a stream cipher based on the mathematical structure underlying the 3 body problem

1. Stream ciphers based on the mathematical
structure underlying the 3 body problem.

2. Author : Samir Bouftass , Independant crypto researcher
E-mail : crypticator@gmail.com

3. Stream cipher ?

4. 1 - Key Stream Generation :

5. 2 - Encryption :

6. 3 – Decryption :

7. A KeyStream K should fulfill this condition :

8. The three body problem :

9. If we perform measurements after a time intervall T :

10. A modélisation of three body problem could be useful to crypto
algorithmes design.

11. A General Stream cipher algorithm :
Input Data :
Secret Key : Sk.
Plain Text : M = [ m_0….m_n] .
Intermediate Data :
Internal State : S = [s_0…..s_n] .
KeyStream : K = [k_0….k_n].
Output Data :
Cipher Text : C = [c_0…..c_n] .

12. State Initialization :
s_0 = Initialize( Sk ).
Encryption/Decryption loop :
For i = 0 to n do
s_(i+1) = F ( s_i )
k_i = G ( s_(i+1) )
c_i = k_i xor p_i

13. The mathematical structure underlying the three body problem.

14. The mathematical structure :
Set S ={ [(p1_0,f1_0),(p2_0,f2_0),(p3_0,f3_0)] …
[(p1_i,f1_i),(p2_i,f2_i),(p3_i,f3_i)] …
[(p1_n,f1_n),(p2_n,f2_n),(pi_n,fi_n)] }
Caracterized by following rules :
f1_i+1 = F1(p2_i , p3_i )
f2_i+1 = F2(p3_i , p1_i )
f3_i+1 = F3(p1_i , p2_i )
p1_i+1 = G1(f1_i+1 )
p2_i+1 = G2(f2_i+1 )
p3_i+1 = G3(f3_i+1 )

15. A General Stream cipher algorithm based on the mathematical
structure underlying the three body problem :
Input Data :
Secret Key : Sk.
Plain Text : M = [ m_0….m_n] .
Intermediate Data :
Internal State : S = { [(p1_0,f1_0),(p2_0,f2_0),(p3_0,f3_0)] …..
[(p1_n,f1_n),(p2_n,f2_n),(p3_n,f3_n)] }
Key Stream : K = [k_0….k_n].
Output Data :
Cipher Text: C = [c_0…..c_n].

16. State Initialization :
s_0 =[(p1_0,f1_0),(p2_0,f2_0),(p3_0,f3_0)] = Initialize(Cs ).
Encryption/Decryption loop :
For i = 0 to n do
f1_i+1 = F1(p2_i , p3_i )
f2_i+1 = F2(p3_i , p1_i )
f3_i+1 = F3(p1_i , p2_i )
p1_i+1 = G1(f1_i+1 )
p2_i+1 = G2(f2_i+1 )
p3_i+1 = G3(f3_i+1 )
k_i = p1_i+1
c_i = k_i xor m_i

17. 3XRC4 : A RC4 variant based on the mathematical structure
underlying The 3 body problème .

18. RC4 Algorithm:
Secret key Sk, a componed of L bytes : K[0], …, K[L-1].
Initialization :
for i : 0 to 255
S[i] : i.
j : 0
for i : 0 to 255
j : ( j + S[i] + K[i mod L ] ) mod 256
swap S[i] and S[j]

19. Encryption / Decryption Loop :
i : 0 , j : 0
for i : 0 to n
i : ( i+1 ) mod 256
j : ( j+S[i] ) mod 256
swap S[i] and S[j].
k_i+1 = S[( S[i] + S[j] ) mod 256 ]
c_i = k_i+1 xor m_i.

21. L’algorithme 3XRC4 :

22. Secret Key Sk =[K1, K2, K3] , Ki=1..3 a componed of L bytes : ki[0], …, ki[L-1].
Initialization :
for i : 0 to 255
S1[i] : S2[i] : S3[i] : i .
u : v : w : 0
for i : 0 to 255
u : ( u + S2[S3[i]] + K1[ i mod L ] ) mod 256
swap S1[i] and S1[u]
v : ( v + S3[S1[i]] + K2[ i mod L ] ) mod 256
swap S2[i] and S2[v]
w : ( w + S3[S1[i]] + K3[ i mod L ] ) mod 256
swap S3[i] and S3[v]

23. Encryption / Decryption Loop :
i : u : v : w :0
for i = 0 to n
i : ( i+1 ) mod 256
u : ( u+S2[S3[i]] ) mod 256
swap S1[i] and S1[u]
v : ( v+S3[S1[i]] ) mod 256
swap S2[i] and S2[v]
w : ( w+S1[S2[i]] ) mod 256
swap S3[i] and S3[w]
k_i+1 = S2[S3[( S1[i] + S1[u] ) mod 256 ]]
c_i = k_i+1 xor m_i.