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Datei: maa.vdmsl Sprache: VDM

Original von: VDM^©

\subsection*{B.1 The Specification}

{-- -- \bf 3.2 Technical}
\begin{vdm_al}

values
   Word_Length = 32;
   Maximum_Number_Size = 2 ** Word_Length - 1;
   Maximum_Number_Size_plus_1 = Maximum_Number_Size + 1;
   Maximum_Number_Size_plus_1_div_2 = Maximum_Number_Size_plus_1 div 2;
   Maximum_No_of_Message_blocks = 1000000;

\end{vdm_al}
{\ \\[-0.3cm]}
{-- -- \bf 4.2.2 The main loop}
\begin{vdm_al}

   A = 2 * 2 **24 + 4 * 2 **16 + 8 * 2 **8 + 1;
   B = 0 * 2 **24 + 128 * 2 **16 + 64 * 2 **8 + 33;
   C = 191 * 2 **24 + 239 * 2 **16 + 127 * 2 **8 + 223;
   D = 125 * 2 **24 + 254 * 2 **16 + 251 * 2 **8 + 255;

\end{vdm_al}
\newpage
{\ \\[-0.3cm]}
{-- -- \bf 5 Specification of the mode of operation}
\begin{vdm_al}

   Maximum_No_of_blocks_for_MAC = 1024 div 4;
   Maximum_No_of_blocks_for_MAC_plus_1 = Maximum_No_of_blocks_for_MAC + 1

\end{vdm_al}
{\ \\[-0.3cm]}
{-- -- \bf 3.2 Technical}
\begin{vdm_al}

types
   Number = nat
   inv N == N  < Maximum_Number_Size_plus_1;

   Bit = nat
   inv b == b in set {0,1};

   Message_in_bits = seq of Bit
   inv M ==
    if  len M mod Word_Length  = 0
    then ( len M div Word_Length  <= Maximum_No_of_Message_blocks) and
       ( len M  > 0)
    else  len M div Word_Length + 1  <= Maximum_No_of_Message_blocks;

   Message_in_blocks_plus_empty_Message = seq of Number
   inv M ==  len M  <= Maximum_No_of_Message_blocks;

   Message_in_blocks = Message_in_blocks_plus_empty_Message
   inv M == 1  <=  len M;

\end{vdm_al}
{\ \\[-0.3cm]}
{-- -- \bf 3.2 Technical\\}
{-- -- \bf 4.1.1 General definitions}
\begin{vdm_al}

   Double_Number = seq of Number
   inv d ==  len d  = 2;

   Key = Double_Number;

\end{vdm_al}
{\ \\[-0.3cm]}
{-- -- \bf 4.2.1 The prelude}
\begin{vdm_al}

   Key_Constant :: X0 : Number
       Y0 : Number
       V0 : Number
       W : Number
       S : Number
       T : Number

\end{vdm_al}
\newpage
{\ \\[-0.3cm]}
{-- -- \bf 3.2 Technical}
\begin{vdm_al}

functions
   Pad_out_Message: Message_in_bits -> Message_in_bits
   Pad_out_Message(M) ==
   let No_Extra_bits = Word_Length -  len M mod Word_Length in
    if No_Extra_bits  = Word_Length
    then M
    else M  ^  Get_Application_defined_bits(M,No_Extra_bits);

   Get_Application_defined_bits(M: Message_in_bits, No_bits: nat)
                               Extra : Message_in_bits
   pre No_bits  < Word_Length
   post  len Extra  = No_bits;

   Form_Message_into_blocks: Message_in_bits -> Message_in_blocks
   Form_Message_into_blocks(M) ==
    if  len M  = Word_Length
    then [Form_Number(M)]
    else [Form_Number(Get_head_in_bits(M,Word_Length))] ^
         Form_Message_into_blocks(Get_tail_in_bits(M,Word_Length))
   pre ( len M  >= Word_Length) and ( len M mod Word_Length  = 0);

   Form_Number: Message_in_bits -> Number
   Form_Number(M) ==
    if  len M  = 1 then  hd M
    else  hd M + 2 * Form_Number( tl M)
   pre  len M  <= Word_Length;

\end{vdm_al}
{\ \\[-0.3cm]}
{-- -- \bf 4  The segment algorithm\\}
{-- -- \bf 4.1 Definition of the functions used in the algorithm\\}
{-- -- \bf 4.1.1 General definitions}
\begin{vdm_al}

   CYC: Number -> Number
   CYC(X) ==
   ADD(X,X) + CAR(X,X);

   AND: Number * Number -> Number
   AND(X,Y) ==
    if (X  = 0) or (Y  = 0)
    then 0
    else X mod 2 * Y mod 2 + 2 * AND(X div 2,Y div 2);

   OR: Number * Number -> Number
   OR(X,Y) ==
    if (X  = 0) or (Y  = 0)
    then X + Y
    else max(X mod 2,Y mod 2) + 2 * OR(X div 2,Y div 2);

   max: int * int -> int
   max(X,Y) ==
    if X  >= Y
    then X
    else Y;

   XOR: Number * Number -> Number
   XOR(X,Y) ==
    if (X  = 0) or (Y  = 0)
    then X + Y
    else (X + Y) mod 2 + 2 * XOR(X div 2,Y div 2);

   ADD: Number * Number -> Number
   ADD(X,Y) ==
   (X + Y) mod Maximum_Number_Size_plus_1;

   CAR: Number * Number -> Number
   CAR(X,Y) ==
   (X + Y) div Maximum_Number_Size_plus_1;

\end{vdm_al}
{\ \\[-0.3cm]}
{-- -- \bf 4.1.2 Definition of multiplication functions\\}
{-- -- \bf 4.1.2.1 To calculate MUL1(X,Y)}
\begin{vdm_al}

   MUL1: Number * Number -> Number
   MUL1(X,Y) ==
   let L = (X * Y) mod Maximum_Number_Size_plus_1,
   U = X * Y div Maximum_Number_Size_plus_1 in
   let S = ADD(U,L),
   C = CAR(U,L) in ADD(S,C);


\end{vdm_al}
{\ \\[-0.3cm]}
{-- -- \bf 4.1.2.2 To calculate MUL2(X,Y)}
\begin{vdm_al}

   MUL2: Number * Number -> Number
   MUL2(X,Y) == let L = (X * Y) mod Maximum_Number_Size_plus_1,
   U = X * Y div Maximum_Number_Size_plus_1 in let D = ADD(U,U),
   E = CAR(U,U) in let F = ADD(D,2 * E) in let S = ADD(F,L),
   C = CAR(F,L) in ADD(S,2 * C);

\end{vdm_al}
\newpage
{\ \\[-0.3cm]}
{-- -- \bf 4.1.2.3 To calculate MUL2A(X,Y)}
\begin{vdm_al}

   MUL2A: Number * Number -> Number
   MUL2A(X,Y) ==
   let L = (X * Y) mod Maximum_Number_Size_plus_1,
   U = X * Y div Maximum_Number_Size_plus_1 in
   let D = ADD(U,U) in
   let S = ADD(D,L),
   C = CAR(D,L) in ADD(S,2 * C)
   pre (X div Maximum_Number_Size_plus_1_div_2  = 0) or
       (Y div Maximum_Number_Size_plus_1_div_2  = 0);

\end{vdm_al}
{\ \\[-0.3cm]}
{-- -- \bf 4.1.3 Definitions of the functions BYT[X,Y] and PAT[X,Y]}
\begin{vdm_al}

   BYT: Double_Number -> Double_Number
   BYT(K) ==
   let X =  hd K,
       Y =  hd  tl K in
   let X' = [Byte(X,3),Byte(X,2),Byte(X,1),Byte(X,0)],
       Y' = [Byte(Y,3),Byte(Y,2),Byte(Y,1),Byte(X,0)] in
   let XY = X' ^ Y',
       P = 0 in
   let XY' = Condition_Sequence(XY,P) in
   let X'' = Get_head_in_blocks(XY',4),
       Y'' = Get_tail_in_blocks(XY',4) in
   [Convert_Bytes_to_Number(X'')]  ^  [Convert_Bytes_to_Number(Y'')];

   Byte: Number * nat -> Number
   Byte(N,B) ==
    if B  = 0
    then N mod 2 **8
    else Byte(N div 2 **8,B - 1)
   pre (B  >= 0) and (B  <= 3);

   Condition_Sequence: Message_in_blocks * Number -> Message_in_blocks
   Condition_Sequence(M,P) ==
    if  len M  = 1
    then [Condition_value( hd M,P)]
    else [Condition_value( hd M,P)] ^
         Condition_Sequence( tl M,Changes( hd M,P));

   Condition_value: Number * Number -> Number
   Condition_value(B,P) ==
   let P' = 2 * P in
   let P'' = P' + 1 in
    if B  = 0 then P''
    else  if B  = 2 **8 - 1
        then 2 **8 - 1 - P''
        else B;

   Changes: Number * Number -> Number
   Changes(B,P) ==
   let P' = 2 * P in
   let P'' = P' + 1 in
    if (B  = 0) or (B  = 2 **8 - 1)
    then P''
    else P';

   Convert_Bytes_to_Number: Message_in_blocks -> Number
   Convert_Bytes_to_Number(M) ==
    if  len M  = 1
    then  hd M
    else Convert_Bytes_to_Number( tl M) +  hd M * 2 **(8 * ( len M - 1));

   PAT: Double_Number -> Number
   PAT(D) ==
   let X =  hd D,
       Y =  hd  tl D in
   let X' = [Byte(X,3),Byte(X,2),Byte(Y,1),Byte(Y,0)],
       Y' = [Byte(Y,3),Byte(Y,2),Byte(Y,1),Byte(Y,0)] in
   let XY = X' ^ Y',
       P = 0 in
   Record_Changes(XY,P);

   Record_Changes: Message_in_blocks * Number -> Number
   Record_Changes(M,P) ==
    if  len M  = 1
    then Changes( hd M,P)
    else Record_Changes( tl M,Changes( hd M,P));

\end{vdm_al}
\newpage
{\ \\[-0.3cm]}
{-- -- \bf 4.2 Specification of the algorithm\\}
{-- -- \bf 4.2.1 The prelude}
\begin{vdm_al}

   Prelude: Key -> Key_Constant
   Prelude(K) ==
   let J1K1 = BYT(K) in
   let J1 =  hd J1K1,
       K1 =  hd  tl J1K1,
       P = PAT(K),
       Q = (1 + P) * (1 + P) in
   let J12 = MUL1(J1,J1),
       J22 = MUL2(J1,J1) in
   let J14 = MUL1(J12,J12),
       J24 = MUL2(J22,J22) in
   let J16 = MUL1(J12,J14),
       J26 = MUL2(J22,J24) in
   let J18 = MUL1(J12,J16),
       J28 = MUL2(J22,J26) in
   let H4 = XOR(J14,J28),
       H6 = XOR(J16,J26),
       H8 = XOR(J18,J28) in
   let K12 = MUL1(K1,K1),
       K22 = MUL2(K1,K1) in
   let K14 = MUL1(K12,K12),
       K24 = MUL2(K22,K22) in
   let K15 = MUL1(K1,K14),
       K25 = MUL2(K1,K24) in
   let K17 = MUL1(K12,K15),
       K27 = MUL2(K22,K25) in
   let K19 = MUL1(K12,K17),
       K29 = MUL2(K22,K27) in
   let H' = XOR(K15,K25) in
   let H5 = MUL2(H',Q),
       H7 = XOR(K17,K27),
       H9 = XOR(K19,K29) in
   let X0Y0 = BYT([H4,H5]),
       V0W = BYT([H6,H7]),
       ST = BYT([H8,H9]) in
   mk_Key_Constant( hd X0Y0, hd  tl X0Y0, hd V0W, hd  tl V0W, hd ST, hd  tl ST);

\end{vdm_al}
\newpage
{\ \\[-0.3cm]}
{-- -- \bf 4.2.2 The main loop}
\begin{vdm_al}

   Main_loop: Message_in_blocks_plus_empty_Message * Key_Constant -> Number
   Main_loop(M,KC) ==
   let mk_Key_Constant(X,Y,V,W,S,T) = KC in
    if  len M  = 0
    then XOR(X,Y)
    else let Mi =  hd M in
         let V' = CYC(V) in
         let E = XOR(V',W),
             X' = XOR(X,Mi),
             Y' = XOR(Y,Mi) in
         let F = ADD(E,Y'),
             G = ADD(E,X') in
         let F' = OR(F,A),
             G' = OR(G,B) in
         let F'' = AND(F',C),
             G'' = AND(G',D) in
         let X'' = MUL1(X',F''),
             Y'' = MUL2A(Y',G'') in
   Main_loop( tl M,mk_Key_Constant(X'',Y'',V',W,S,T));


\end{vdm_al}
{\ \\[-0.3cm]}
{-- -- \bf 4.2.3 The coda}
\begin{vdm_al}

   Z: Message_in_blocks * Key -> Number
   Z(M,K) ==
   let KC = Prelude(K) in
   let S = KC.S,
       T = KC.T in
   let M' = M ^ [S] ^ [T] in
   Main_loop(M',KC);

\end{vdm_al}
{\ \\[-0.3cm]}
{-- -- \bf 5 Specification of the mode of operation}
\begin{vdm_al}

   MAC: Message_in_bits * Key -> Number
   MAC(M,K) ==
   let M' = Pad_out_Message(M) in
   let M'' = Form_Message_into_blocks(M') in
    if  len M''  <= Maximum_No_of_blocks_for_MAC
    then Z(M'',K)
    else let M''' =
               [Z(Get_head_in_blocks(M'',Maximum_No_of_blocks_for_MAC),K)]
                 ^  Get_tail_in_blocks(M'',Maximum_No_of_blocks_for_MAC) in
       Z_of_SEG(M''',K,Maximum_No_of_blocks_for_MAC_plus_1);

   Z_of_SEG: Message_in_blocks * Key * nat -> Number
   Z_of_SEG(M,K,No_blocks) ==
    if  len M  <= No_blocks
    then Z(M,K)
    else let M' = [Z(Get_head_in_blocks(M,No_blocks),K)] ^
               Get_tail_in_blocks(M,No_blocks) in
       Z_of_SEG(M',K,No_blocks);

\end{vdm_al}
{\ \\[-0.3cm]}
{-- -- \bf Auxiliary functions\\}
{-- -- (These are not directly derived from the main text of the standard)}
\begin{vdm_al}

   Get_tail_in_bits: Message_in_bits * nat -> Message_in_bits
   Get_tail_in_bits(M,No_bits) ==
    if No_bits  = 0
    then M
    else Get_tail_in_bits( tl M,No_bits - 1)
   pre  len M  >= No_bits;

   Get_head_in_bits: Message_in_bits * nat -> Message_in_bits
   Get_head_in_bits(M,No_bits) ==
    if No_bits  = 0
    then [ hd M]
    else [ hd M]  ^  Get_head_in_bits( tl M,No_bits - 1)
   pre ( len M  >= No_bits) and (No_bits  >= 1);

   Get_tail_in_blocks: Message_in_blocks * nat -> Message_in_blocks
   Get_tail_in_blocks(M,No_blocks) ==
    if No_blocks  = 0
    then M
    else Get_tail_in_blocks( tl M,No_blocks - 1)
   pre  len M  >= No_blocks;

   Get_head_in_blocks: Message_in_blocks * nat -> Message_in_blocks
   Get_head_in_blocks(M,No_blocks) ==
    if No_blocks  = 0
    then [ hd M]
    else [ hd M]  ^  Get_head_in_blocks( tl M,No_blocks - 1)
   pre ( len M  >= No_blocks) and (No_blocks  >= 1)

\end{vdm_al}

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