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A block encryption processing idea taken from linear algebra
Posted:
Jun 18, 2013 3:00 AM


The iterative solution of a system of n linear equations can be 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 socalled singlestep or Seidel 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 nonlinear functions, the r's are pseudorandom numbers and the assignments are performed sequentially (the f's and the r's are (secret) keydependent and different for different rounds, if more than one rounds are used, computation is mod 2**m for block size of m bits):
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 nonlinear and further the r's are pseudorandom. Note also that the effect of blockchaining 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



