10 2 ( ) Vol. 10 No. 2

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1 10 2 () Vol. 10No Journal of J imei University(Natural Science) Jun [ ] (2005) , 2 (1., ; 2., ) [ ], Hopfield,,,,, D iffie2hellman.,,,. [] ; ; ; [] TP [] A 0 [ 1 ],.,,,,., [ 2 ]. 1976W. D iffiem. Hellman [ 3 ],,,.,, [ 4 ],.,, L. M. Pecora T. L. Carroll [ 5 ], K. R. Crounse V. M ilanovie [ 6, 7 ],,,,., [ 8 ], Hopfield Hamm ing,,,.,, 28, [ ] [ ] ( ; ) ; (A ) [ ] ( ),,,.

2 126 () M IPS Years. D iffie2hellman,,,. 1., D iffie - Hellman 1. 1 Hopfield (Hopfield Neural Networks, HNN)J. J. Hopfield20 80 [ 9 ],. Hopfield, MHD (M inimum Hamm ing D istance, ),, HNN., N HNN,, N.,,,, HNN OHNN (Overstoraged HNN). HopfieldN, 01, S i ( t + 1), :, f ( x) N - 1 S i ( t + 1) = f T ij S ( t) + i, i = 0, 1, 2,, N - 1. (1) j =0 T ij ij, i i, f ( x), =( x), : ( x) = 1, x 0 0, x < 0 HNN, i 0 ( i = 0, 1, 2,, N - 1), T ij T. (1) (2), :, S ( t + 1) t: S ( t + 1) = F T (S ( t) ) =(S ( t) T) (3) = { S 0 ( t), S 1 ( t), S 2 ( t),, S N - 1 ( t) }, t : S ( t) = F T (S ( t - 1) ) = F t T (S (0) ) (4) E ( t) = - (2) 1 2 ij T ij S i ( t) S j ( t) (5) J. J. Hopfield [ 10 ],,,., MHD, [ 8 ] ; T, [ 8 ]. H, S S S S : u S = S 3 H (6) S = S 3 H (7)

3 2 : D iffe2hellman [ 11 ], Tn, n H,, T = HTH, T, n.,,, H 1 H 2, H 1 3 H 2 = H 2 3 H 1. D iffie2hellman, T 0, n. H a, T a n, A = H a T 0 H a, H a H a, H a, T a. A B, T = H a T b H a = H b T a H b, A (B ). T a T b T H a H b, n.,, D iffie2hellman, [ 12 ]. 2 : (S ) ( T ij ),., D iffie2hellman, , T 0, : 1) T 0 ; 2), - 1, [ 13 ]., n, T 0, T 0

4 128 () 10. T 0,,, M., i H i, T i n = H i T 0 H i ( 1), H i, T i,,,,, D iffie2hellman, ) H s, T r, T = H s T r H s = H s H r T 0 H r H s, T r T. 2) M T S, YYx = { S }. 3) 01, OHNN T S (0), (1) (2), S ( ), S ; S ( ) S, S (0) S, S (0) Y X,. S ( ) S,,, S ( ) S,, ),,,,,,,. 2) H r T s, T = H r T s H r = H r H s T 0 H s H r S, ( 6) ( 7), ;,, X, (1), S ( ), S ( ) S, Yx = { S } ;, M, Yx = { S } Y,,. 8 OHNN, T 0 : T 0 =

5 2 : 129 H s H r : H s = T s T r : T s = H s T 0 H s =, H r = , T r = H r T 0 H r = ; ( T ) : T = Hs H r T 0 H r H s = H r H s T 0 H s H r = T 0 T s T r T,, T, S 0 = ( ) S 1 = ( ) S 2 = ( ) S 3 = ( ) S 4 = ( ) S 5 = ( ) S 6 = ( ) S 7 = ( ). 3 M,

6 130 () 10 : 0000, 0011, 0102, 0113, 1004, 1015, 1106, 1117, 8 S , S = ( ), S 6 ( ).,,,, T,, ( ), ( 1) T, ( ), S 6 ; 3 M,. 6110, 3,,. 3. 1, OHNN.,,, OHNN, T 0, T 0 T r T s T, T 0 H r H s T r T s T, T r T s T H r H s, :,. : T =B CT, R (R - 1. R ), T (B R - 1 ) (R C), R - 1 R, T r T s H r H s. QR ( ) LU ( )., T 0 T r T s T, QR., T 0 T r T s T n ( n > 64),, :, [ 14 ] ;,, [ 15 ], (2 + 2 m - n +1 m, m > n, r = m - 1 n + 2 r (2 k! ) 2m - nk - 1 ), k =2 ;, : 1) ; 2)., n,,., LU,,, ;, LU T = HT 0 H., T 0,, T i H i ; H i, H i, H i T 0 H i T i,, nh i 01,, 2 n2, O (2 n2 ), n,

7 2 : 131 n,,., H i, T 0 Hessenberg, T i Hessenberg, T i T 0 Hessenberg, T i T 0 Hessenberg, H i,, Hessenberg, O ( n 3 ), n ;, Hessenberg [ 14 ], 2 n., n(128), Hessenberg., T s T r, T^.,, T s (T r ) H s (H r ) T,, n, ; T s T r T, X, : T s T r (8), T = Ts X T r ( 8 ) T = Hs T 0 H s XH r T 0 H r ( 9 ) T = H s H r T 0 H r H s = H r H s T 0 H s H r. (10) T 0, T T s T r,, X (9) (10), H,,,,,..,., N OHNN, p,, HN!, N!., H, N!, 10 6 H H, H N, 2. N = 32, H10 20 M IPS Years, M IPS Years.,, A,,.,,.. OHNN N, N,

8 132 () 10,, N = 8, = 20, N = 32, = A,,,. 3. 2,,,., D iffie2hellman,,,,.,.,,., N OHNN, n, p = 2 n OHNNn(: OHNN 2 n, ), N, : e = N / n = N / log 2 p (11) (11) OHNN, OHNN N,,, ; OHNN,. 4,,,,,,,. [ ] [ 1 ] Marcus Goncalves, Kitty N iles. IPv6 [M ].,,. :, [ 2 ] Simon Haykin. Neural Networks 2A Comp rehensive Foundation ( Second Edition) [M ]. :, [ 3 ] D iffie W, Hellman M. New D irections In Cryp tography [ J ]. IEEE Transactions on Information Theory, 1976, 22 ( 6) : [ 4 ] W illiam Stallings. Cryp tography and Network Security: Princip les and Practice ( 2nd) [M ]. New Jersey: Prentice2Hall [ 5 ] Pecora L M, Carrol l T L. Synchronization in Chaotic System s [ J ]. Physical Review Letters, 1990, 64 ( 8 ) : [ 6 ] Crounse K R, Yang T, Chua L O. Pseudo2random sequence generation using the CNN universal machine with app lications to cryp tography [A ]. IEEE. Proceedings of the IEEE InternationalWorkshop on Cellular Neural Networks and their Ap2 p lications [ C ]. Singapore: World Scientific Press, [ 7 ] Veljko M ilanovic, Mona E. Zaqhloul. Synchronization of chaotic neural networks for secure communications [ J ]. IEEE International Symposium on Circuits and System s, 1996, 3: [ 8 ] Donghui Guo, Zheng L M, Zheng L L. A New Symmetric Probabilistic Encryp tion Scheme Based on Chaotic A ttractors of

9 2 : 133 Neural Networks [ J ]. App lied Intelligence, 1999, 10 ( 1) : [ 9 ] Hopfield J J. Neural Networks and Physical System s with Emergent Collective Computational Abilities [ J ]. Proceedings of the National Academy of Science, 1982, 79 ( 4) : [ 10 ] Hopfield J J. Neurons, Dynam ics and Computation [ J ]. Physics Today, 1994, 47: [ 11 ],. [M ]. :, [ 12 ] B resson Emmanuel, Chevassut O livier, Pointcheval David, et al. Probably Authenticated Group D iffie2hellman Key Ex2 change [A ]. Pierangela Samarati Proceedings of the 8 th ACM Conference on Computer and Communications Security [ C ]. New York: ACM Press, [ 13 ] Gardner E. Maximum Storage Capacity in Neural Networks [ J ]. Europhys Lett, 1987, 4 ( 4) : [ 14 ]. [M ]. :, [ 15 ],. m[ J ]., 1997, 24 (1) : A New Public2key Cryptography Ba sed on Chaotic A ttractors of Neura l Networks L IU N ian2sheng 1, GUO Dong2hui 2 ( 1. School of Computer Engineering, J imei University, Xiamen , China; 2. School of Physics and Mechanical & Electrical Engineering, Xiamen University, Xiamen , China) Abstract: A new public2key cryp tography based on chaotic attractors of neural networks is described. There is a one2way function between chaotic attractors and initial states in an Overstoraged Hopfield Neural Networks (OHNN), and each attractor and its corresponding dom ain of attraction are changed w ith permuta2 tion operations on the neural synap tic matrix. If the neural synap tic matrix is used as a trap door and changed by commutative random permutation matrix, a new cryp tography technique according to D iffie2hellman public2 key cryp tosystem is p roposed. By keep ing the random permutation operation of the neural synap tic matrix as the secret key, and the neural synap tic m atrix after permutation as public2key, a new encryp tion scheme for a public2key cryp tosystem is introduced. Security of the new scheme is discussed. Key words: neural networks; public2key cryp tosystem; chaotic attractor; matrix decomposition ( )