Page 163 - ISCI’2017

p the length of symbol in binary representation will be l and the number of possible modes will
1 p
1
be defined as 2 p l 1 . Where degree of alphabet extension can be assessed as

µ 2 = 2 p l 1 2 m l = 2 p l 1 m l − .

During transformation according to module p the length of symbol in binary representation will

p l
be l , and the number of possible modes will be defined as 2 . Degree of alphabet extension during
p
switching to transformation according to p will be:

µ 1 = 2 p l 2 1 p l = 2 p l 1 p l − .

Correspondingly the possibility of guessing an alphabet symbol according to module p is defined
1
as

P = 1 = 2 m l 1 p l − . (24)
1 p µ 2

The possibility of guessing an alphabet symbol according to module p is defined as

P = 1 µ 1 = 2 p l 1 p l − . (25)
p

Thus theorem is proved. The general possibility of guessing an alphabet P symbol according to
G
module p during switching from m -ary source to p-ary will be defined with multiplication of events

P (24) and P (25), i.e.:
1 p p

P = P ⋅ P = 2 m l p l − 1 ⋅ 2 p l 1 p l − = 2 m l p l − . (26)
p
G
p 1
Using (26) the one can define complexity I of one alphabet symbol according to module p
G
during switching from m -ary source to p-ary as

I = 1 P в = 2 p l m l − .
G

Thus while applying of generator scheme without hashing the complexity I KR of key

reconstruction X = K 0 i + is determined with a formula:

( ) ). (27)
1
( ) ln
I KR = I G ⋅ I DL = 2 p l − m l ⋅exp (ε ln p υ ln p ( υ− )
For a case of applying of generator schemes with guessing a field element, discrete logarithm
solution and hashing the complexity I KRH of key reconstruction X = K 0 i + is determined with a

formula:

υ
( ln p
I KRH = I G ⋅ I DL ⋅ I H = 2 p l − m l ⋅ exp ε ( ) ln ln p (1−υ ) n 2 . (28)
( ) ) 2⋅

163