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3XRC4

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Samir Crypticus
Samir Crypticus

The document proposes a stream cipher based on the mathematical structure underlying the three-body problem. It describes modeling the three-body problem as a set of state variables that are iteratively updated according to specific rules. This mathematical structure is then used in the design of a stream cipher, where the internal state consists of state variables modeled on the three-body problem. The keystream is generated by updating these state variables in each round. It also presents an algorithm called 3XRC4 that extends the RC4 stream cipher based on this three-body problem structure.

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Stream ciphers based on the mathematical structure underlying the 3 body problem.
Author : Samir Bouftass , Independant crypto researcher E-mail : crypticator@gmail.com
Stream cipher ?
1 - Key Stream Generation :
2 - Encryption :
3 ¨C Decryption :
A KeyStream K should fulfill this condition :
The three body problem :
If we perform measurements after a time intervall T :
A mod¨¦lisation of three body problem could be useful to crypto algorithmes design.
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] .
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
The mathematical structure underlying the three body problem.
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 )
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].
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
3XRC4 : A RC4 variant based on the mathematical structure underlying The 3 body probl¨¨me .
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]
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.
3XRC4
L¡¯algorithme 3XRC4 :
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]
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.

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3XRC4

  • 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
  • 4. 1 - Key Stream Generation :
  • 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 mod¨¦lisation 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 probl¨¨me .
  • 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.
  • 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.