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• For restoration of secret message X in this case it is enough to calculate secret key - common divisor on the basis of elements from 480-th and 90-th lines (it
Message 1 of 1 , Apr 3, 2002
For restoration of secret message X in this case it is enough to
calculate secret key - common divisor on the basis of elements from
480-th and 90-th lines (it means y=X^480mod481 and y=X^90mod91), for
P=481 on a line 480 there are numbers 248,417. From these numbers it
is possible to find (248-1) = 247 and (417-1) =416 which has the
greatest common divisor 13 with 481, that is the first common divisor
of the modulo. Further does not represent difficulties of calculation
of the second common divisor 37 modules that allows to calculate
function Euler (481) =12*36=432 on the basis of which there is a
secret key i2 for restoration of message X with use algorithm of
Euclidean as i2=i1-1 mod 432. Here, i1 an open key with which use
message X is encrypted.

Such numbers have the some properties as Carmichael numbers, but
unique difference of Carmichael numbers consists that Carmichael
numbers are the product of least three distinct primes and necessary
what P-1 divides n-1 for every prime divisor P of n, n={P1, P2,
P3...Pk}. For new kind of numbers it is not necessary.

If P =P1*P2*P3 (1) or P1*P2 (2) is modulo of function Y = Xi mod P ,
value of

function Y = X(P1-1)+(P2-1)+(P3-1) mod P= X(P-1)mod P (1) or

Y = X (P1-1) +(P2-1)mod P = X(P-1)mod P (2) will true.

This best regards
Hislat
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