    
 <!- --------------------------------------------------------- From:   (kvn@win.wplus.net) -------------------------------------------------------------- -->

                
 __________________________________

 " "



 


   ,

     

   



  1993

 .
 -----------


 1.   ................................

 2.  ................................

 3.      ........

 4.    
      
 .....................................................


 1.  " "
 ---------------------------------

      60-70 
  ,    
   .   -
       
       . -
    .
       1977   
    "  "
 , , " ".
        -
 ,    ()   , 
  ()  ,    -
  .
 "" -      -
   ""     -
  ,      .
     -  
 -    ,   -
     -    -
   .    -
        
  .

 2.   
 --------------------------------

         -
     RSA,   1977  
        -
    : R.Rivest, A.Shamir, L.Adleman. -
         
  .
  -    
 ""    M.   ("-") -
       .,
   " "  .
 ""        
  d       
     n, ..,:

 d
 S =  mod n

  -    [M,S]. 
       d.
       -
   e,     


 e ?
 S = M mod n
 .
  ,    S  -
       d,    e 
 d  ,      n.  
   n   ,   
     150-170  .
  "-"    
  ,  RSA.
        
   .
         El
 Gamal T., A public-key cryptosysems and signature schme based
 on discret logarithms,IEEE Trans.Inform Theory, v.31 N4, pp.
 469-472 (1985).
      -
        4  
     ,  
    1000 .


 3.    
 -----------------------------------------


       -
 .     
   .
        .-
   ..,   1976 .   
    

 x
 f(x) = a (mod p)

  ,  -  ,   . -
  ,    ,    f(x), -
     .
        
 X X
 ,     a (mod p)     .
      .   -
    .      
      .   -  Y   
   .    Y    -
    . X Y
      ,    -  .
 X Y
          
 X Y
    . ,     
       .
  ,        
 Y YX
    ,     .   
 XY
 a ,       
  .
 .

    
 ̻ ̻
  Ŀ X   X 
  |X|->a -------------------------------------> a 
     
  Y   Y Ŀ 
  a <------------------------------------ a <-|Y| 
     
  Ŀ   Ŀ 
  Ŀ Y | YX|   Ŀ X | XY| 
  |X|+a ->|a |   |Y|+a ->|a | 
        
 ̼ ̼

       
        ,
      ,    -
       .     -
   ,    
  .
       
   RSA.
    " " 
     RSA   ,  
        . -
         RSA  
    ,     -
     .  ,  
  RSA      ,  
      ,   -
       .
       -
         .  -
         
    . ,    -
          
 !
      RSA 
    RSA    
    .
  RSA     
   - .   , 
      !
    "-"  -
   1000      RSA.
    () 
 "-" 200/.  -286,  
     DES.
 .

 4.     
 ------------------------------------------------------
      
 -------------------------------------------


    N+1  :   N -
 .   i = 1, 2,..., N.   
    .      
    (,   ..). 
       .  -
      .    
        .


    - .

 I .     .
 II .       () -
     . X
        
 Y
  . XY
 III .        
  . XY
 IV .        
  .
 V .   .


  
 +====================+ +========================+
 II . +-+ +-+ I I I
 I | |->| | I I I
 I +-+ +sign I I I
 I I I I
 III . X I I X II .I
 I X->a ------------------------> a I
 I Y I I Y I
 I a <----------------------- a <-Y I
 IIII . I I III .I
 I +-----+ I I +-----+ I
 I+-+ |+-+ | I I |+-+ | I
 I| |----->|| | |----- ---------------->|| | | I
 I+sign | |+sign| I I |+sign| I
 I | +-----+ I I +-----+ I
 I  XY I I | IV .I
 I   a I I | -  XYI
 I I I |   a I
 I I I | I
 I I I |  I
 I I I +-+ | +-+ I
 I I I | |------>| | I
 I I I +sign +-+ I
 +====================+ +========================+
        -
 .



 .
  ,   

   i     x  
 Xi
   .         -
    ,    . 
         ,
   ,    .

  i 
 +---+ +---+ +---+-+
 Xi | Xi| | Xi| Xi | X1| |
 Xi->a ->|a |---------------------------->|a |--->a -->|a |1|
 | +-sign +-sign | | X2| |
 | | |a |2|
   |: |:|
  |: |:|
 | XN| |
 |a |N|
 +---+-+
 |
 ----->|
 |
 +-------+
 ----------------------|+---+-+|
 ----------------------|| X1| ||
 ----------------------||a |1||
 ----------------------|| X2| ||
 ----------------------||a |2||
 ----------------------||: |:||
 ----------------------||: |:||
 ----------------------|| XN| ||
 ----------------------||a |N||
 ----------------------|+---+-+|
 ----------------------+--sign-+
 |||||||||
  |||||||||
   |||||||||
 |||||||||

       , -
    j,  i    -
 Xj
     j       -
 Xj XiXj
  Xi        .   i
     .      -
 XiXj
      i   
  j .      
 ,        ,
    .     j
    .
  ,    j, -
    ,   . -
          -
 XY
     (.  "   -
 ").  ,    j  .
 .
 ,      
   i,      , 
       ,   
   - .    j   , 
      j.   
        i,
        
 XiXj XiXj
 a .    i     
 XiX1 XiX2 XiXN
 N-1      ,  , ... ,a .
            -
   ,     
 .
     . 
  i      j  
 :
 Ŀ
 | * * * | i | * * * | s | * * * | j |  j |
 
 i - ,   ( i)
 j - ,   ( j)
  j -    j
 S -  ,  i    j  
  j
 * -     .
  ,   S  i    j, 
   ,   j,
   .  " j"    -
 .
 Ŀ
 | * * * | i | * * * | s | * * * | j |  j |
 
 |  i  
 | " j"   -
 |       
 | 
 Ŀ
 | * * * | i | * * * | s | * * * | j |  j | k |
 
 | 
 |
 |  i   
 |    
 | XY
 |   (. " -
 |  -.")
 |
 | XY
  a 
   
  Ŀ 
  | * * * | i | * * * | s | * * * | j |  j | k | sign | 
   
   
 
 |    
 | XY
 |   ,   
 |  .
 |
 | 
 Ŀ
 | * * * | i | * * * | s | * * * | j |  j | k |
 
 | 
 |
 |  i    -
 |    Z  W, -
 |   
 | Z W
 |     .  -
 | ZW
 |      .
 |     
 | ZW
 |  ,    -
 |  j.
 |
 ZW
  a 
   
  Ŀ 
  | * * * | i | * * * | s | * * * | j |  j | k | sign | 
   
   
 
 |
 |  j   -
 | ZW
 |   ,  , -
 |    i,   -
 | ,  
 | XiXj
 |   ,   -
 |     
 |  " j"
 |
 +---------------------+------------------------+
 | * * * | i | * * * | s | * * * | j |  j |
 +----------------------------------------------+


  i      
 Xi,          
  j.     i  j  
  ,      -
    .    -
        -
  i  j.
   " ",   -
 ,        .,  37 -
      ,     .

 .


 " "

      
    " ".

       . 
  -    [ ]  -
         
      ,
     .

  ,      ,   
 ,        
       , 
        .

      ,   
     ,      -
   ,   "   "   
  ,        -
 .

         -
      .


  



 .
 
 " "


 

 1.   () ............. 4

 1.1.  RSA....................................... 5

 1.2.   ................................ 7

 1.3.   " "................... 9

 1.4.    ..............11

 2.  ................................13

 2.1.  DES....................................14

 2.2.   28147-89.........................15

 2.3.   "".....................16

 2.4.   .....................17

 3.   ........................19

 3.1.   - .........................20

 3.2.  .........................21

 3.3.    ""............22

 3.4.  ................................23

 ..............................................24

 ..............................................25

  .
   " "...........................26

  .
     "H "...29

  .
     ...........35

  .
      ..........40

  .
    .....................47



 1.   () .


        
 .  . [ 1 ]  1976 .     
     
         ,
       .

        , 
       
  ,       , 
        (
 ),       -
 ,   ,   -
  ,      -,  
   .

       
  ,    -  
   (, , ...)  
  ,    .

     ,    
       , ..
     .

 ,      -
  "    ",    
 ,       -
  ,     ,   -
    .

  "  "   
         -
 .

   ,      -
       -
 (   "  ")
      -  
         -
       .
 1.1.  RSA.


       
 "  RSA",   1977   -
    ()    -
      : R.Rivest, A.Shamir,
 L.Adleman [ 4 ].

     , ,    
  n,      
  ,  n = P*Q,     
  e, d, 1<e,d<n, ,   n  e,  d  
 ,      n, ..  P  Q.

       ,   
        (, 100 
 200),     .    
     ,     ,
      ,  
  ,       -
 .

 ,       
 n  200      100  -
    100   CRAY,  -
      .

   RSA     
    .

       ,
 100
 , P Q 10 ,    n = P*Q  
   e, d , 1<e,d<n.

  n  e    ,   d 
       .

   ( , , , ...) 
     
 m = h(M) , 1<h(m)<n ,
     "",   h -
 -)      M  
  d  
 d
 s = m mod n,

 ..   m     d   
    n.

  [M, s]    .


    s   M 
 
 e ?
 s mod n = h(M),
 ..   s        -
     n    
        h,    -
    .
    ,   
   s      , -
         d, -
     (" ").   -
 ,      n  e  d 
    ,    n   P 
 Q.       .

      
  RSA     " "  -
      :

 -  RSA    # 4 405 829    -
     (   -
  )      - -
  "RSA Data Security Inc." ();

 -       
  , ,     -
   ( DES),    -
    n, e, d  ,   360  ( 108 -
  ) ,     -
  ;

 -    n   e, d  -
 ,  ,    ( 
  ),     
     ;

 -  RSA       -
    ,   
        
  ,
 , 

 2 = h( )  s -    ,
 1 1 1
 3 = h( )  s -    , 
 2 2 2
 6 = h( ),  s *s -    ,
 3 1 2 3
   ,     .

 ,       
         -
  .


 1.2.   .


         -
    T.ElGamal  1985 . [ 5 ].

     ,    
       -
     ,   
     -   -
 .

   ,      
 P, (,  100  150 ),     
 g, x, 1<g,x<P-1,     
 x
 b = g (mod P),
 
 ..     g  P,   
    b  P (   ),  
   g,  , ..  
  1 000  ,       -
  P (       ).

  ,       RSA,
        -
      .

      -
  .

       -
 154
      g, P,, g P 10 ,

    .

      (!!!) 
  , 1<x<P-1,        
 x
    ,   b = g (mod P).

  b - "     ," 
   .

     ""   

 m = h(M) , 1<m<P-1,

    

 r, 1 < r < P-1,

   u,
 r
 u = g mod P,

         v

 v = (m-ux)/r (mod P-1),

    u     (u , v) 
  .
     [M, u, v]  -
  m = h(M),       b  -
        -
 
 m ? u v
 g mod P = b * u mod P.

    ,   -
       ,  
 (u, v)        
  ,       b.

     ,  -
        
 ,      ,      
   .

     RSA     
 :

 - -,      -
    ,     -
 ,    30% ,   
    2      
  ;

 - -,    P  ,
   ,     P-1    -
 ;

 - -,       -
     RSA     -
     .

   ,    , 
    ""    
         
   1991      -
       (DSS)
   .

  ,     
     ,  -
          
  10    .



 1.3.   " ".


         -
   [ 5 ]     
 /     
    .

      
       -
   ( know-how  " "), -
       -/286 
  1,5       ,  -
     .

        -
          
 300    200 ,      500 
 1 000 .

         
 ","  :
 -      
    ,
 -     ,
 -  ()  ,
 -     
      .

       
   "".

  "  "   
       P ,  
 P-1   2*Q,   Q   , " "
    P ,  P-1   
 160
 Q( Q 2 ),   x   

 1< x < Q.

  " "  ,   
        u ,   -
          Q ( 
    R.Shnorr).


   PC/AT-286 , 16MHz ,  -  .
  P -  512 . +----------------------------------------------------------------------------+ | |   |   |   | | +-----------------------------------------------------------+ | |  |  |  |  |  |  | +----------------------------------------------------------------------------+ |  | 1.0101 | 0.9885 | 0.3223 | 0.2975 | 0.2966 | 0.2899 | +----------------------------------------------------------------------------+ |  | | | | | | | |  | 2.2286 | 2.7520 | 0.8153 | 1.0166 | 0.5065 | 0.7050 | |  | | | | | | | +----------------------------------------------------------------------------+ |  |  | |  | |  | | |   |  | 4.9641 |  | 1.6883 |  | 0.9643 | |  |  | |  | |  | | +----------------------------------------------------------------------------+ |  | 2.2973 |  | 0.7076 |  | 0.2832 |  | |  | |  | |  | |  | +----------------------------------------------------------------------------+ |   | 128 | 192  | 40 | 104  | 27 | 91  | |   | | 128 | | 40 | | 27 | +----------------------------------------------------------------------------+ |  | | | | | | | |  | 0.9013 | 4.5693 | 0.2777 | 2.1599 | 0.2903 | 1.6585 | |  | | | | | | | +----------------------------------------------------------------------------+ |  |  | |  | |  | | |  |  | 4.0532 |  | 1.4292 |  | 1.6388 | |   |  | |  | |  | | +----------------------------------------------------------------------------+

  " "    
    ,     -
    .

  ,     -
      ,  
         -
 .       .

      " "
    .


 1.4.    .


    " "    -
       
     x   
 x
   x    b = g mod P
   , ..  " -
 "  b   g   P,   -
     .

       -
        P ( -
   )   

 +-------------------------------------------------------------------------+
 |  P |  |  |    |
 | (  ) |  |  |   |
 | |  x |  |  CRAY (10**9 /c) |
 | | | (  ) | |
 +-------------------------------------------------------------------------+
 | | 12 | 6 | |
 | 128 | 2*10 | 7*10 |   |
 | | 16 | 8 | |
 | 200 | 10 | 10 |   |
 | | 17 | 9 | |
 | 256 | 9*10 | 10 |    |
 | | 24 | 12 | |
 | 512 | 4*10 | 3*10 |  |
 | | 34 | 17 | | |
 | 1024 | 10 | 10 | | |
 | | 41 | 20 | |  100  |
 | 1500 | 10 | 8*10 |   |
 | | 47 | 24 | |  |
 | 2000 | 7*10 | 10 | | |
 | | 50 | 25 | | |
 | 2200 | 10 | 10 |  |
 | | | | |
 +-------------------------------------------------------------------------+

        -
         -
       " " , 
       ,   -
        -
         10
  ( CRYPTO`82-92, EUROCRYPT`84-92,  ..).

   ,  " " 
       
 P  200 - 250   ,     -
    ,     -
      30 - 50 , , , -
     P  500 .  
   1000 (  1500)     
 ""     
  .


      ,
        -
   ,     -   , 
         ,
   x .

      ,     "-
 "  ""  .

        
        -
  ,       -22
         
  ,    .



 2.  .


     ,   -
  "" (1976 ),  ,
        
 .

 ,     70-    
      ( , -
   ..)      .
        , 
      .

       
    ,    -
    " ". ,   
         -
  ,        ,
 -      .
 ,      
 ,      
 ,   ,    .

        -
       
         ,  
 1973         -
      (DES).

          
 ,      .

 "   " ,   1989    
         ,
     28147-89.
 ,      "", ..
   "  ".    -
   , ,     , 
        .

    1989       -
     FEAL-4  FEAL-8.

         -
       .



 2.1.  DES.


    DES   
   64 .    .  -
       -
        
  .

        
 DES, ..         
  (., , [ 7 ] )    
     [ 2,3 ].

  ,  DES   1973   -
         -
        . 
  ,       DES -
     56 ,      
    "".

     " "  -
 ,     DES.

 , DES       -
  :

 -         
   (      -
   DES     PC-AT/286 -
     50 /.);

 -     DES  
 " " (      -
 ,       
   64 ),      
       -
  ).



 2.2.   28147-89.


       ( 28147-89) 
     1989 .    
   ,     
     (DES).   28147-89 
      64 . 
       
       -
    2,      32 .
        
   .      
    DES       -
 .

     28147-89  
     , ..    -
    .     
        
   ,      
 ,     
 , ..     -
          
  .

  ,       -
       -
 ,     ,  
    ,   .

 ,  " "    
     , 
      28147-89 
 -/286  ,      -
       
        -
  DES,       .

        -
   ,      
   ,      
     .


 2.3.   "".


     , 
  " "   ,  
   DES,      
 28147-89.

       ,    -
       -
     2     -
     ,   -
 /    -
      2   .

       
 :

 -     (    
       
 :     );

 -       -
   (       
    :   -
  ""  /  
 PC/AT-286       - 200-220
 /);

 -      
  :      
 .

         (  -
       ) -
 :       
    .

  ,      
    " "   -
    3.

    / -
  ""    .


 2.4.   .


   , ..    -
        
     (" ")  
           -
     -   -
 .

   ,    (DES)   -
  20          
         
 18
    10 ,       

  .

   ,    
 DES     CRAY,   
 9
    (10 ) ,   30 
 7
   (  -  3*10 ).

      -
  ,      
 12
    ,  CRAY, ..  10 ./.,
  ,   ""  DES 
     .    
 ,   1973      
   -,     
    DES    .

         (
 28147-89)    1989      
 1990-1991   -  
 "Kami"  ""      -
 .  1992        "-
     ",  
  ""     "  ",   
        ,   DES.

 ,      
 28147-89      ,  ,  
    ()     
  ,        -
 -      .

   ,    -
         
     .


     " "   
    28147-89     -
  ,      ,  
 50
    10 ,      
  .

     "", 
     DES   28147,  -
 70
    ,      10  
    .

       
 50
   DES ,   10 ,   
 70
 - 10 ,       
  ,    .

         -
    ,     
        -
  .



 3.   .


       ,   
       
      (
 )       -
   .

       
      ,  -
       
 .

  ,     -
      ,  -
        
   :     
  (, ,  )  -
 ,       
  .

  ,      
 ,          
      -
 .    , , ... 
  ,      .

 ,  1976      .  .,
       1,    
      .

    ,    ,   -
       ,
     ,    
    .

         -
 (  ),       -
         -
 .      "".

      (  ),
     , , " RSA".

   -   
       
  (   ).





 3.1.   - .


        , 
         -
           
 / ,   
 .

        -
     ( )
   .

      -  -
   ,     .

       
       , (!)
      ,  
  ,      .

        
           
 .

  ,      -
   -      
 /     .

       - 
   ().

      -
          
   - .

     ,       ,
        
  P.

  ,   
  1976 ,   ,   ,  ,
       ,  -
     ,  -
       1.4.

      -
    " "    .



 3.2.  .


   ,      -
   -     
         , 
 ,    .

       
     ,  
 :       
    (    
  -  ),  ,   
        
  , ..   , 
        .

       -
         2.
  ,      -
     .

       
     :  
   ,   , ... .

     -    -
     , -  -
    .


 3.3.    "".


    " "  -
    -   
      ,  
  .

      ,
        ,  -
         P,   -
    -   
        -
  ,       1.3.

        -
     -    ,  -
     .

      P    -
     ,  
   .

       P , 
    (DSS),   -
     P ,      
 16
     2   ,  
        .

       
  ,   P   ,   -
    "" .  
        
  P ,       , 
      .    ,  
     P     -
      ""  P.

        -
  .


 3.4.  .


    , ..    
         -
  ,   ,  
  ""     -
 .

   -  " "  -
        
   1976 .

        
       -
   .   -
       1.4.

        -
      .




 

    ,
    " ".


 1.   ,    ,
      P  510 ,
     ,
 24
  10
 (   1000      100 -
 9
   10 ./. ).

 2.   ( ) -
         
   :
 18
 -    (DES) - 10 ,
 50
 -   28147-89 - 10 ,
 70
 -  "" - 10 .

 3.     
 ( )   

 -   P  510 ,   
  ,

 -   ,    -
   ,
 24
  10 .



 .
 
 " "











 

 1. W. Diffie, M. Hellman, " New directions in cryptography ",
 IEEE Trans. Informat. Theory, vol. IT-22, pp.644-654,
 Nov.1976.

 2. Federal Register , "Encryption algorithm for computer data
 protection", vol.40, no.52, pp.12134-12139, Mar.17, 1975.

 3. "Data encryption standard(DES)", National Bureau of Standards
 (U.S.), Federal information Service, Springfield , VA ,
 Federal Information Processing Standard Publication,46,
 Apr. 1977.

 4. R. Rivest, A. Shamir, L. Adleman, "A method for obtaining di-
 gital signatures and public key cryptosystems", Commun. ACM,
 vol. 21, no. 2, pp. 120-126, Feb. 1978.

 5. T. ElGamal, "A Public Key Cryptosystem and a Signature Scheme
 Based on Discrete Logarithms", IEEE Trans. Informat. Theory ,
 vol. IT-31, no.4, pp.469-472, Nov.1985.

 6. Federal Register , "A Proposed Federal Information Processing
 Standard for Digital Signature Standard(DSS), vol.56, no.169,
 Aug.30, 1991.

 7. A. Konheim , " Cryptography. A Primer"
 John Wiley & Sons, New York, 1983.







  *  * 







     . 

       

     
  ()   ......... .


   ,       .............................. ( - ),   ,
     
       ;
       
         -
 ;
     
 ,       
  ;
       -
      
 ,   --
  ;
    , -
  ,  ,    -
  ,
                 :

       I.

          ()   ,             () .

       II.

  ,        () , -       -  () ,     .

       III.

       -  ,        () ,       ............... .
   ,       ............................ ,  -   ,     -    () , -    ,    .
           ............................        .
        -      .................     ,  , -   .

       IV.

  ,    ,     .
      .
        ()     -     ,       .......................... .

       V.

       ... ,  ,    , -  .
   ..........................              .




  ___________________________________________________

 __________________________ (_____________)
 ..


  ___________________________________________________

 __________________________ (_____________)
 ..
 . . . . . . . . . . . . . . . . . . . . . . . . .


    

 

    
 ,   ,  
 ,     -
      -
     
  .......................... .


 1.             ,     ,     ,        (          ),     .
        ,         ,  -       -      ,          .
 2.                -  (        ),  -           .
  .58       /1/   ,   ,       ,       .  ,        ,         - ,             .
 ____________________

 /1/     31  1991 . (-         . 1991. N 26. .733).     -      14  1992 . " -       -  " (          . 1992. N 30. .1800)         3  1992 . ( - ).

 ,    ,          .  -          ,  ,    - , , ,   ..,  ,    (.2 .58 ).       -          .
      - ,     --  ,   ,         ,  -    .   ,           ,  .           -  ,  ,   ,  -       .      1974       , -        -   ,    - - ./1/  .2  -       ,      ,    ,         ,          - .
  ,   ,                --  ,      -  ,     . ____________________

 /1/       29  1979 . N -1-4 "    -     ,    - .// -          . . 1989. .47-49.

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   ,    .7      .27          ,   ,             ,   
 _______________________________

 1.         . 1991, N30, .1013;           . 1992, N34, . 1965. (    ).
 2.     5  1992.//          .1991, N16, .836. ( - )      . ,  -            ,   (.157).
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 _________________________________

 1.     -      .1992, N30, .1790. ( -     ).

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