A New Security on Neural Cryptography with Queries

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1 Int. J. of Advanced etworng and Applcatons 437 Volume:, Issue:, Pages: () A ew Securty on eural Cryptograpy wt Queres Dr.. Prabaaran Department of Computer Applcatons, Rajalasm ngneerng College, Cenna- 6 5, Inda -mal: prabaaran_om@yaoo.com Dr. P. Vveanandan Professor & Drector, nowledge Data Centre, Anna Unversty, Cenna-6 5, Inda -mal: vve@annaunv.edu ASTRACT We can generate a secret ey usng neural cryptograpy, wc s based on syncronzaton of Tree Party Macnes (TPMs) by mutual learnng. In te proposed TPMs random nputs are replaced wt queres wc are consdered. Te queres depend on te current state of A and TPMs. Ten, TPMs dden layer of eac output vectors are compared. Tat s, te output vectors of dden unt usng Hebban learnng rule, left-dynamc dden unt usng Random wal learnng rule and rgtdynamc dden unt usng Ant-Hebban learnng rule are compared. Among te compared values, one of te best values s receved by te output layer. Te queres fx te securty aganst majorty flppng and geometrc attacs are sown n ts paper. Te new parameter H can accompls a ger level of securty for te neural ey-excange protocol wtout alterng te average syncronzaton tme. eywords: Majorty attacs, eural Syncronzaton, Queres, Tree Party Macnes Date of Submsson: October, 9 Date of Acceptance: May 7, Introducton eural cryptograpy s a metod for generatng secret nformaton over a publc cannel. efore two partners A and can excange a secret message over a publc communcaton cannel, tey ave to agree on a secret encrypton ey. Usng number teoretc metods, suc eys can be constructed over publc cannels wtout prevous secret agreements of te two partners. Te algortm as well as te complete nformaton passed between te partners s nown to a possble attacer ; neverteless te fnal ey s secret, t s nown to te two partners A and only and an attacer wt lmted computer power cannot calculate te ey. Te metod s based on te computatonal dffculty of factorzng large numbers or calculatng te dscrete logartm of large numbers [8]. Two dentcal dynamcal systems, startng from dfferent ntal condtons can be syncronzed by a common nput values wc are coupled to te two systems. Two networs wc are traned on ter mutual output can syncronze to a tme-dependent state of dentcal synaptc wegts. Te networs receve a common nput vector after calculatng ter outputs and update ter wegt vectors accordng to te matc between ter mutual outputs n every tme step. Te nput or output relatons are excanged troug a publc cannel untl ter wegt vectors are dentcal and can be used as a secret ey for encrypton and decrypton of secret messages. Te random nputs are replaced by queres n ts networ. It s based on excangng nputs between A and wc s correlated to wegt vectors of te two networs []. Te propertes of te syncronzaton process depend on te synaptc dept L and queres. Hence tere s an addtonal parameter H, wc fxes te absolute value of te local felds n te TPMs generatng te current query. As te predcton error of a dden unt, left-dynamc dden unt and rgt-dynamc dden unt s a functon of bot te overlap ρ and te local feld, te partners modfy te probablty of repulsve steps P r (ρ) f tey cange H. Te partners A and are able to adjust te dffculty of neural syncronzaton and learnng [7]. Ts paper s organzed as follows. In Secton, te basc algortm for neural syncronzaton wt queres s gven. Also lower layer spy unt, upper layer spy unt, defnton of te order parameters and transton probabltes of proposed TPMs are explaned. In Secton 3, te generaton of queres s descrbed. Syncronzaton tme of two TPMs s brefly explaned n Secton 4. Te securty aganst nown attacs wt success probablty of an attacer s presented n Sec. 5. In Secton 6, te advanced attacs are dscussed. Fnally, concluson s sown n Secton 7.

2 Int. J. of Advanced etworng and Applcatons 438 Volume:, Issue:, Pages: (). eural Syncronzaton Te wegt vectors of te two neural networs begn wt random numbers, wc are generated by Pseudo-Random umber Generators (PRGs). In tese networs, random nputs are replaced by queres. Tat s A and coose alternatvely accordng to ter own wegt vectors. Te partners A and receve a common nput vector at eac tme; ter outputs are computed and ten communcated over publc cannel [8]. If tey agree on te mappng between te current nput and te output, ter wegts are updated accordng to te learnng rule.. A Structure of Tree Party Macne Te TPMs consst of -dden unts, Y-left dynamc dden unts [3] and Z-rgt dynamc dden unts [4], eac of tem beng a perceptron wt an -dmensonal wegt vector w. Te lower layer spy unt (ϑ) s assocated wt te -nput unts x. Te upper layer spy unt (ξ) s assocated wt te dden unts (σ) [5], left-dynamc dden unts (δ), rgt-dynamc dden unts (γ) and output unt (τ). Te lower layer spy unt receves te nput from te -nput unts. Te upper layer spy unt receves te nput from te Y-left dynamc dden unts, Z-rgt dynamc dden unts, -dden unts and output unt. Te networ structure of ts TPM s sown n fg.. Upper Layer Spy Unt τ ξ Te components of te nput vectors x are bnary: [9] x j { -, +}, x m {-, +}, x { -, +} () and te wegts are dscrete numbers between -L and +L : w j { -L, -L+,,,, L-, L }, w m { -L, -L+,,,, L-, L }, w { -L, -L+,,,, L-, L }. () were L s te depts of te wegts of te networs []. Te TPM receves te nput vectors usng queres. Tese nput vectors are correlated wt te present wegt vector w () t. At odd tme steps, te partner A generates an nput vector wc as a certan overlap to ts wegts w A. At even tme steps, te partner generates an nput vector wc as a certan overlap to ts wegts w. It s based on te queres to mprove te securty of te systems. Te ndex =,.., denotes te t dden unt of TPM, m=,..,y denotes te m t left-dynamc dden unt of te TPM, =,..,Z denotes t rgt-dynamc dden unt of te TPM and j =,., denotes te nput unts [6]. Te dfferent transfer functons for dden layer are gven below: σ = sgn wj xj (3) j= δ = tan wm xm (4) m= δ σ γ γ = arctan w x (5) = x w were equaton (3) s te transfer functon of te dden unt, te equaton (4) te transfer functon of te leftdynamc dden unt and te equaton (5) te transfer functon of te rgt-dynamc dden unt. Te transfer functons for lower layer spy unt and upper layer spy unt are gven below: Lower Layer Spy Unt Fg. : A structure of Tree Party Macne wt =3, Y=3, Z=3, ϑ=, ξ= and =4. ϑ ϑ = - sgn xj w = j = j (6) ξ - sgn = δ σ γ τ (7) =

3 Int. J. of Advanced etworng and Applcatons 439 Volume:, Issue:, Pages: () were te equaton (6) s te transfer functon of te lower layer spy unt and equaton (7) te transfer functon of te upper layer spy unt. Te -dden unts of σ, Y-left dynamc dden unts [3] of δ and Z-rgt dynamc dden unts [4] of γ defne common output bt of dden layer of te networ and are gven by: β β β a b c = σ (8) = Y = Z = δ (9) = = γ () were equaton (8) s te output for te dden unts, equaton (9) te output for te left-dynamc dden unts and equaton () te output for te rgt-dynamc dden unts. Te two TPMs compare te dden layer s output bts (dden, left-dynamc and rgt-dynamc dden unts) and ten update te wegt vector to te output unt as well as partners A and tat are tryng to syncronze ter wegt vectors: A, = comp( a, b, c) () A A A A φ = wj x j τ ψ ψ β β β φ () A = w τ ψ (3) j x j were equaton () represents comparson of te output of dden, left-dynamc and rgt-dynamc dden unts of A and. Te equaton () and (3) represent output of dden, left and rgt-dynamc dden unts of A and respectvely. τ = φ (4) = Te equaton (4) represents te output vector of te output unt of te TPM.. Learnng Rules Te partners A and ntalze ter random number of wegt vectors before te start of te tranng perod. At eac tme step t, a publc nput vector s generated and te bts τ A and τ are swtced over te publc cannel. In te case of ndstngusable output bts τ A = τ, eac TPM adjust tose of ts wegt vectors for wc te dden unt, left-dynamc dden unt and rgt-dynamc dden unt s A / A / dentcal to te outputφ = τ. Tese wegts are adjusted accordng to a gven learnng rules. Tey are (a) Hebban Learnng rule for dden unts: A A A A A A w ( t+ ) = w ( t) + x τ Θ( τ φ ) Θ ( τ τ ) A w ( t+ ) = w ( t) + x τ Θ( τ φ ) Θ ( τ τ ) (5) were Θ s te Heavsde step functon, f te nput s postve ten te output s and f nput s negatve ten te functon evaluates to. (b) Random wal learnng for left-dynamc dden unts: A A A A A w ( t + ) = w ( t) + x Θ( τ φ ) Θ ( τ τ ) A w ( t + ) = w ( t) + x Θ( τ φ ) Θ ( τ τ ) (6) (c) Ant-Hebban learnng for rgt-dynamc dden unts: A A A A A w ( t + ) = w ( t) - φx Θ( τ φ ) Θ ( τ τ ) A w ( t + ) = w ( t) - φ xθ( τ φ ) Θ ( τ τ ) (7).3 Order Parameters Te process of syncronzaton tself can be descrbed by standard order parameters. Tese order parameters are: A A A A A A W W Wj W j W Q =, Q =, Q = W j (8) Te equaton (8) represents wegt dstrbuton of dden unts, left-dynamc dden unts and rgt-dynamc dden unts of A s TPM. (9) W W R R R A A A A, W W A, j j A, W W =, =, = j Te equaton (9) represents overlap between two dden unts, two left-dynamc dden unts and two rgtdynamc dden unts of A and respectvely. Te dstance between two correspondng dden unt, left-dynamc dden unts and rgt-dynamc dden unts are defned by te overlap s gven below: R ρ = + + () A, A, A, A, R j R j A A A QQ QQ j j QQ

4 Int. J. of Advanced etworng and Applcatons 44 Volume:, Issue:, Pages: ().4 Transton Probabltes A repulsve step can only occur n te t dden unt, j t left-dynamc dden unt and t rgt-dynamc dden unt, f te two correspondng outputs φ are dfferent. Te probablty for ts event s gven by te well-nown generalzaton error for te perceptron: [] p arccos ( ρ ) j π = () were equaton () represents te generalzaton error for dden, left-dynamc and rgt-dynamc unt of two TPMs. Te quantty s a measure of te dstance ρ between te wegt vectors of te correspondng dden unts, left-dynamc dden unts and rgt-dynamc dden unts and tese values are ndependent. Te values ρ determne te condtonal probablty P r for a repulsve step and P a for a attractve step between two dden unts, leftdynamc dden unts and rgt-dynamc dden unts gven dentcal output bts of te two TPMs. In te case of dentcal dstances ρ =, te values of, Y and Z are found as =3, Y=3 and Z=3. P a P r 9 3 ( ) + 3( ) = 9 3 ( ) + 9( ) 3 6( ) = 9 3 3( ) + 9( ) () (3) Te equaton () and (3) represent probablty of attractve and repulsve steps between two dden unts, two left-dynamc dden unts and two rgt-dynamc dden unts of A and respectvely. Te attacer can assgn a confdence level to eac output σ, δ and ϒ of ts dden unts, left-dynamc dden unts and rgt-dynamc dden unts. For ts tas te local feld s gven by: j w x w j x j w x = + + (4) were equaton (4) represents te local feld of dden unt, left-dynamc and rgt-dynamc dden unts of an attacer s TPM. From te below Fg., we are able to predct te probablty of repulsve steps occur more frequently n s TPM tan n A s and s for equal overlap < ρ <. So, te partners A and ave a clear advantage over a smple attac n neural cryptograpy. Pr =3,Y=3,Z=3 =4,Y=4,Z=4 =5,Y=5,Z=5 =6,Y=6,Z= Fg. : Probablty Pr ( ρ ) of repulsve steps for syncronzaton wt mutual nteracton under te condtonτ A = τ. Ten te predcton error of te probablty of dfferent output bts for an nput vectors x nducng a local feld j s gven below: ( ρj, j ) = erf ρ j ( ρj ) j Q (5) were equaton (5) represents predcton error of te local feld of dden unts, left-dynamc and rgtdynamc dden unts of an attacer s TPM. 3. Generaton of Queres As bot nputs x,m and wegts w,m are dscrete, tere are only (L+) possble solutons for te product x,m w,m [7]. Terefore, a set of nput vectors consstng of all permutaton, wc do not excange, can be depcted by countng te number c,l of products wt x,m w,m = l. Ten te local feld s gven by: ( ( ) ( ) ( ),,,,,, ) (6) L = l c c + l c c + l c c j l l j l j l l l l = were equaton (6) s te number of nputs and wegts n te local feld of TPM. Te sum njl, = cl, + c, l+ cjl, + cj, l+ cl, + c s equal to, l te number of wegts wt w, j = l and tus ndependent of x. Accordngly, one can wrte j as a functon of only L varables, because te generaton of queres cannot cange w :

5 Int. J. of Advanced etworng and Applcatons 44 Volume:, Issue:, Pages: () ( ( ) ( ) ( ),,,,,, ) (7) L = l c n + l c n + l c n j l l j l j l l l l = were equaton (7) s te nputs and sum of current wegts vectors of te local feld of two TPMs. Te nputs vectors x s connected wt zero wegts wc are selected by randomly, because tey do not determne te local feld. Te oter nput bts x,m are dvded nto L groups accordng to te absolute value l = w m, of ter correspondng wegt. In eac group, c,l nputs are selected randomly and set to x,m =sgn(w,m ). Te remanng n, l c nput bts are set to x, l,m = -sgn (w,m ). Te maxmum possbltes of te wegt vectors of an attacer s TPM s gven by: Ten l = L + (8) ( ) max (4 ) + Y + Z ln ( l ) = ( + Y + Z) ln(4l+) (9) max durng queres syncronzaton. Te partner A or use te generaton of queres (sown n Secton 3) at eac tme step A/ to generate nput vectors x, wc result n j ± H. ot partners calculate te output of ter TPMs wt queres. Ten te excange of τ A and τ te Hebban learnng rule s used to update te wegts. Ts leads to syncronzaton after t sync steps. Te dependence on te synaptc dept L of t sync, wc s nduced by two effects. Te control sgnals σ, δ, γ and τ are neglected, f wegt vectors can be descrbed as random wal wt reflecton boundares: [8]. t L (3) sync. Te probablty of repulsve steps P r depends on te overlap ρ j, but also on te quantty j / Q usng queres. Assumng unformly dstrbuted wegts as gven below: Q w L L L 3 3 j = w j j = ( + ) (3) ln [lmax] L= L=4 L=6 L=8 L= Terefore te lengt of te wegt vectors mproves proportonal to L. H = αl+ ( ϑ+ ξ ) (3) were α s te rescaled feld of H/L. L s te synaptc dept of TPM, ϑ s te lower layer spy unt vector and ξ s te upper layer spy unt vector. In eqn. (3) and (3), we can rescale t sync n order to obtan functons f L (α), wc are nearly ndependent of te synaptc dept n te case L >>: Fg. 3: Te possble values of l max of te wegt vectors of an attac s TPM. t sync = L H fl L (33) From te above fg.3, we are able to determne a large number of possbltes of wegt vectors aganst an attacer, n wc wegts are selected by queres. In eac tme step, eter A or generates te nput vectors. Te attacer cannot easly gan te wegt vector and useful nformaton from analyzng queres. 4. Syncronzaton Tme Te generaton of te nputs vectors x j as to be canged Te functon of f L (α) converges to a unversal scalng functon f(α) n te lmt L. f( α) = lm f ( α) (34) L L Te dstance fl( α) f( α) srns proporton to (L ). Terefore te unversal functon f L (α) can be found out by fnte-sze scalng.

6 Int. J. of Advanced etworng and Applcatons 44 Volume:, Issue:, Pages: () 5. Securty aganst nown Attacs Te neural ey excange protocol usng queres s effcent, secure aganst te attacs. Te dfferent values of te local feld nfluence te securty of te system. Te results mpose furter restrctons upon te usable range of te parameter H [7, 8]. 5. Success Probablty Te two partners A and use queres for te neural ey-excange. Te success probablty strongly depends on te parameter H []. Ts model s sutable for bot te majorty and te geometrc attac. P = (35) + exp( β ( H µ )) two parameter β and µ s a sutable fttng functons for descrbng P as a functon of H. P L=3 L=5 L=7 L= H Fg. 4: Success probablty of te Majorty Flppng Attac as a functon of H. Symbols denote te smulatons results for = 3, Y=3, Z=3, η=3, M = and = (H= η+.45 * L). Te poston µ of te smoot step ncreases lnearly wt te synaptc dept L as gven below: µ = α L + η s (36) Te parameter α s s te maxmum α for securty and η s te number of dden layer (dden unt, leftdynamc and rgt-dynamc dden unts) and depend on te learnng rule and te attac. Combned eqn. (35) and (36) yelds: for te success probablty of any nown attac. P α=.3 α=.35 α= L Fg. 5: Success probablty of te Majorty Flppng Attacs depends on te synaptc dept of L. Symbols denote te smulatons results for = 3, Y=3, Z=3, η=3, M = and = (α s =.45, =.4, β=8.). Te partner A and coose α = H / L accordng to te condtonα < α, P s vanses for L, ts asymptotc beavor s gven by: ( s ) L P e βη e β α α (38) Hence te success probablty P decreases exponentally wt ncreasng synaptc dept of L y ( L L ) P e (39) 6. Advanced Attacs Te attacer may mprove te success probablty P by usng addtonal nformaton exposed troug te algortm generatng te nput vectors. Te two approaces are. Te attacer could use to mprove te nternal representaton of te absolute local feld ( σ, σ,..., σ ), ( δ, δ,..., δ ) and ( γ, γ,..., γ ) n te geometrc correcton.. ac wegt vector w s correlated to te correspondng nput vector x of te generatng networ. P = (37) + exp( βη ) exp( β( α α)) L s

7 Int. J. of Advanced etworng and Applcatons 443 Volume:, Issue:, Pages: () P average of P Fg. 6: te Average success probablty of Majorty Flppng Attac as a functon of H. Symbols denote te smulatons results for = 3, Y=3, Z=3, η=3, M = and =. 6. nown Local Feld Te absolute local feld value n eter A s or s dden unts, left-dynamc unts and rgt-dynamc unts s gven by H usng queres and nows te local felds n attacer s TPM. A A 6ρ H P( φ φ ) = + exp η A ( ρ ) Q L A Q (4) From te Fg.6, we are able predct te predcton error of te local feld s ncreased aganst te geometrc attac. As tere s no qualtatve dfference compared to syncronzaton wt random nputs, t s not possble to mprove te geometrc attac by usng H as addtonal nformaton. P H=. -5 Fg. 6: Predcton error P as a functon of te local feld for A Q =, Q = and ρ= Informaton about wegt vectors Te queres gve as addtonal nformaton about te wegt vectors n A s and s TPMs wle H cannot be used drectly n te geometrc attac [7]. Te absolute local feld j for syncronzaton wt queres s lower tat te average value. j η Q j =.78 Q (4) j π observed for random nputs. Terefore te overlap: ρ j, n wj xj j = = w w x x 3 Q j j j j j (4) between nput vector and wegt vector converges to zero n te lmt, even f queres wt < j < are used. Consequently, x j and w j are nearly perpendcular to eac oter, so tat te nformaton exposed by queres s mnmzed. A gven value of H te number of wegt vectors, wc are consstent wt a gven query s stll exponentally large. 7. Concluson In te proposed TPMs, we traned te tree transfer functons n te dden layer. Tat s, dden unt usng Hebban learnng rule, left-dynamc dden unt usng Random wal rule and rgt-dynamc dden unt usng Ant- Hebban learnng rule ncluded wt queres. Also, te queres ncrease te probablty of repulsve steps for an attacer durng te syncronzaton. In addton, te metod receves a new parameter, wc can be adapted to gve optmal securty. Te attacer s at a dsadvantage compared to A and because needs a ger absolute value of te local feld tan te partners n order to syncronze on average. Hence, t s possble to adjust te new parameter H combned wt lower layer spy unt vector and upper layer spy unt vector. Te partners A and syncronze fast, but s not successful regardless of te attac metod. References [] A. ngel and C.Van Den roec, Statstcal Mecancs of Learnng, (Cambrdge Unversty Press, Cambrdge ). [] W. nzeland I. anter, Interactng neural networs and cryptograpy, Advances n Sold State Pyscs, by. ramer (Sprnger, erln), 4,, [condmat/3].

8 Int. J. of Advanced etworng and Applcatons 444 Volume:, Issue:, Pages: () [3]. Prabaaran, P.Loganatan and P. Vveanandan, eural Cryptograpy wt Multple Transfer functon and Multple Learnng Rule, Internatonal Journal of Soft Computng, 3(3), 8, [4]. Prabaaran, P. aruppucamy and P. Vveanandan, A ew approac on eural Cryptograpy wt Dynamc and Spy unts usng multple transfer functons and learnng rules. Asan Journal of Informaton Tecnology, 7(7), 8, [5]. Prabaaran, P. Saravanan and P. Vveanandan P (8), A ew tecnque on eural Cryptograpy wt Securng of lectronc Medcal Records n Telemedcne System, Internatonal Journal of Soft Computng, 3(5), 8, [6] A. Ruttor, W. nzel, L. Sacam and I. anter, eural cryptograpy wt feedbac, Pyscal Revew dton, 69, 4,- 8. [7] A. Ruttor, eural Syncronzaton and Cryptograpy, P. D. Tess,6. [8] A. Ruttor, W. nzel and I. anter, eural cryptograpy wt queres, Pyscs Rev.. 5(), 5, -. [9] A. Ruttor, I. anter and W. nzel, Dynamcs of neural cryptograpy, Py. Rev.., 75(5), 6, [] A.Ruttor, I. anter, R. ae and W. nzel, Genetc attac on neural cryptograpy, Py. Rev.., 73(3), 6, [] R. Mslovaty,. len, I. anter and W. nzel, Publc cannel cryptograpy by syncronzaton of neural networs and caotc maps, Py. Revew Letter 9(), 3, 87. Unversty from 978. He vsted Sngapore, Malaysa, Japan, anglades, Sultanate of Oman and USA for presentng researc papers and Carng Sessons. He as publsed more tan 8 researc papers n natonal and nternatonal journals. Hs areas of researc are eural etwors, Internet Securty and Software Relablty. Autors ograpy PRAAARA, e as receved M.Sc n Computer Scence from Anna Unversty. He ad done P.D from Anna Unversty. Hs areas of nterest are networ securty, vsual programmng and wreless tecnology. He ad publsed Sx Internatonal Journals and two papers n atonal Journals and e as presented fve papers n nternatonal conferences. VIVAADA PRIYASAMY receved s Master of Scence n Appled Matematcs from Madras Unversty n 978 and Doctor of Plosopy from Anna Unversty n 987. Also, e obtaned s postgraduate degree n Master of ngneerng n Computer Scence and ngneerng from Anna Unversty n 995. He s worng as Professor of Matematcs, Department of Matematcs n Anna