Mathematical Methods of Cryptography
On characteristics of a three-stage key generator with an alternating step modified with key generator “stop-forward”
V. M. Fomichevabcd,
D. M. Kolesovaa a Financial University under the Government of the Russian Federation, Moscow
b National Engineering Physics Institute "MEPhI", Moscow
c Federal Research Center "Computer Science and Control" of Russian Academy of Sciences, Moscow
d "Security Code", Moscow
Abstract:
The generator
$G$ named in the title of the paper consists of five binary linear feedback shift registers (LFSRs) of maximal periods divided into three cascades. The first cascade is a filter generator
$X$ based on a LFSR of a length
$n$. Each of the second and third cascades consists of two LFSRs
$Y,Z$ and
$U,V$ of lengths
$m,\mu$ and
$r,\rho$ respectively. The registers
$Y,Z$ are controlled by the output
$x$ of the filter generator
$X$, the registers
$U,V$ – by the sum
$y\oplus z$ of the outputs
$y,z$ of the registers
$Y,Z$ respectively. The control is made in such a way: if a controlling signal is 1, then one of the controlled registers shifts but another does not change its state; otherwise their behaviour is just opposite. The output of the generator
$G$ is the sum
$u\oplus v$ of the outputs of registers
$U,V$. It is shown, that if the numbers
$n,m,\mu,r,\rho$ are relatively prime, then the period
$t$ of the sequence produced by
$G$ equals the product of the (maximal) periods of its registers. In the cyclic group of order
$t$ of the generator
$G$, there is a linear subgroup of order
$(2^r-1)(2^\rho-1)$. Local exponents
$i,(p+1)-\exp\Gamma$ of the mixing digraph
$\Gamma$ of
$G$ are equal to
$n+2$ if
$i\in\{1,\dots,n\}$, to
$\max(m,\mu)+1$ if
$i\in\{n+1,\dots,n+m+\mu\}$, and to
$\max(r,\rho)$ if
$i\in\{n+m+\mu+1,\dots,p+1\}$ where
$p=n+m+\mu+r+\rho$. Consequently, for
$G$ the length of “free running” is recommended to be at least
$\max\{n+2,\max(m,\mu)+1,\max(r,\rho)\}$.
Keywords:
key generator, linear shift register, length of period, mixing properties, local primitivity of mixing digraph.
UDC:
519.1
DOI:
10.17223/2226308X/10/40