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**posted on**

- Mok-Kong Shen

June 17, 2013, 7:44 am

formulated as follows:

x1 := a11

***x1 + a12***x2 + ... + a1n*xn + b1

x2 := a21

***x1 + a22***x2 + ... + a2n*xn + b2

.....................

xn := an1

***x1 + an2***x2 + ... + ann*xn + bn

where (in the so-called single-step method) the assignments are

performed sequentially. See V. N. Faddeeva, Computational Methods of

Linear Algebra, p.117, Dover Publ., 1959. (Note that many textbooks

of linear algebra present however a different, in fact less general,

formulation.)

Using this as a hint, we propose to do for block encryption processing

of n blocks, x1, x2, ... xn, the follwoing, where the f's are

invertible non-linear functions, the r's are pseudo-random numbers and

the assignments are performed sequentially (the f's and the r's are

(secret) key-dependent and different for different rounds, if more

then one rounds are used):

x1 := f1(x1 + x2 ... + xn + r1)

x2 := f2(x1 + x2 ... + xn + r2)

................

xn := fn(x1 + x2 ... + xn + rn)

Note that we have left out the multiplication with a's, which is

deemed a justifiable simplicity since the f's are non-linear and

further the r's are pseudo-random. Note also that the effect of

block-chaining in the use of the common block ciphers is intrinsically

present in our scheme. A viable variant of the scheme is to employ

^r instead of +r.

M. K. Shen

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