:: Algebraic group on Fixed-length bit integer and its adaptation to {IDEA} Cryptography
:: by Yasushi Fuwa and Yoshinori Fujisawa
::
:: Received September 7, 1998
:: Copyright (c) 1998-2021 Association of Mizar Users

Lm1: for i, j being Nat st i <> 0 & i < j & j is prime holds
i,j are_coprime

proof end;

Lm2: for i, j, k being Nat st j is prime & i < j & k < j & i <> 0 holds
ex a being Nat st (a * i) mod j = k

proof end;

theorem Th1: :: IDEA_1:1

for i, j, k being Nat st j is prime & i < j & k < j & i <> 0 holds
ex a being Nat st
( (a * i) mod j = k & a < j )

proof end;

theorem Th2: :: IDEA_1:2

for n, k1, k2 being Nat st n <> 0 & k1 mod n = k2 mod n & k1 <= k2 holds
ex t being Nat st k2 - k1 = n * t

proof end;

theorem Th3: :: IDEA_1:3

for a, b being Nat holds a - b <= a

proof end;

theorem Th4: :: IDEA_1:4

for b1, b2, c being Nat st b2 <= c holds
b2 - b1 <= c

proof end;

theorem Th5: :: IDEA_1:5

for a, b, c being Nat st 0 < a & 0 < b & a < c & b < c & c is prime holds
(a * b) mod c <> 0

proof end;

:: In this section,we construct an algebraic group on
:: Fixed-length bit Integer.
:: IDEA's Basic operators are 'xor', ADD_MOD and MUL_MOD.
:: 'xor' is 16-bitwise XOR. ADD_MOD is addition mod 2^{16}.
:: MUL_MOD is multiplication mod 2^(16)+1. And, we define
:: two functions NEG_MOD and INV_MOD that give inverse
:: element of ADD_MOD and MUL_MOD respectively.

definition

let n be Nat;

func ZERO n -> Tuple of n, BOOLEAN equals :: IDEA_1:def 1
n |-> FALSE;

correctness
coherence
n |-> FALSE is Tuple of n, BOOLEAN ;
;

end;

:: deftheorem defines ZERO IDEA_1:def 1 :
for n being Nat holds ZERO n = n |-> FALSE;

definition

let n be Nat;
let x, y be Tuple of n, BOOLEAN ;

func x 'xor' y -> Tuple of n, BOOLEAN means :Def2: :: IDEA_1:def 2
for i being Nat st i in Seg n holds
it /. i = (x /. i) 'xor' (y /. i);

existence
ex b₁ being Tuple of n, BOOLEAN st
for i being Nat st i in Seg n holds
b₁ /. i = (x /. i) 'xor' (y /. i)

proof end;

uniqueness
for b₁, b₂ being Tuple of n, BOOLEAN st ( for i being Nat st i in Seg n holds
b₁ /. i = (x /. i) 'xor' (y /. i) ) & ( for i being Nat st i in Seg n holds
b₂ /. i = (x /. i) 'xor' (y /. i) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def2 defines 'xor' IDEA_1:def 2 :
for n being Nat
for x, y, b₄ being Tuple of n, BOOLEAN holds
( b₄ = x 'xor' y iff for i being Nat st i in Seg n holds
b₄ /. i = (x /. i) 'xor' (y /. i) );

theorem Th6: :: IDEA_1:6

for n being Nat
for x being Tuple of n, BOOLEAN holds x 'xor' x = ZERO n

proof end;

theorem Th7: :: IDEA_1:7

for n being Nat
for x, y being Tuple of n, BOOLEAN holds x 'xor' y = y 'xor' x

proof end;

definition

let n be Nat;
let x, y be Tuple of n, BOOLEAN ;
:: original: 'xor'
redefine func x 'xor' y -> Tuple of n, BOOLEAN ;
commutativity
for x, y being Tuple of n, BOOLEAN holds x 'xor' y = y 'xor' x by Th7;

end;

theorem Th8: :: IDEA_1:8

for n being Nat
for x being Tuple of n, BOOLEAN holds (ZERO n) 'xor' x = x

proof end;

theorem Th9: :: IDEA_1:9

for n being Nat
for x, y, z being Tuple of n, BOOLEAN holds (x 'xor' y) 'xor' z = x 'xor' (y 'xor' z)

proof end;

definition

let n, i be Nat;

pred i is_expressible_by n means :: IDEA_1:def 3
i < 2 to_power n;

end;

:: deftheorem defines is_expressible_by IDEA_1:def 3 :
for n, i being Nat holds
( i is_expressible_by n iff i < 2 to_power n );

theorem :: IDEA_1:10

for n being Nat holds n -BinarySequence 0 = ZERO n

proof end;

definition

let n be Nat;
let i, j be Nat;

func ADD_MOD (i,j,n) -> Nat equals :: IDEA_1:def 4
(i + j) mod (2 to_power n);

coherence
(i + j) mod (2 to_power n) is Nat ;

end;

:: deftheorem defines ADD_MOD IDEA_1:def 4 :
for n, i, j being Nat holds ADD_MOD (i,j,n) = (i + j) mod (2 to_power n);

definition

let n, i be Nat;
assume A1: i is_expressible_by n ;

func NEG_N (i,n) -> Nat equals :Def5: :: IDEA_1:def 5
(2 to_power n) - i;

coherence
(2 to_power n) - i is Nat

proof end;

end;

:: deftheorem Def5 defines NEG_N IDEA_1:def 5 :
for n, i being Nat st i is_expressible_by n holds
NEG_N (i,n) = (2 to_power n) - i;

definition

let n, i be Nat;

func NEG_MOD (i,n) -> Nat equals :: IDEA_1:def 6
(NEG_N (i,n)) mod (2 to_power n);

coherence
(NEG_N (i,n)) mod (2 to_power n) is Nat ;

end;

:: deftheorem defines NEG_MOD IDEA_1:def 6 :
for n, i being Nat holds NEG_MOD (i,n) = (NEG_N (i,n)) mod (2 to_power n);

theorem Th11: :: IDEA_1:11

for i, n being Nat st i is_expressible_by n holds
ADD_MOD (i,(NEG_MOD (i,n)),n) = 0

proof end;

theorem :: IDEA_1:12

for i, j, n being Nat holds ADD_MOD (i,j,n) = ADD_MOD (j,i,n) ;

theorem Th13: :: IDEA_1:13

for i, n being Nat st i is_expressible_by n holds
ADD_MOD (0,i,n) = i by NAT_D:24;

theorem Th14: :: IDEA_1:14

for i, j, k, n being Nat holds ADD_MOD ((ADD_MOD (i,j,n)),k,n) = ADD_MOD (i,(ADD_MOD (j,k,n)),n)

proof end;

theorem Th15: :: IDEA_1:15

for i, j, n being Nat holds ADD_MOD (i,j,n) is_expressible_by n by NAT_D:1, POWER:34;

theorem :: IDEA_1:16

for i, n being Nat holds NEG_MOD (i,n) is_expressible_by n by NAT_D:1, POWER:34;

definition

let n, i be Nat;

func ChangeVal_1 (i,n) -> Nat equals :Def7: :: IDEA_1:def 7
2 to_power n if i = 0
otherwise i;

coherence
( ( i = 0 implies 2 to_power n is Nat ) & ( not i = 0 implies i is Nat ) ) ;
correctness
consistency
for b₁ being Nat holds verum;
;

end;

:: deftheorem Def7 defines ChangeVal_1 IDEA_1:def 7 :
for n, i being Nat holds
( ( i = 0 implies ChangeVal_1 (i,n) = 2 to_power n ) & ( not i = 0 implies ChangeVal_1 (i,n) = i ) );

theorem Th17: :: IDEA_1:17

for i, n being Nat st i is_expressible_by n holds
( ChangeVal_1 (i,n) <= 2 to_power n & ChangeVal_1 (i,n) > 0 )

proof end;

theorem Th18: :: IDEA_1:18

for n, a1, a2 being Nat st a1 is_expressible_by n & a2 is_expressible_by n & ChangeVal_1 (a1,n) = ChangeVal_1 (a2,n) holds
a1 = a2

proof end;

definition

let n, i be Nat;

func ChangeVal_2 (i,n) -> Nat equals :Def8: :: IDEA_1:def 8
0 if i = 2 to_power n
otherwise i;

correctness
coherence
( ( i = 2 to_power n implies 0 is Nat ) & ( not i = 2 to_power n implies i is Nat ) );
consistency
for b₁ being Nat holds verum;
;

end;

:: deftheorem Def8 defines ChangeVal_2 IDEA_1:def 8 :
for n, i being Nat holds
( ( i = 2 to_power n implies ChangeVal_2 (i,n) = 0 ) & ( not i = 2 to_power n implies ChangeVal_2 (i,n) = i ) );

theorem :: IDEA_1:19

for i, n being Nat st i is_expressible_by n holds
ChangeVal_2 (i,n) is_expressible_by n by Def8;

theorem Th20: :: IDEA_1:20

for n, a1, a2 being Nat st a1 <> 0 & a2 <> 0 & ChangeVal_2 (a1,n) = ChangeVal_2 (a2,n) holds
a1 = a2

proof end;

definition

let n be Nat;
let i, j be Nat;

func MUL_MOD (i,j,n) -> Nat equals :: IDEA_1:def 9
ChangeVal_2 ((((ChangeVal_1 (i,n)) * (ChangeVal_1 (j,n))) mod ((2 to_power n) + 1)),n);

coherence
ChangeVal_2 ((((ChangeVal_1 (i,n)) * (ChangeVal_1 (j,n))) mod ((2 to_power n) + 1)),n) is Nat ;

end;

:: deftheorem defines MUL_MOD IDEA_1:def 9 :
for n, i, j being Nat holds MUL_MOD (i,j,n) = ChangeVal_2 ((((ChangeVal_1 (i,n)) * (ChangeVal_1 (j,n))) mod ((2 to_power n) + 1)),n);

definition

let n be non zero Nat;
let i be Nat;
assume that
A1: i is_expressible_by n and
A2: (2 to_power n) + 1 is prime ;

func INV_MOD (i,n) -> Nat means :Def10: :: IDEA_1:def 10
( MUL_MOD (i,it,n) = 1 & it is_expressible_by n );

existence
ex b₁ being Nat st
( MUL_MOD (i,b₁,n) = 1 & b₁ is_expressible_by n )

proof end;

uniqueness
for b₁, b₂ being Nat st MUL_MOD (i,b₁,n) = 1 & b₁ is_expressible_by n & MUL_MOD (i,b₂,n) = 1 & b₂ is_expressible_by n holds
b₁ = b₂

proof end;

end;

:: deftheorem Def10 defines INV_MOD IDEA_1:def 10 :
for n being non zero Nat
for i being Nat st i is_expressible_by n & (2 to_power n) + 1 is prime holds
for b₃ being Nat holds
( b₃ = INV_MOD (i,n) iff ( MUL_MOD (i,b₃,n) = 1 & b₃ is_expressible_by n ) );

theorem :: IDEA_1:21

for i, j, n being Nat holds MUL_MOD (i,j,n) = MUL_MOD (j,i,n) ;

theorem Th22: :: IDEA_1:22

for i, n being Nat st i is_expressible_by n holds
MUL_MOD (1,i,n) = i

proof end;

theorem Th23: :: IDEA_1:23

for i, j, k, n being Nat st (2 to_power n) + 1 is prime & i is_expressible_by n & j is_expressible_by n & k is_expressible_by n holds
MUL_MOD ((MUL_MOD (i,j,n)),k,n) = MUL_MOD (i,(MUL_MOD (j,k,n)),n)

proof end;

theorem Th24: :: IDEA_1:24

for i, j, n being Nat holds MUL_MOD (i,j,n) is_expressible_by n

proof end;

theorem :: IDEA_1:25

for i, n being Nat st ChangeVal_2 (i,n) = 1 holds
i = 1

proof end;

:: We define three IDEA's operations IDEAoperationA, IDEAoperationB
:: and IDEAoperationC using the basic operators. IDEA Cryptogram
:: encrypts 64-bit plain text to 64-bit ciphertext. These texts
:: are divided into 4 16-bits text blocks. Here, we use m as the
:: text block sequence. And, IDEA's operations use key sequence
:: explains k in this section. n is bit-length of text blocks.
:: Definiton of IDEA Operation A

definition

let n be Nat;
let m, k be FinSequence of NAT ;

func IDEAoperationA (m,k,n) -> FinSequence of NAT means :Def11: :: IDEA_1:def 11
( len it = len m & ( for i being Nat st i in dom m holds
( ( i = 1 implies it . i = MUL_MOD ((m . 1),(k . 1),n) ) & ( i = 2 implies it . i = ADD_MOD ((m . 2),(k . 2),n) ) & ( i = 3 implies it . i = ADD_MOD ((m . 3),(k . 3),n) ) & ( i = 4 implies it . i = MUL_MOD ((m . 4),(k . 4),n) ) & ( i <> 1 & i <> 2 & i <> 3 & i <> 4 implies it . i = m . i ) ) ) );

existence
ex b₁ being FinSequence of NAT st
( len b₁ = len m & ( for i being Nat st i in dom m holds
( ( i = 1 implies b₁ . i = MUL_MOD ((m . 1),(k . 1),n) ) & ( i = 2 implies b₁ . i = ADD_MOD ((m . 2),(k . 2),n) ) & ( i = 3 implies b₁ . i = ADD_MOD ((m . 3),(k . 3),n) ) & ( i = 4 implies b₁ . i = MUL_MOD ((m . 4),(k . 4),n) ) & ( i <> 1 & i <> 2 & i <> 3 & i <> 4 implies b₁ . i = m . i ) ) ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of NAT st len b₁ = len m & ( for i being Nat st i in dom m holds
( ( i = 1 implies b₁ . i = MUL_MOD ((m . 1),(k . 1),n) ) & ( i = 2 implies b₁ . i = ADD_MOD ((m . 2),(k . 2),n) ) & ( i = 3 implies b₁ . i = ADD_MOD ((m . 3),(k . 3),n) ) & ( i = 4 implies b₁ . i = MUL_MOD ((m . 4),(k . 4),n) ) & ( i <> 1 & i <> 2 & i <> 3 & i <> 4 implies b₁ . i = m . i ) ) ) & len b₂ = len m & ( for i being Nat st i in dom m holds
( ( i = 1 implies b₂ . i = MUL_MOD ((m . 1),(k . 1),n) ) & ( i = 2 implies b₂ . i = ADD_MOD ((m . 2),(k . 2),n) ) & ( i = 3 implies b₂ . i = ADD_MOD ((m . 3),(k . 3),n) ) & ( i = 4 implies b₂ . i = MUL_MOD ((m . 4),(k . 4),n) ) & ( i <> 1 & i <> 2 & i <> 3 & i <> 4 implies b₂ . i = m . i ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def11 defines IDEAoperationA IDEA_1:def 11 :
for n being Nat
for m, k, b₄ being FinSequence of NAT holds
( b₄ = IDEAoperationA (m,k,n) iff ( len b₄ = len m & ( for i being Nat st i in dom m holds
( ( i = 1 implies b₄ . i = MUL_MOD ((m . 1),(k . 1),n) ) & ( i = 2 implies b₄ . i = ADD_MOD ((m . 2),(k . 2),n) ) & ( i = 3 implies b₄ . i = ADD_MOD ((m . 3),(k . 3),n) ) & ( i = 4 implies b₄ . i = MUL_MOD ((m . 4),(k . 4),n) ) & ( i <> 1 & i <> 2 & i <> 3 & i <> 4 implies b₄ . i = m . i ) ) ) ) );

definition

let n be non zero Nat;
let m, k be FinSequence of NAT ;

func IDEAoperationB (m,k,n) -> FinSequence of NAT means :Def12: :: IDEA_1:def 12
( len it = len m & ( for i being Nat st i in dom m holds
( ( i = 1 implies it . i = Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)))) ) & ( i = 2 implies it . i = Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)),n)))) ) & ( i = 3 implies it . i = Absval ((n -BinarySequence (m . 3)) 'xor' (n -BinarySequence (MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)))) ) & ( i = 4 implies it . i = Absval ((n -BinarySequence (m . 4)) 'xor' (n -BinarySequence (ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)),n)))) ) & ( i <> 1 & i <> 2 & i <> 3 & i <> 4 implies it . i = m . i ) ) ) );

existence
ex b₁ being FinSequence of NAT st
( len b₁ = len m & ( for i being Nat st i in dom m holds
( ( i = 1 implies b₁ . i = Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)))) ) & ( i = 2 implies b₁ . i = Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)),n)))) ) & ( i = 3 implies b₁ . i = Absval ((n -BinarySequence (m . 3)) 'xor' (n -BinarySequence (MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)))) ) & ( i = 4 implies b₁ . i = Absval ((n -BinarySequence (m . 4)) 'xor' (n -BinarySequence (ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)),n)))) ) & ( i <> 1 & i <> 2 & i <> 3 & i <> 4 implies b₁ . i = m . i ) ) ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of NAT st len b₁ = len m & ( for i being Nat st i in dom m holds
( ( i = 1 implies b₁ . i = Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)))) ) & ( i = 2 implies b₁ . i = Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)),n)))) ) & ( i = 3 implies b₁ . i = Absval ((n -BinarySequence (m . 3)) 'xor' (n -BinarySequence (MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)))) ) & ( i = 4 implies b₁ . i = Absval ((n -BinarySequence (m . 4)) 'xor' (n -BinarySequence (ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)),n)))) ) & ( i <> 1 & i <> 2 & i <> 3 & i <> 4 implies b₁ . i = m . i ) ) ) & len b₂ = len m & ( for i being Nat st i in dom m holds
( ( i = 1 implies b₂ . i = Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)))) ) & ( i = 2 implies b₂ . i = Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)),n)))) ) & ( i = 3 implies b₂ . i = Absval ((n -BinarySequence (m . 3)) 'xor' (n -BinarySequence (MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)))) ) & ( i = 4 implies b₂ . i = Absval ((n -BinarySequence (m . 4)) 'xor' (n -BinarySequence (ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)),n)))) ) & ( i <> 1 & i <> 2 & i <> 3 & i <> 4 implies b₂ . i = m . i ) ) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def12 defines IDEAoperationB IDEA_1:def 12 :
for n being non zero Nat
for m, k, b₄ being FinSequence of NAT holds
( b₄ = IDEAoperationB (m,k,n) iff ( len b₄ = len m & ( for i being Nat st i in dom m holds
( ( i = 1 implies b₄ . i = Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)))) ) & ( i = 2 implies b₄ . i = Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)),n)))) ) & ( i = 3 implies b₄ . i = Absval ((n -BinarySequence (m . 3)) 'xor' (n -BinarySequence (MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)))) ) & ( i = 4 implies b₄ . i = Absval ((n -BinarySequence (m . 4)) 'xor' (n -BinarySequence (ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(MUL_MOD ((ADD_MOD ((MUL_MOD ((Absval ((n -BinarySequence (m . 1)) 'xor' (n -BinarySequence (m . 3)))),(k . 5),n)),(Absval ((n -BinarySequence (m . 2)) 'xor' (n -BinarySequence (m . 4)))),n)),(k . 6),n)),n)))) ) & ( i <> 1 & i <> 2 & i <> 3 & i <> 4 implies b₄ . i = m . i ) ) ) ) );

:: Definiton of IDEA Operation C

definition

let m be FinSequence of NAT ;

func IDEAoperationC m -> FinSequence of NAT means :Def13: :: IDEA_1:def 13
( len it = len m & ( for i being Nat st i in dom m holds
it . i = IFEQ (i,2,(m . 3),(IFEQ (i,3,(m . 2),(m . i)))) ) );

existence
ex b₁ being FinSequence of NAT st
( len b₁ = len m & ( for i being Nat st i in dom m holds
b₁ . i = IFEQ (i,2,(m . 3),(IFEQ (i,3,(m . 2),(m . i)))) ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence of NAT st len b₁ = len m & ( for i being Nat st i in dom m holds
b₁ . i = IFEQ (i,2,(m . 3),(IFEQ (i,3,(m . 2),(m . i)))) ) & len b₂ = len m & ( for i being Nat st i in dom m holds
b₂ . i = IFEQ (i,2,(m . 3),(IFEQ (i,3,(m . 2),(m . i)))) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def13 defines IDEAoperationC IDEA_1:def 13 :
for m, b₂ being FinSequence of NAT holds
( b₂ = IDEAoperationC m iff ( len b₂ = len m & ( for i being Nat st i in dom m holds
b₂ . i = IFEQ (i,2,(m . 3),(IFEQ (i,3,(m . 2),(m . i)))) ) ) );

theorem Th26: :: IDEA_1:26

for n being Nat
for m, k being FinSequence of NAT st len m >= 4 holds
( (IDEAoperationA (m,k,n)) . 1 is_expressible_by n & (IDEAoperationA (m,k,n)) . 2 is_expressible_by n & (IDEAoperationA (m,k,n)) . 3 is_expressible_by n & (IDEAoperationA (m,k,n)) . 4 is_expressible_by n )

proof end;

theorem Th27: :: IDEA_1:27

for m, k being FinSequence of NAT
for n being non zero Nat st len m >= 4 holds
( (IDEAoperationB (m,k,n)) . 1 is_expressible_by n & (IDEAoperationB (m,k,n)) . 2 is_expressible_by n & (IDEAoperationB (m,k,n)) . 3 is_expressible_by n & (IDEAoperationB (m,k,n)) . 4 is_expressible_by n )

proof end;

Lm3: for m being FinSequence of NAT
for i being Nat st i = 2 & i in dom m holds
(IDEAoperationC m) . i = m . 3

proof end;

Lm4: for m being FinSequence of NAT
for i being Nat st i = 3 & i in dom m holds
(IDEAoperationC m) . i = m . 2

proof end;

Lm5: for m being FinSequence of NAT
for i being Nat st i <> 2 & i <> 3 & i in dom m holds
(IDEAoperationC m) . i = m . i

proof end;

theorem :: IDEA_1:28

for n being Nat
for m being FinSequence of NAT st len m >= 4 & m . 1 is_expressible_by n & m . 2 is_expressible_by n & m . 3 is_expressible_by n & m . 4 is_expressible_by n holds
( (IDEAoperationC m) . 1 is_expressible_by n & (IDEAoperationC m) . 2 is_expressible_by n & (IDEAoperationC m) . 3 is_expressible_by n & (IDEAoperationC m) . 4 is_expressible_by n )

proof end;

theorem Th29: :: IDEA_1:29

for n being non zero Nat
for m, k1, k2 being FinSequence of NAT st (2 to_power n) + 1 is prime & len m >= 4 & m . 1 is_expressible_by n & m . 2 is_expressible_by n & m . 3 is_expressible_by n & m . 4 is_expressible_by n & k1 . 1 is_expressible_by n & k1 . 2 is_expressible_by n & k1 . 3 is_expressible_by n & k1 . 4 is_expressible_by n & k2 . 1 = INV_MOD ((k1 . 1),n) & k2 . 2 = NEG_MOD ((k1 . 2),n) & k2 . 3 = NEG_MOD ((k1 . 3),n) & k2 . 4 = INV_MOD ((k1 . 4),n) holds
IDEAoperationA ((IDEAoperationA (m,k1,n)),k2,n) = m

proof end;

theorem Th30: :: IDEA_1:30

for n being non zero Nat
for m, k1, k2 being FinSequence of NAT st (2 to_power n) + 1 is prime & len m >= 4 & m . 1 is_expressible_by n & m . 2 is_expressible_by n & m . 3 is_expressible_by n & m . 4 is_expressible_by n & k1 . 1 is_expressible_by n & k1 . 2 is_expressible_by n & k1 . 3 is_expressible_by n & k1 . 4 is_expressible_by n & k2 . 1 = INV_MOD ((k1 . 1),n) & k2 . 2 = NEG_MOD ((k1 . 3),n) & k2 . 3 = NEG_MOD ((k1 . 2),n) & k2 . 4 = INV_MOD ((k1 . 4),n) holds
IDEAoperationA ((IDEAoperationC (IDEAoperationA ((IDEAoperationC m),k1,n))),k2,n) = m

proof end;

theorem Th31: :: IDEA_1:31

for n being non zero Nat
for m, k1, k2 being FinSequence of NAT st len m >= 4 & m . 1 is_expressible_by n & m . 2 is_expressible_by n & m . 3 is_expressible_by n & m . 4 is_expressible_by n & k2 . 5 = k1 . 5 & k2 . 6 = k1 . 6 holds
IDEAoperationB ((IDEAoperationB (m,k1,n)),k2,n) = m

proof end;

theorem :: IDEA_1:32

for m being FinSequence of NAT st len m >= 4 holds
IDEAoperationC (IDEAoperationC m) = m

proof end;

:: For making a model of IDEA, we define sequences of IDEA's
:: operations IDEA_P, IDEA_PS, IDEA_PE, IDEA_Q, IDEA_QS and
:: IDEA_QE. And, we define MESSAGES as plain and cipher text.

definition

func MESSAGES -> set equals :: IDEA_1:def 14
NAT * ;

coherence
NAT * is set ;

end;

:: deftheorem defines MESSAGES IDEA_1:def 14 :
MESSAGES = NAT * ;

registration

cluster MESSAGES -> non empty ;

coherence
not MESSAGES is empty ;

end;

registration

cluster MESSAGES -> functional ;

coherence
MESSAGES is functional ;

end;

registration

cluster -> FinSequence-like for Element of MESSAGES ;

coherence
for b₁ being Element of MESSAGES holds b₁ is FinSequence-like ;

end;

definition

let n be non zero Nat;
let k be FinSequence of NAT ;

func IDEA_P (k,n) -> Function of MESSAGES,MESSAGES means :Def15: :: IDEA_1:def 15
for m being FinSequence of NAT holds it . m = IDEAoperationA ((IDEAoperationC (IDEAoperationB (m,k,n))),k,n);

existence
ex b₁ being Function of MESSAGES,MESSAGES st
for m being FinSequence of NAT holds b₁ . m = IDEAoperationA ((IDEAoperationC (IDEAoperationB (m,k,n))),k,n)

proof end;

uniqueness
for b₁, b₂ being Function of MESSAGES,MESSAGES st ( for m being FinSequence of NAT holds b₁ . m = IDEAoperationA ((IDEAoperationC (IDEAoperationB (m,k,n))),k,n) ) & ( for m being FinSequence of NAT holds b₂ . m = IDEAoperationA ((IDEAoperationC (IDEAoperationB (m,k,n))),k,n) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def15 defines IDEA_P IDEA_1:def 15 :
for n being non zero Nat
for k being FinSequence of NAT
for b₃ being Function of MESSAGES,MESSAGES holds
( b₃ = IDEA_P (k,n) iff for m being FinSequence of NAT holds b₃ . m = IDEAoperationA ((IDEAoperationC (IDEAoperationB (m,k,n))),k,n) );

definition

let n be non zero Nat;
let k be FinSequence of NAT ;

func IDEA_Q (k,n) -> Function of MESSAGES,MESSAGES means :Def16: :: IDEA_1:def 16
for m being FinSequence of NAT holds it . m = IDEAoperationB ((IDEAoperationA ((IDEAoperationC m),k,n)),k,n);

existence
ex b₁ being Function of MESSAGES,MESSAGES st
for m being FinSequence of NAT holds b₁ . m = IDEAoperationB ((IDEAoperationA ((IDEAoperationC m),k,n)),k,n)

proof end;

uniqueness
for b₁, b₂ being Function of MESSAGES,MESSAGES st ( for m being FinSequence of NAT holds b₁ . m = IDEAoperationB ((IDEAoperationA ((IDEAoperationC m),k,n)),k,n) ) & ( for m being FinSequence of NAT holds b₂ . m = IDEAoperationB ((IDEAoperationA ((IDEAoperationC m),k,n)),k,n) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def16 defines IDEA_Q IDEA_1:def 16 :
for n being non zero Nat
for k being FinSequence of NAT
for b₃ being Function of MESSAGES,MESSAGES holds
( b₃ = IDEA_Q (k,n) iff for m being FinSequence of NAT holds b₃ . m = IDEAoperationB ((IDEAoperationA ((IDEAoperationC m),k,n)),k,n) );

definition

let r, lk be Nat;
let n be non zero Nat;
let Key be Matrix of lk,6,NAT;

func IDEA_P_F (Key,n,r) -> FinSequence means :Def17: :: IDEA_1:def 17
( len it = r & ( for i being Nat st i in dom it holds
it . i = IDEA_P ((Line (Key,i)),n) ) );

existence
ex b₁ being FinSequence st
( len b₁ = r & ( for i being Nat st i in dom b₁ holds
b₁ . i = IDEA_P ((Line (Key,i)),n) ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence st len b₁ = r & ( for i being Nat st i in dom b₁ holds
b₁ . i = IDEA_P ((Line (Key,i)),n) ) & len b₂ = r & ( for i being Nat st i in dom b₂ holds
b₂ . i = IDEA_P ((Line (Key,i)),n) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def17 defines IDEA_P_F IDEA_1:def 17 :
for r, lk being Nat
for n being non zero Nat
for Key being Matrix of lk,6,NAT
for b₅ being FinSequence holds
( b₅ = IDEA_P_F (Key,n,r) iff ( len b₅ = r & ( for i being Nat st i in dom b₅ holds
b₅ . i = IDEA_P ((Line (Key,i)),n) ) ) );

registration

let r, lk be Nat;
let n be non zero Nat;
let Key be Matrix of lk,6,NAT;

cluster IDEA_P_F (Key,n,r) -> Function-yielding ;

coherence
IDEA_P_F (Key,n,r) is Function-yielding

proof end;

end;

definition

let r, lk be Nat;
let n be non zero Nat;
let Key be Matrix of lk,6,NAT;

func IDEA_Q_F (Key,n,r) -> FinSequence means :Def18: :: IDEA_1:def 18
( len it = r & ( for i being Nat st i in dom it holds
it . i = IDEA_Q ((Line (Key,((r -' i) + 1))),n) ) );

existence
ex b₁ being FinSequence st
( len b₁ = r & ( for i being Nat st i in dom b₁ holds
b₁ . i = IDEA_Q ((Line (Key,((r -' i) + 1))),n) ) )

proof end;

uniqueness
for b₁, b₂ being FinSequence st len b₁ = r & ( for i being Nat st i in dom b₁ holds
b₁ . i = IDEA_Q ((Line (Key,((r -' i) + 1))),n) ) & len b₂ = r & ( for i being Nat st i in dom b₂ holds
b₂ . i = IDEA_Q ((Line (Key,((r -' i) + 1))),n) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def18 defines IDEA_Q_F IDEA_1:def 18 :
for r, lk being Nat
for n being non zero Nat
for Key being Matrix of lk,6,NAT
for b₅ being FinSequence holds
( b₅ = IDEA_Q_F (Key,n,r) iff ( len b₅ = r & ( for i being Nat st i in dom b₅ holds
b₅ . i = IDEA_Q ((Line (Key,((r -' i) + 1))),n) ) ) );

registration

let r, lk be Nat;
let n be non zero Nat;
let Key be Matrix of lk,6,NAT;

cluster IDEA_Q_F (Key,n,r) -> Function-yielding ;

coherence
IDEA_Q_F (Key,n,r) is Function-yielding

proof end;

end;

definition

let k be FinSequence of NAT ;
let n be Nat;

func IDEA_PS (k,n) -> Function of MESSAGES,MESSAGES means :Def19: :: IDEA_1:def 19
for m being FinSequence of NAT holds it . m = IDEAoperationA (m,k,n);

existence
ex b₁ being Function of MESSAGES,MESSAGES st
for m being FinSequence of NAT holds b₁ . m = IDEAoperationA (m,k,n)

proof end;

uniqueness
for b₁, b₂ being Function of MESSAGES,MESSAGES st ( for m being FinSequence of NAT holds b₁ . m = IDEAoperationA (m,k,n) ) & ( for m being FinSequence of NAT holds b₂ . m = IDEAoperationA (m,k,n) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def19 defines IDEA_PS IDEA_1:def 19 :
for k being FinSequence of NAT
for n being Nat
for b₃ being Function of MESSAGES,MESSAGES holds
( b₃ = IDEA_PS (k,n) iff for m being FinSequence of NAT holds b₃ . m = IDEAoperationA (m,k,n) );

definition

let k be FinSequence of NAT ;
let n be Nat;

func IDEA_QS (k,n) -> Function of MESSAGES,MESSAGES means :Def20: :: IDEA_1:def 20
for m being FinSequence of NAT holds it . m = IDEAoperationA (m,k,n);

existence
ex b₁ being Function of MESSAGES,MESSAGES st
for m being FinSequence of NAT holds b₁ . m = IDEAoperationA (m,k,n)

proof end;

uniqueness
for b₁, b₂ being Function of MESSAGES,MESSAGES st ( for m being FinSequence of NAT holds b₁ . m = IDEAoperationA (m,k,n) ) & ( for m being FinSequence of NAT holds b₂ . m = IDEAoperationA (m,k,n) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def20 defines IDEA_QS IDEA_1:def 20 :
for k being FinSequence of NAT
for n being Nat
for b₃ being Function of MESSAGES,MESSAGES holds
( b₃ = IDEA_QS (k,n) iff for m being FinSequence of NAT holds b₃ . m = IDEAoperationA (m,k,n) );

definition

let n be non zero Nat;
let k be FinSequence of NAT ;

func IDEA_PE (k,n) -> Function of MESSAGES,MESSAGES means :Def21: :: IDEA_1:def 21
for m being FinSequence of NAT holds it . m = IDEAoperationA ((IDEAoperationB (m,k,n)),k,n);

existence
ex b₁ being Function of MESSAGES,MESSAGES st
for m being FinSequence of NAT holds b₁ . m = IDEAoperationA ((IDEAoperationB (m,k,n)),k,n)

proof end;

uniqueness
for b₁, b₂ being Function of MESSAGES,MESSAGES st ( for m being FinSequence of NAT holds b₁ . m = IDEAoperationA ((IDEAoperationB (m,k,n)),k,n) ) & ( for m being FinSequence of NAT holds b₂ . m = IDEAoperationA ((IDEAoperationB (m,k,n)),k,n) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def21 defines IDEA_PE IDEA_1:def 21 :
for n being non zero Nat
for k being FinSequence of NAT
for b₃ being Function of MESSAGES,MESSAGES holds
( b₃ = IDEA_PE (k,n) iff for m being FinSequence of NAT holds b₃ . m = IDEAoperationA ((IDEAoperationB (m,k,n)),k,n) );

definition

let n be non zero Nat;
let k be FinSequence of NAT ;

func IDEA_QE (k,n) -> Function of MESSAGES,MESSAGES means :Def22: :: IDEA_1:def 22
for m being FinSequence of NAT holds it . m = IDEAoperationB ((IDEAoperationA (m,k,n)),k,n);

existence
ex b₁ being Function of MESSAGES,MESSAGES st
for m being FinSequence of NAT holds b₁ . m = IDEAoperationB ((IDEAoperationA (m,k,n)),k,n)

proof end;

uniqueness
for b₁, b₂ being Function of MESSAGES,MESSAGES st ( for m being FinSequence of NAT holds b₁ . m = IDEAoperationB ((IDEAoperationA (m,k,n)),k,n) ) & ( for m being FinSequence of NAT holds b₂ . m = IDEAoperationB ((IDEAoperationA (m,k,n)),k,n) ) holds
b₁ = b₂

proof end;

end;

:: deftheorem Def22 defines IDEA_QE IDEA_1:def 22 :
for n being non zero Nat
for k being FinSequence of NAT
for b₃ being Function of MESSAGES,MESSAGES holds
( b₃ = IDEA_QE (k,n) iff for m being FinSequence of NAT holds b₃ . m = IDEAoperationB ((IDEAoperationA (m,k,n)),k,n) );

theorem Th33: :: IDEA_1:33

for n being non zero Nat
for m, k1, k2 being FinSequence of NAT st (2 to_power n) + 1 is prime & len m >= 4 & m . 1 is_expressible_by n & m . 2 is_expressible_by n & m . 3 is_expressible_by n & m . 4 is_expressible_by n & k1 . 1 is_expressible_by n & k1 . 2 is_expressible_by n & k1 . 3 is_expressible_by n & k1 . 4 is_expressible_by n & k2 . 1 = INV_MOD ((k1 . 1),n) & k2 . 2 = NEG_MOD ((k1 . 3),n) & k2 . 3 = NEG_MOD ((k1 . 2),n) & k2 . 4 = INV_MOD ((k1 . 4),n) & k2 . 5 = k1 . 5 & k2 . 6 = k1 . 6 holds
((IDEA_Q (k2,n)) * (IDEA_P (k1,n))) . m = m

proof end;

theorem Th34: :: IDEA_1:34

for n being non zero Nat
for m, k1, k2 being FinSequence of NAT st (2 to_power n) + 1 is prime & len m >= 4 & m . 1 is_expressible_by n & m . 2 is_expressible_by n & m . 3 is_expressible_by n & m . 4 is_expressible_by n & k1 . 1 is_expressible_by n & k1 . 2 is_expressible_by n & k1 . 3 is_expressible_by n & k1 . 4 is_expressible_by n & k2 . 1 = INV_MOD ((k1 . 1),n) & k2 . 2 = NEG_MOD ((k1 . 2),n) & k2 . 3 = NEG_MOD ((k1 . 3),n) & k2 . 4 = INV_MOD ((k1 . 4),n) holds
((IDEA_QS (k2,n)) * (IDEA_PS (k1,n))) . m = m

proof end;

theorem Th35: :: IDEA_1:35

for n being non zero Nat
for m, k1, k2 being FinSequence of NAT st (2 to_power n) + 1 is prime & len m >= 4 & m . 1 is_expressible_by n & m . 2 is_expressible_by n & m . 3 is_expressible_by n & m . 4 is_expressible_by n & k1 . 1 is_expressible_by n & k1 . 2 is_expressible_by n & k1 . 3 is_expressible_by n & k1 . 4 is_expressible_by n & k2 . 1 = INV_MOD ((k1 . 1),n) & k2 . 2 = NEG_MOD ((k1 . 2),n) & k2 . 3 = NEG_MOD ((k1 . 3),n) & k2 . 4 = INV_MOD ((k1 . 4),n) & k2 . 5 = k1 . 5 & k2 . 6 = k1 . 6 holds
((IDEA_QE (k2,n)) * (IDEA_PE (k1,n))) . m = m

proof end;

theorem Th36: :: IDEA_1:36

for n being non zero Nat
for lk being Nat
for Key being Matrix of lk,6,NAT
for k being Nat holds IDEA_P_F (Key,n,(k + 1)) = (IDEA_P_F (Key,n,k)) ^ <*(IDEA_P ((Line (Key,(k + 1))),n))*>

proof end;

theorem Th37: :: IDEA_1:37

for n being non zero Nat
for lk being Nat
for Key being Matrix of lk,6,NAT
for k being Nat holds IDEA_Q_F (Key,n,(k + 1)) = <*(IDEA_Q ((Line (Key,(k + 1))),n))*> ^ (IDEA_Q_F (Key,n,k))

proof end;

theorem Th38: :: IDEA_1:38

for n being non zero Nat
for lk being Nat
for Key being Matrix of lk,6,NAT
for k being Nat holds IDEA_P_F (Key,n,k) is FuncSeq-like FinSequence

proof end;

theorem Th39: :: IDEA_1:39

for n being non zero Nat
for lk being Nat
for Key being Matrix of lk,6,NAT
for k being Nat holds IDEA_Q_F (Key,n,k) is FuncSeq-like FinSequence

proof end;

theorem Th40: :: IDEA_1:40

for n being non zero Nat
for lk being Nat
for Key being Matrix of lk,6,NAT
for k being Nat st k <> 0 holds
IDEA_P_F (Key,n,k) is FuncSequence of MESSAGES , MESSAGES

proof end;

theorem Th41: :: IDEA_1:41

for n being non zero Nat
for lk being Nat
for Key being Matrix of lk,6,NAT
for k being Nat st k <> 0 holds
IDEA_Q_F (Key,n,k) is FuncSequence of MESSAGES , MESSAGES

proof end;

theorem Th42: :: IDEA_1:42

for n being non zero Nat
for M, m, k being FinSequence of NAT st M = (IDEA_P (k,n)) . m & len m >= 4 holds
( len M >= 4 & M . 1 is_expressible_by n & M . 2 is_expressible_by n & M . 3 is_expressible_by n & M . 4 is_expressible_by n )

proof end;

theorem Th43: :: IDEA_1:43

for n being non zero Nat
for lk being Nat
for Key being Matrix of lk,6,NAT
for r being Nat holds
( rng (compose ((IDEA_P_F (Key,n,r)),MESSAGES)) c= MESSAGES & dom (compose ((IDEA_P_F (Key,n,r)),MESSAGES)) = MESSAGES )

proof end;

theorem :: IDEA_1:44

for n being non zero Nat
for lk being Nat
for Key being Matrix of lk,6,NAT
for r being Nat holds
( rng (compose ((IDEA_Q_F (Key,n,r)),MESSAGES)) c= MESSAGES & dom (compose ((IDEA_Q_F (Key,n,r)),MESSAGES)) = MESSAGES )

proof end;

theorem Th45: :: IDEA_1:45

for n being non zero Nat
for lk being Nat
for Key being Matrix of lk,6,NAT
for r being Nat
for M, m being FinSequence of NAT st M = (compose ((IDEA_P_F (Key,n,r)),MESSAGES)) . m & len m >= 4 & m . 1 is_expressible_by n & m . 2 is_expressible_by n & m . 3 is_expressible_by n & m . 4 is_expressible_by n holds
( len M >= 4 & M . 1 is_expressible_by n & M . 2 is_expressible_by n & M . 3 is_expressible_by n & M . 4 is_expressible_by n )

proof end;

:: IDEA encryption algorithm is as follows:
:: IDEA_PS -> IDEA_P -> IDEA_P -> ... -> IDEA_P -> IDEA_PE
:: IDEA decryption algorithm is as follows:
:: IDEA_QE -> IDEA_Q -> IDEA_Q -> ... -> IDEA_Q -> IDEA_QS
:: Theorem 49 ensures that the ciphertext that is encrypted by IDEA
:: encryption algorithm can be decrypted by IDEA decryption algorithm.

theorem Th46: :: IDEA_1:46

for n being non zero Nat
for lk being Nat
for Key1, Key2 being Matrix of lk,6,NAT
for r being Nat
for m being FinSequence of NAT st lk >= r & (2 to_power n) + 1 is prime & len m >= 4 & m . 1 is_expressible_by n & m . 2 is_expressible_by n & m . 3 is_expressible_by n & m . 4 is_expressible_by n & ( for i being Nat st i <= r holds
( Key1 * (i,1) is_expressible_by n & Key1 * (i,2) is_expressible_by n & Key1 * (i,3) is_expressible_by n & Key1 * (i,4) is_expressible_by n & Key1 * (i,5) is_expressible_by n & Key1 * (i,6) is_expressible_by n & Key2 * (i,1) is_expressible_by n & Key2 * (i,2) is_expressible_by n & Key2 * (i,3) is_expressible_by n & Key2 * (i,4) is_expressible_by n & Key2 * (i,5) is_expressible_by n & Key2 * (i,6) is_expressible_by n & Key2 * (i,1) = INV_MOD ((Key1 * (i,1)),n) & Key2 * (i,2) = NEG_MOD ((Key1 * (i,3)),n) & Key2 * (i,3) = NEG_MOD ((Key1 * (i,2)),n) & Key2 * (i,4) = INV_MOD ((Key1 * (i,4)),n) & Key1 * (i,5) = Key2 * (i,5) & Key1 * (i,6) = Key2 * (i,6) ) ) holds
(compose (((IDEA_P_F (Key1,n,r)) ^ (IDEA_Q_F (Key2,n,r))),MESSAGES)) . m = m

proof end;

theorem :: IDEA_1:47

for n being non zero Nat
for lk being Nat
for Key1, Key2 being Matrix of lk,6,NAT
for r being Nat
for ks1, ks2, ke1, ke2, m being FinSequence of NAT st lk >= r & (2 to_power n) + 1 is prime & len m >= 4 & m . 1 is_expressible_by n & m . 2 is_expressible_by n & m . 3 is_expressible_by n & m . 4 is_expressible_by n & ( for i being Nat st i <= r holds
( Key1 * (i,1) is_expressible_by n & Key1 * (i,2) is_expressible_by n & Key1 * (i,3) is_expressible_by n & Key1 * (i,4) is_expressible_by n & Key1 * (i,5) is_expressible_by n & Key1 * (i,6) is_expressible_by n & Key2 * (i,1) is_expressible_by n & Key2 * (i,2) is_expressible_by n & Key2 * (i,3) is_expressible_by n & Key2 * (i,4) is_expressible_by n & Key2 * (i,5) is_expressible_by n & Key2 * (i,6) is_expressible_by n & Key2 * (i,1) = INV_MOD ((Key1 * (i,1)),n) & Key2 * (i,2) = NEG_MOD ((Key1 * (i,3)),n) & Key2 * (i,3) = NEG_MOD ((Key1 * (i,2)),n) & Key2 * (i,4) = INV_MOD ((Key1 * (i,4)),n) & Key1 * (i,5) = Key2 * (i,5) & Key1 * (i,6) = Key2 * (i,6) ) ) & ks1 . 1 is_expressible_by n & ks1 . 2 is_expressible_by n & ks1 . 3 is_expressible_by n & ks1 . 4 is_expressible_by n & ks2 . 1 = INV_MOD ((ks1 . 1),n) & ks2 . 2 = NEG_MOD ((ks1 . 2),n) & ks2 . 3 = NEG_MOD ((ks1 . 3),n) & ks2 . 4 = INV_MOD ((ks1 . 4),n) & ke1 . 1 is_expressible_by n & ke1 . 2 is_expressible_by n & ke1 . 3 is_expressible_by n & ke1 . 4 is_expressible_by n & ke2 . 1 = INV_MOD ((ke1 . 1),n) & ke2 . 2 = NEG_MOD ((ke1 . 2),n) & ke2 . 3 = NEG_MOD ((ke1 . 3),n) & ke2 . 4 = INV_MOD ((ke1 . 4),n) & ke2 . 5 = ke1 . 5 & ke2 . 6 = ke1 . 6 holds
((IDEA_QS (ks2,n)) * ((compose ((IDEA_Q_F (Key2,n,r)),MESSAGES)) * ((IDEA_QE (ke2,n)) * ((IDEA_PE (ke1,n)) * ((compose ((IDEA_P_F (Key1,n,r)),MESSAGES)) * (IDEA_PS (ks1,n))))))) . m = m

proof end;