MASH-1

MASH-1 (Modular Arithmetic Secure Hash) is a hash function based on modular arithmetic.

History

Despite many proposals, few hash functions based on modular arithmetic have withstood attack, and most that have tend to be relatively inefficient. MASH-1 evolved from a long line of related proposals successively broken and repaired.

Standard

Committee Draft ISO/IEC 10118-4 (Nov 95)

Description

MASH-1 involves use of an RSA-like modulus ${\displaystyle N}$, whose bitlength affects the security. ${\displaystyle N}$ is a product of two prime numbers and should be difficult to factor, and for ${\displaystyle N}$ of unknown factorization, the security is based in part on the difficulty of extracting modular roots.

Let ${\displaystyle L}$ be the length of a message block in bit. ${\displaystyle N}$ is chosen to have a binary representation a few bits longer than ${\displaystyle L}$, typically ${\displaystyle L<|N|\leq L+16}$.

The message is padded by appending the message length and is separated into blocks ${\displaystyle D_{1},\cdots ,D_{q}}$ of length ${\displaystyle L/2}$. From each of these blocks ${\displaystyle D_{i}}$, an enlarged block ${\displaystyle B_{i}}$ of length ${\displaystyle L}$ is created by placing four bits from ${\displaystyle D_{i}}$ in the lower half of each byte and four bits of value 1 in the higher half. These blocks are processed iteratively by a compression function:

${\displaystyle H_{0}=IV}$

${\displaystyle H_{i}=f(B_{i},H_{i-1})=((((B_{i}\oplus H_{i-1})\vee E)^{e}{\bmod {N}}){\bmod {2}}^{L})\oplus H_{i-1};\quad i=1,\cdots ,q}$

Where ${\displaystyle E=15\cdot 2^{L-4}}$ and ${\displaystyle e=2}$. ${\displaystyle \vee }$ denotes the bitwise OR and ${\displaystyle \oplus }$ the bitwise XOR.

From ${\displaystyle H_{q}}$ are now calculated more data blocks ${\displaystyle D_{q+1},\cdots ,D_{q+8}}$ by linear operations (where ${\displaystyle \|}$ denotes concatenation):

${\displaystyle H_{q}=Y_{1}\,\|\,Y_{3}\,\|\,Y_{0}\,\|\,Y_{2};\quad |Y_{i}|=L/4}$

${\displaystyle Y_{i}=Y_{i-1}\oplus Y_{i-4};\quad i=4,\cdots ,15}$

${\displaystyle D_{q+i}=Y_{2i-2}\,\|\,Y_{2i-1};\quad i=1,\cdots ,8}$

These data blocks are now enlarged to ${\displaystyle B_{q+1},\cdots ,B_{q+8}}$ like above, and with these the compression process continues with eight more steps:

${\displaystyle H_{i}=f(B_{i},H_{i-1});\quad i=q+1,\cdots ,q+8}$

Finally the hash value is ${\displaystyle H_{q+8}{\bmod {p}}}$, where ${\displaystyle p}$ is a prime number with ${\displaystyle 7\cdot 2^{L/2-3}<p<2^{L/2}}$.[1]

MASH-2

There is a newer version of the algorithm called MASH-2 with a different exponent. The original ${\displaystyle e=2}$ is replaced by ${\displaystyle e=2^{8}+1}$. This is the only difference between these versions.

References

1. Smashing MASH-1, Vladimir Antipkin

• A. Menezes, P. van Oorschot, S. Vanstone, Handbook of Applied Cryptography, ISBN 0-8493-8523-7