cat chinesert
Introduction

Four numbers are published:
        a = 7657728029668
        b = 11945445123579
        c = 11032393478962
        d = 11562013642173
Four pairs of two digit numbers are to be encrypted.
The sum w(x,y,z,u) = a * x + b * y + c * z + d * u generates the encrypted value.
w = w(12,34,56,78) =  2017688969469068
This result, transmitted publicly to the originator of the public key code, may
not easily be decomposed into its components. This is an example of a one way
function.
The originator of the code, however has access to a trapdoor which enables the
decryption of w.
If      u = 4978449787536
and     v = 17838619569241
Let     o = w * u % v =  10549129864
now     o % p = 12
        o % q = 34
        o % r = 56
        o % s = 78
Mathematical foundation

Let p, q, and r be three mutually prime numbers.
Let     m = p * q * r
        i * i( m / p % p , p )  where i(x,y) represents the modular inverse of
                                x modulo y
        j = i( m / q % q , q )
        k = i( m / r % r , r )
        a = i * m / p
        b = j * m / q
        c = k * m / r
if      w = a * x + b * y + c * z
then    x = w % p
        y = w % q
        z = w % r


Operational arrangement

In setting up the public key arrangement, values of p, q , and r are chosen and
kept secret.  They are used to calculate a , b, and c.  They will also be used
later to invert the encryption of x , y , and z.
Values for a , b , and c are published. To encode values of x , y , and z the
equation w = a * x + b * y + c * z is used to form a sum of products, w.  The
value of w is transmitted.  Any receiver who knows p , q , and r can recover
the values of x , y , and z  using the relationships x = w % p , y = w % q ,
and z = w % r.  Anyone knowing only  w , a , b , and c can  determine x , y , and
z only by and extended and difficult analysis.

Numerical example
If      p = 97
        q = 101
        r = 103
Then    w = 967479 * x + 629433 * y + 421271 * z
Suppose x = 25
        y = 75
        z = 50
Then    w = 92458000
Checking:
        x = 92458000 % 97 = 25
        y = 92458000 % 101 = 75
        z = 92458000 % 103 = 50

Transcript of bc script

define i(x,y) {
auto a,  z, w
w = y
z = 1
while( x < w) {
                a = ( y / x ) + 1
                w = x
                x = a * x % y
                z = z * a % y
}
return(z)
}
p = 97
q = 101
r = 103
s = 107
m = p * q * r * s
i = i( m / p % p , p )
j = i( m / q % q , q )
k = i( m / r % r , r )
l = i( m / s % s , s )
a = i * m / p
b = j * m / q
c = k * m / r
d = l * m / s
u = 2231244 ^ 2
v = 4223579 ^ 2
t = i(u,v)
e = a * t % v
f = b * t % v
g = c * t % v
n = d * t % v
define w(x,y,z,h) {
auto p , q , r , s , m , i , j , k , l , a , b , c , d , w , e , f , g , n , t , u , v
p = 97
q = 101
r = 103
s = 107
m = p * q * r * s
i = i( m / p % p , p )
j = i( m / q % q , q )
k = i( m / r % r , r )
l = i( m / s % s , s )
a = i * m / p
b = j * m / q
c = k * m / r
d = l * m / s
u = 2231244 ^ 2
v = 4223579 ^ 2
t = i(u,v)
e = a * t % v
f = b * t % v
g = c * t % v
n = d * t % v
w = e * x + f * y + g * z + n * h
return(w)
}

Trace of values produced running bc script

p = 97
q = 101
r = 103
s = 107
a = 21149299
b = 65211257
c = 92248552
d = 37336367
e = 7657728029668
f = 11945445123579
g = 11032393478962
n = 11562013642173
w = w(12,34,56,78) =  2017688969469068
u = 4978449787536
v = 17838619569241
t = 9074132988486
o = w * u % v =  10549129864
o % p = 12
o % q = 34
o % r = 56
o % s = 78
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