}$  32-word array message blocks ${\displaystyle \{{\textsf {M}}^{(i)}\}_{i=0}^{t-1}}$ , which are generated from a message ${\displaystyle m}$  in the initialization stage, are compressed by iteration of compression functions. The compression function ${\displaystyle {\textrm {CF}}:{\mathcal {W}}^{16}\times {\mathcal {W}}^{32}\rightarrow {\mathcal {W}}^{16}}$  has two inputs; the ${\displaystyle i}$ -th 16-word chaining variable ${\displaystyle {\textsf {CV}}^{(i)}}$  and the ${\displaystyle i}$ -th 32-word message block ${\displaystyle {\textsf {M}}^{(i)}}$ . And it returns the ${\displaystyle (i+1)}$ -th 16-word chaining variable ${\displaystyle {\textsf {CV}}^{(i+1)}}$ . Here and subsequently, ${\displaystyle {\mathcal {W}}^{t}}$  denotes the set of all ${\displaystyle t}$ -word arrays for ${\displaystyle t\geq 1}$ .

The following four functions are used in a compression function:

• Message expansion function ${\displaystyle {\textrm {MsgExp}}:{\mathcal {W}}^{32}\rightarrow {\mathcal {W}}^{16(Ns+1)}}$ 
• Message addition function ${\displaystyle {\textrm {MsgAdd}}:{\mathcal {W}}^{16}\times {\mathcal {W}}^{16}\rightarrow {\mathcal {W}}^{16}}$ 
• Mix function ${\displaystyle {\textrm {Mix}}_{j}:{\mathcal {W}}^{16}\rightarrow {\mathcal {W}}^{16}}$ 
• Word-permutation function ${\displaystyle {\textrm {WordPerm}}:{\mathcal {W}}^{16}\rightarrow {\mathcal {W}}^{16}}$ 

The overall structure of the compression function is shown in the following figure.

 

Compression function of LSH

In a compression function, the message expansion function ${\displaystyle {\textrm {MsgExp}}}$  generates ${\displaystyle (N_{s}+1)}$  16-word array sub-messages ${\displaystyle \{{\textsf {M}}_{j}^{(i)}\}_{j=0}^{N_{s}}}$  from given ${\displaystyle {\textsf {M}}^{(i)}}$ . Let ${\displaystyle {\textsf {T}}=(T[0],\ldots ,T[15])}$  be a temporary 16-word array set to the ${\displaystyle i}$ -th chaining variable ${\displaystyle {\textsf {CV}}^{(i)}}$ . The ${\displaystyle j}$ -th step function ${\displaystyle {\textrm {Step}}_{j}}$  having two inputs ${\displaystyle {\textsf {T}}}$  and ${\displaystyle {\textsf {M}}_{j}^{(i)}}$  updates ${\displaystyle {\textsf {T}}}$ , i.e., ${\displaystyle {\textsf {T}}\leftarrow {\textrm {Step}}_{j}\left({\textsf {T}},{\textsf {M}}_{j}^{(i)}\right)}$ . All step functions are proceeded in order ${\displaystyle j=0,\ldots ,N_{s}-1}$ . Then one more ${\displaystyle {\textrm {MsgAdd}}}$  operation by ${\displaystyle {\textsf {M}}_{N_{s}}^{(i)}}$  is proceeded, and the ${\displaystyle (i+1)}$ -th chaining variable ${\displaystyle {\textsf {CV}}^{(i+1)}}$  is set to ${\displaystyle {\textsf {T}}}$ . The process of a compression function in detail is as follows.

function Compression function ${\displaystyle {\textrm {CF}}}$  input: The ${\displaystyle i}$ -th chaining variable ${\displaystyle {\textsf {CV}}^{(i)}\in {\mathcal {W}}^{16}}$  and the ${\displaystyle i}$ -th message block ${\displaystyle {\textsf {M}}^{(i)}\in {\mathcal {W}}^{32}}$  output: The ${\displaystyle (i+1)}$ -th chaining variable ${\displaystyle {\textsf {CV}}^{(i+1)}\in {\mathcal {W}}^{16}}$  procedure ${\displaystyle \qquad }$ ${\displaystyle \{{\textsf {M}}_{j}^{(i)}\}_{j=0}^{N_{s}}\leftarrow {\textrm {MsgExp}}\left({\textsf {M}}^{(i)}\right)}$  ${\displaystyle \qquad }$ ${\displaystyle {\textsf {T}}\leftarrow {\textsf {CV}}^{(i)}}$  ${\displaystyle \qquad }$ for ${\displaystyle j=0}$  to ${\displaystyle (N_{s}-1)}$  do ${\displaystyle \qquad }$ ${\displaystyle \qquad }$ ${\displaystyle {\textsf {T}}\leftarrow {\textrm {Step}}_{j}\left({\textsf {T}},{\textsf {M}}_{j}^{(i)}\right)}$  ${\displaystyle \qquad }$ end for ${\displaystyle \qquad }$ ${\displaystyle {\textsf {CV}}^{(i+1)}\leftarrow {\textrm {MsgAdd}}\left({\textsf {T}},{\textsf {M}}_{N_{s}}^{(i)}\right)}$  ${\displaystyle \qquad }$ return ${\displaystyle {\textsf {CV}}^{(i+1)}}$

Here the ${\displaystyle j}$ -th step function ${\displaystyle {\textrm {Step}}_{j}:{\mathcal {W}}^{16}\times {\mathcal {W}}^{16}\rightarrow {\mathcal {W}}^{16}}$  is as follows.

${\displaystyle {\textrm {Step}}_{j}:={\textrm {WordPerm}}\circ {\textrm {Mix}}_{j}\circ {\textrm {MsgAdd}}}$  ${\displaystyle (0\leq j\leq (N_{s}-1))}$ 

The following figure shows the ${\displaystyle j}$ -th step function ${\displaystyle {\textrm {Step}}_{j}}$  of a compression function.

 

The ${\displaystyle j}$ -th step function ${\displaystyle {\textrm {Step}}_{j}}$ 

Message Expansion Function MsgExp

Let ${\displaystyle {\textsf {M}}^{(i)}=(M^{(i)}[0],\ldots ,M^{(i)}[31])}$  be the ${\displaystyle i}$ -th 32-word array message block. The message expansion function ${\displaystyle {\textrm {MsgExp}}}$  generates ${\displaystyle (N_{s}+1)}$  16-word array sub-messages ${\displaystyle \{{\textsf {M}}_{j}^{(i)}\}_{j=0}^{N_{s}}}$  from a message block ${\displaystyle {\textsf {M}}^{(i)}}$ . The first two sub-messages ${\displaystyle {\textsf {M}}_{0}^{(i)}=(M_{0}^{(i)}[0],\ldots ,M_{0}^{(i)}[15])}$  and ${\displaystyle {\textsf {M}}_{1}^{(i)}=(M_{1}^{(i)}[0],\ldots ,M_{1}^{(i)}[15])}$  are defined as follows.

• ${\displaystyle {\textsf {M}}_{0}^{(i)}\leftarrow (M^{(i)}[0],M^{(i)}[1],\ldots ,M^{(i)}[15])}$ 
• ${\displaystyle {\textsf {M}}_{1}^{(i)}\leftarrow (M^{(i)}[16],M^{(i)}[17],\ldots ,M^{(i)}[31])}$ 

The next sub-messages ${\displaystyle \{{\textsf {M}}_{j}^{(i)}=(M_{j}^{(i)}[0],\ldots ,M_{j}^{(i)}[15])\}_{j=2}^{N_{s}}}$  are generated as follows.

• ${\displaystyle {\textsf {M}}_{j}^{(i)}[l]\leftarrow {\textsf {M}}_{j-1}^{(i)}[l]\boxplus {\textsf {M}}_{j-2}^{(i)}[\tau (l)]}$  ${\displaystyle (0\leq l\leq 15,\ 2\leq j\leq N_{s})}$ 

Here ${\displaystyle \tau }$  is the permutation over ${\displaystyle \mathbb {Z} _{16}}$  defined as follows.

The permutation ${\displaystyle \tau :\mathbb {Z} _{16}\rightarrow \mathbb {Z} _{16}}$ 

${\displaystyle l}$	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
${\displaystyle \tau (l)}$	3	2	0	1	7	4	5	6	11	10	8	9	15	12	13	14

Message Addition Function MsgAdd

For two 16-word arrays ${\displaystyle {\textsf {X}}=(X[0],\ldots ,X[15])}$  and ${\displaystyle {\textsf {Y}}=(Y[0],\ldots ,Y[15])}$ , the message addition function ${\displaystyle {\textrm {MsgAdd}}:{\mathcal {W}}^{16}\times {\mathcal {W}}^{16}\rightarrow {\mathcal {W}}^{16}}$  is defined as follows.

${\displaystyle {\textrm {MsgAdd}}({\textsf {X}},{\textsf {Y}}):=(X[0]\oplus Y[0],\ldots ,X[15]\oplus Y[15])}$ 

Mix Function Mix

The ${\displaystyle j}$ -th mix function ${\displaystyle {\textrm {Mix}}_{j}:{\mathcal {W}}^{16}\rightarrow {\mathcal {W}}^{16}}$  updates the 16-word array ${\displaystyle {\textsf {T}}=(T[0],\ldots ,T[15])}$  by mixing every two-word pair; ${\displaystyle T[l]}$  and ${\displaystyle T[l+8]}$  for ${\displaystyle (0\leq l<8)}$ . For ${\displaystyle 0\leq j<N_{s}}$ , the mix function ${\displaystyle {\textrm {Mix}}_{j}}$  proceeds as follows.

${\displaystyle (T[l],T[l+8])\leftarrow {\textrm {Mix}}_{j,l}(T[l],T[l+8])}$  ${\displaystyle (0\leq l<8)}$ 

Here ${\displaystyle {\textrm {Mix}}_{j,l}}$  is a two-word mix function. Let ${\displaystyle X}$  and ${\displaystyle Y}$  be words. The two-word mix function ${\displaystyle {\textrm {Mix}}_{j,l}:{\mathcal {W}}^{2}\rightarrow {\mathcal {W}}^{2}}$  is defined as follows.

function Two-word mix function ${\displaystyle {\textrm {Mix}}_{j,l}}$  input: Words ${\displaystyle X}$  and ${\displaystyle Y}$  output: Words ${\displaystyle X}$  and ${\displaystyle Y}$  procedure ${\displaystyle \qquad }$ ${\displaystyle X\leftarrow X\boxplus Y}$ ;${\displaystyle \qquad X\leftarrow X^{\lll \alpha _{j}}}$ ; ${\displaystyle \qquad }$ ${\displaystyle X\leftarrow X\oplus SC_{j}[l]}$ ; ${\displaystyle \qquad }$ ${\displaystyle Y\leftarrow X\boxplus Y}$ ;${\displaystyle \qquad Y\leftarrow Y^{\lll \beta _{j}}}$ ; ${\displaystyle \qquad }$ ${\displaystyle X\leftarrow X\boxplus Y}$ ;${\displaystyle \qquad Y\leftarrow Y^{\lll \gamma _{l}}}$ ; ${\displaystyle \qquad }$ return ${\displaystyle X}$ , ${\displaystyle Y}$ ;

The two-word mix function ${\displaystyle {\textrm {Mix}}_{j,l}}$  is shown in the following figure.

 

Two-word mix function ${\displaystyle {\textrm {Mix}}_{j,l}(X,Y)}$ 

The bit rotation amounts ${\displaystyle \alpha _{j}}$ , ${\displaystyle \beta _{j}}$ , ${\displaystyle \gamma _{l}}$  used in ${\displaystyle {\textrm {Mix}}_{j,l}}$  are shown in the following table.

Bit rotation amounts ${\displaystyle \alpha _{j}}$ , ${\displaystyle \beta _{j}}$ , and ${\displaystyle \gamma _{l}}$ 

${\displaystyle w}$	${\displaystyle j}$	${\displaystyle \alpha _{j}}$	${\displaystyle \beta _{j}}$	${\displaystyle \gamma _{0}}$	${\displaystyle \gamma _{1}}$	${\displaystyle \gamma _{2}}$	${\displaystyle \gamma _{3}}$	${\displaystyle \gamma _{4}}$	${\displaystyle \gamma _{5}}$	${\displaystyle \gamma _{6}}$	${\displaystyle \gamma _{7}}$
32	even	29	1	0	8	16	24	24	16	8	0
	odd	5	17
64	even	23	59	0	16	32	48	8	24	40	56
	odd	7	3

The ${\displaystyle j}$ -th 8-word array constant ${\displaystyle {\textsf {SC}}_{j}=(SC_{j}[0],\ldots ,SC_{j}[7])}$  used in ${\displaystyle {\textrm {Mix}}_{j,l}}$  for ${\displaystyle 0\leq l<8}$  is defined as follows. The initial 8-word array constant ${\displaystyle {\textsf {SC}}_{0}=(SC_{0}[0],\ldots ,SC_{0}[7])}$  is defined in the following table. For ${\displaystyle 1\leq j<N_{s}}$ , the ${\displaystyle j}$ -th constant ${\displaystyle {\textsf {SC}}_{j}=(SC_{j}[0],\ldots ,SC_{j}[7])}$  is generated by ${\displaystyle SC_{j}[l]\leftarrow SC_{j-1}[l]\boxplus SC_{j-1}[l]^{\lll 8}}$  for ${\displaystyle 0\leq l<8}$ .

Initial 8-word array constant ${\displaystyle {\textsf {SC}}_{0}}$ 

	${\displaystyle w=32}$	${\displaystyle w=64}$
${\displaystyle SC_{0}[0]}$	917caf90	97884283c938982a
${\displaystyle SC_{0}[1]}$	6c1b10a2	ba1fca93533e2355
${\displaystyle SC_{0}[2]}$	6f352943	c519a2e87aeb1c03
${\displaystyle SC_{0}[3]}$	cf778243	9a0fc95462af17b1
${\displaystyle SC_{0}[4]}$	2ceb7472	fc3dda8ab019a82b
${\displaystyle SC_{0}[5]}$	29e96ff2	02825d079a895407
${\displaystyle SC_{0}[6]}$	8a9ba428	79f2d0a7ee06a6f7
${\displaystyle SC_{0}[7]}$	2eeb2642	d76d15eed9fdf5fe

Word-Permutation Function WordPerm

Let ${\displaystyle {\textsf {X}}=(X[0],\ldots ,X[15])}$  be a 16-word array. The word-permutation function ${\displaystyle {\textrm {WordPerm}}:{\mathcal {W}}^{16}\rightarrow {\mathcal {W}}^{16}}$  is defined as follows.

${\displaystyle {\textrm {WordPerm}}({\textsf {X}})=(X[\sigma (0)],\ldots ,X[\sigma (15)])}$ 

Here ${\displaystyle \sigma }$  is the permutation over ${\displaystyle \mathbb {Z} _{16}}$  defined by the following table.

The permutation ${\displaystyle \sigma :\mathbb {Z} _{16}\rightarrow \mathbb {Z} _{16}}$ 

${\displaystyle l}$	0	1	2	3	4	5	6	7	8	9	10	11	12	13	14	15
${\displaystyle \sigma (l)}$	6	4	5	7	12	15	14	13	2	0	1	3	8	11	10	9

Finalization

The finalization function ${\displaystyle {\textrm {FIN}}_{n}:{\mathcal {W}}^{16}\rightarrow \{0,1\}^{n}}$  returns ${\displaystyle n}$ -bit hash value ${\displaystyle h}$  from the final chaining variable ${\displaystyle {\textsf {CV}}^{(t)}=(CV^{(t)}[0],\ldots ,CV^{(t)}[15])}$ . When ${\displaystyle {\textsf {H}}=(H[0],\ldots ,H[7])}$  is an 8-word variable and ${\displaystyle {\textsf {h}}_{\textsf {b}}=(h_{b}[0],\ldots ,h_{b}[w-1])}$  is a ${\displaystyle w}$ -byte variable, the finalization function ${\displaystyle {\textrm {FIN}}_{n}}$  performs the following procedure.

• ${\displaystyle H[l]\leftarrow CV^{(t)}[l]\oplus CV^{(t)}[l+8]}$  ${\displaystyle (0\leq l\leq 7)}$ 
• ${\displaystyle h_{b}[s]\leftarrow H[\lfloor 8s/w\rfloor ]_{[7:0]}^{\ggg (8s\mod w)}}$  ${\displaystyle (0\leq s\leq (w-1))}$ 
• ${\displaystyle h\leftarrow (h_{b}[0]\|\ldots \|h_{b}[w-1])_{[0:n-1]}}$ 

Here, ${\displaystyle X_{[i:j]}}$  denotes ${\displaystyle x_{i}\|x_{i-1}\|\ldots \|x_{j}}$ , the sub-bit string of a word ${\displaystyle X}$  for ${\displaystyle i\geq j}$ . And ${\displaystyle x_{[i:j]}}$  denotes ${\displaystyle x_{i}\|x_{i+1}\|\ldots \|x_{j}}$ , the sub-bit string of a ${\displaystyle l}$ -bit string ${\displaystyle x=x_{0}\|x_{1}\|\ldots \|x_{l-1}}$  for ${\displaystyle i\leq j}$ .

Security

LSH is secure against known attacks on hash functions up to now. LSH is collision-resistant for ${\displaystyle q<2^{n/2}}$  and preimage-resistant and second-preimage-resistant for ${\displaystyle q<2^{n}}$  in the ideal cipher model, where ${\displaystyle q}$  is a number of queries for LSH structure.^[1] LSH-256 is secure against all the existing hash function attacks when the number of steps is 13 or more, while LSH-512 is secure if the number of steps is 14 or more. Note that the steps which work as security margin are 50% of the compression function.^[1]

Performance

LSH outperforms SHA-2/3 on various software platforms. The following table shows the speed performance of 1MB message hashing of LSH on several platforms.

1MB message hashing speed of LSH (cycles/byte)^[1]

Platform	P1^[a]	P2^[b]	P3^[c]	P4^[d]	P5^[e]	P6^[f]	P7^[g]	P8^[h]
LSH-256-${\displaystyle n}$	3.60	3.86	5.26	3.89	11.17	15.03	15.28	14.84
LSH-512-${\displaystyle n}$	2.39	5.04	7.76	5.52	8.94	18.76	19.00	18.10

1. Intel Core i7-4770K @ 3.5GHz (Haswell), Ubuntu 12.04 64-bit, GCC 4.8.1 with “-m64 -mavx2 -O3”
2. Intel Core i7-2600K @ 3.40GHz (Sandy Bridge), Ubuntu 12.04 64-bit, GCC 4.8.1 with “-m64 -msse4 -O3”
3. Intel Core 2 Quad Q9550 @ 2.83GHz (Yorkfield), Windows 7 32-bit, Visual studio 2012
4. AMD FX-8350 @ 4GHz (Piledriver), Ubuntu 12.04 64-bit, GCC 4.8.1 with “-m64 -mxop -O3”
5. Samsung Exynos 5250 ARM Cortex-A15 @ 1.7GHz dual core (Huins ACHRO 5250), Android 4.1.1
6. Qualcomm Snapdragon 800 Krait 400 @ 2.26GHz quad core (LG G2), Android 4.4.2
7. Qualcomm Snapdragon 800 Krait 400 @ 2.3GHz quad core (Samsung Galaxy S4), Android 4.2.2
8. Qualcomm Snapdragon 400 Krait 300 @ 1.7GHz dual core (Samsung Galaxy S4 mini), Android 4.2.2

The following table is the comparison at the platform based on Haswell, LSH is measured on Intel Core i7-4770k @ 3.5 GHz quad core platform, and others are measured on Intel Core i5-4570S @ 2.9 GHz quad core platform.

Speed benchmark of LSH, SHA-2 and the SHA-3 finalists at the platform based on Haswell CPU (cycles/byte)^[1]

Algorithm	Message size in bytes
	long	4,096	1,536	576	64	8
LSH-256-256	3.60	3.71	3.90	4.08	8.19	65.37
Skein-512-256	5.01	5.58	5.86	6.49	13.12	104.50
Blake-256	6.61	7.63	7.87	9.05	16.58	72.50
Grøstl-256	9.48	10.68	12.18	13.71	37.94	227.50
Keccak-256	10.56	10.52	9.90	11.99	23.38	187.50
SHA-256	10.82	11.91	12.26	13.51	24.88	106.62
JH-256	14.70	15.50	15.94	17.06	31.94	257.00
LSH-512-512	2.39	2.54	2.79	3.31	10.81	85.62
Skein-512-512	4.67	5.51	5.80	6.44	13.59	108.25
Blake-512	4.96	6.17	6.82	7.38	14.81	116.50
SHA-512	7.65	8.24	8.69	9.03	17.22	138.25
Grøstl-512	12.78	15.44	17.30	17.99	51.72	417.38
JH-512	14.25	15.66	16.14	17.34	32.69	261.00
Keccak-512	16.36	17.86	18.46	20.35	21.56	171.88

The following table is measured on Samsung Exynos 5250 ARM Cortex-A15 @ 1.7 GHz dual core platform.

Speed benchmark of LSH, SHA-2 and the SHA-3 finalists at the platform based on Exynos 5250 ARM Cortex-A15 CPU (cycles/byte)^[1]

Algorithm	Message size in bytes
	long	4,096	1,536	576	64	8
LSH-256-256	11.17	11.53	12.16	12.63	22.42	192.68
Skein-512-256	15.64	16.72	18.33	22.68	75.75	609.25
Blake-256	17.94	19.11	20.88	25.44	83.94	542.38
SHA-256	19.91	21.14	23.03	28.13	90.89	578.50
JH-256	34.66	36.06	38.10	43.51	113.92	924.12
Keccak-256	36.03	38.01	40.54	48.13	125.00	1000.62
Grøstl-256	40.70	42.76	46.03	54.94	167.52	1020.62
LSH-512-512	8.94	9.56	10.55	12.28	38.82	307.98
Blake-512	13.46	14.82	16.88	20.98	77.53	623.62
Skein-512-512	15.61	16.73	18.35	22.56	75.59	612.88
JH-512	34.88	36.26	38.36	44.01	116.41	939.38
SHA-512	44.13	46.41	49.97	54.55	135.59	1088.38
Keccak-512	63.31	64.59	67.85	77.21	121.28	968.00
Grøstl-512	131.35	138.49	150.15	166.54	446.53	3518.00

Test vectors

Test vectors for LSH for each digest length are as follows. All values are expressed in hexadecimal form.

LSH-256-224("abc") = F7 C5 3B A4 03 4E 70 8E 74 FB A4 2E 55 99 7C A5 12 6B B7 62 36 88 F8 53 42 F7 37 32

LSH-256-256("abc") = 5F BF 36 5D AE A5 44 6A 70 53 C5 2B 57 40 4D 77 A0 7A 5F 48 A1 F7 C1 96 3A 08 98 BA 1B 71 47 41

LSH-512-224("abc") = D1 68 32 34 51 3E C5 69 83 94 57 1E AD 12 8A 8C D5 37 3E 97 66 1B A2 0D CF 89 E4 89

LSH-512-256("abc") = CD 89 23 10 53 26 02 33 2B 61 3F 1E C1 1A 69 62 FC A6 1E A0 9E CF FC D4 BC F7 58 58 D8 02 ED EC

LSH-512-384("abc") = 5F 34 4E FA A0 E4 3C CD 2E 5E 19 4D 60 39 79 4B 4F B4 31 F1 0F B4 B6 5F D4 5E 9D A4 EC DE 0F 27 B6 6E 8D BD FA 47 25 2E 0D 0B 74 1B FD 91 F9 FE

LSH-512-512("abc") = A3 D9 3C FE 60 DC 1A AC DD 3B D4 BE F0 A6 98 53 81 A3 96 C7 D4 9D 9F D1 77 79 56 97 C3 53 52 08 B5 C5 72 24 BE F2 10 84 D4 20 83 E9 5A 4B D8 EB 33 E8 69 81 2B 65 03 1C 42 88 19 A1 E7 CE 59 6D

Implementations

LSH is free for any use public or private, commercial or non-commercial. The source code for distribution of LSH implemented in C, Java, and Python can be downloaded from KISA's cryptography use activation webpage.^[2]

KCMVP

LSH is one of the cryptographic algorithms approved by the Korean Cryptographic Module Validation Program (KCMVP).^[3]

Standardization

LSH is included in the following standard.

• KS X 3262, Hash function LSH (in Korean)^[4]

References

1. Kim, Dong-Chan; Hong, Deukjo; Lee, Jung-Keun; Kim, Woo-Hwan; Kwon, Daesung (2015). LSH: A New Fast Secure Hash Function Family. Springer International Publishing. pp. 286–313. doi:10.1007/978-3-319-15943-0_18. ISBN 978-3-319-15943-0.
2. "KISA 암호이용활성화 - 암호알고리즘 소스코드". seed.kisa.or.kr.
3. "KISA 암호이용활성화 - 개요". seed.kisa.or.kr.
4. "Korean Standards & Certifications (in Korean)".