Further Research on Registration System with Vandermonde Matrix

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1 IJCSI Iteratioa Joura of Computer Sciece Issues, Vo 10, Issue 1, No 1, Jauary 2013 wwwijcsig 614 Further Research o Registratio System with Vadermode Matrix Nig Huag 1, Xia-tog Huag 2, Jig-i Re 3, Xia-we He 2, Yag Liu 2 1 Ceter of Moder Educatioa Techoogy, Gaa Nma Uiversity Gazhou, , Chia hgzjx@qqcom 2 Coege of Mathematics ad Computer Sciece, Gaa Nma Uiversity Gazhou, , Chia 3 Coege of Commuicatio ad Media, Gaa Nma Uiversity Gazhou, , Chia Abstract We propose a improved software registratio system from our previous research Our improvemets are maiy as foows (1) Chagig basic fied to make the scheme suitabe f a characters (2) Chagig ecryptio ad decryptio fmuae to make the scheme me compex (3) Usig the techique of etter decompositio ad compositio to make the scheme me deceptive to a possibe adversary (4) Usig mobie phoe i the system to ehace the security Experimeta resuts ad aaysis show that the improvemets are successfu ad the scheme is viabe ad secure Keywds: software, registratio, Vadermode matrix 1 Itroductio Copy protectio f computer software started a og cat-ad-mouse strugge betwee pubishers ad crackers These were (ad are) programmers who woud defeat copy protectio o software as a hobby, add their aias to the tite scree, ad the distribute the cracked product to the etwk BBSes Iteret sites that speciaized i distributig uauthized copies of software [1] Research o the topic ever eds I [2], we proposed a scheme of registratio with Vadermode matrix i a Gaois fied GF (p), which is a appicatio of Hi cipher [8] I this paper, we further discuss the improvemets of the method Our improvemets are maiy as foows (1) Usig the Gaois fied GF (2 m ) istead of GF (p) to make the scheme suitabe f a characters (2) Usig matrix equatio V 1 X+C istead of V 1 X to make the scheme me compex (3) Usig the techique of etter decompositio ad compositio to make the scheme me deceptive to a possibe adversary (4) Usig mobie phoe i the system to ehace the security The rest of the paper is gaized as foows I Sectio 2, we briefy itroduce our igia research i GF (p) I Sectio 3, we propose some ove ideas to improve our igia scheme I Sectio 4, we desig the registratio system I Sectio 5, we give experimeta resuts ad aaysis We cocude the paper i Sectio 6 2 Origia Scheme Agree o permissio cotro character strig such as P ROF ESSIONALV ERSION o both sides of the ved ad user The we take differet characters µ 1, µ 2,,, µ, from hard id [7] to create a Vadermode matrix i a Gaois fied GF (p), where p is a prime That is V = V (µ 1, µ 2,, µ ) µ = 1 µ 2 µ mod p (1) µ 1 1 µ 1 2 µ 1 The we obtai a determiat fmua det(v ) = (p + µ i µ j ) mod p (2) 1 i<j Copyright (c) 2013 Iteratioa Joura of Computer Sciece Issues A Rights Reserved

2 IJCSI Iteratioa Joura of Computer Sciece Issues, Vo 10, Issue 1, No 1, Jauary 2013 wwwijcsig 615 It foows from µ i s are differet from each other that V is ivertibe The fast computatio of the iverse of V is A = V 1 (µ 1, µ 2,, µ ) = HL mod p (3) where H is a upper triaguar matrix ad L a ower triaguar oe The eemets of each ca be obtaied from recursive fmuae Let X = (x 1, x 2,, x ) T be the pai text, defie a iear mappig: X (y 1, y 2, y ) T = V X mod p (4) as a ecryptio agithm, which is equivaet to the foowig iear fuctios: y 1 = x 1 + x x y 2 = µ 1 x 1 + µ 2 x µ x mod p (5) y = µ 1 1 x 1 + µ 1 2 x µ 1 x The decryptio agithm is X = V 1 AY mod p (6) which is equivaet to the foowig iear fuctios: x 1 = a 11 y 1 + a 12 y a 1 y x 2 = a 21 y 1 + a 22 y a 2 y mod p (7) x = a 1 y 1 + a 2 y a y I der to miimize the set of ecessary fmuae o the user s side, we use A = V 1 o the ved s side as the ecryptio key whie V o the user s side as the decryptio key O the ved s side, we take the permissio cotro strig as the pai text X We compute V 1 X mod p as the cipher text O the user s side, we use X = V Y mod p to verify the registratio Our igia scheme takes p = 37 as the moduus, choose 10 umbers ad 26 capita etters ad the symbo $ Lower case etters are coverted to upper oes Other symbos are iged The key space is (37 + 1) f a give 3 Improved Scheme 31 Basic fied of the scheme To make the scheme uiversa, we defie our computatio i the fied GF (2 m ) I fact, GF (2 m ) is cogruet to Z 2 [x]/f(x), where Z 2 [x] is the poyomia rig over Z 2, f(x) is a primitive poyomia of Z 2 [x] I Z 2 [x]/f(x), the additio of poyomias α(x) ad β(x) is defied by σ(x) = α(x) + β(x)with the character of Z 2 beig 2, f arbitrary α(x) Z 2 [x], we have α(x) + α(x) = 0The mutipicatio of poyomias ad is defied by π(x) = α(x) + β(x) mod f(x)see me detais i [4, 5, 6] If we take m = 8,GF (2 8 ) is just the etter set of exteded ASCII This meas we ca use every symbo i a pai text However, that wi cause the probem of upritabe etters i the registratio strig We use the decompositio of etters to sove this probem by spittig a etter ito two, each is i the rage of {0, 1, 2,, 9, A, B, C, D, E, F } Furtherme, we pus each vaue with A i the geeratio of a registratio strig F exampe, if a etter with the vaue 0 is used f a registratio strig, we reay see it as A Simiary, we see B f vaue 1, C f vaue 2 through P f vaue F This ca easiy be reaized i C aguage Suppose c is i the cipher text, the decompositio ad coversio is expressed by c 1 = c/0x10 + A c 2 = c%0x10 + A This method aso icreases the compexity of the ecryptio Bravo! It actuay butters both sides of our bread 32 Fast computatio of V 1 = HL We deveop the method i [3] to use it i GF (2 m ) Let L be the matrix whose rows are associated with the coefficiets of the poyomias ψ 1 (s) = 1 (8) ψ i = (s + µ j )ψ i 1 (s) L ca be deoted by L(1, s,, s 1 ) T = (ψ 1 (s), ψ 2 (s),, ψ (s)) T L = (9) 1 2 Let ii = 1, ij = 0(i j) (10) i+1,1 = (µ i ) i,j 1 i+1,j = i,j 1 j = 2, 3,, i, i = 1, 2,, (11) Copyright (c) 2013 Iteratioa Joura of Computer Sciece Issues A Rights Reserved

3 IJCSI Iteratioa Joura of Computer Sciece Issues, Vo 10, Issue 1, No 1, Jauary 2013 wwwijcsig 616 Let the iitia vect h = (c 1, c 2,, c ) T be determied from the partia fractio expasio 1 (s + µ 1 ) (s + µ ) = c c s + µ 1 s + µ Deote H by h ij = Deote d(s) by Let H = (h 1, h 2,, h ) T (12) { 1 ψ j+1 (µi) if i j 0, if i > j (13) d(s) = (µ 1 + s, µ 2 + s,, µ + s) T (14) h i 1 = h i d(µ i ), i =, 1,, 2 (15) edig at h 1 = (1, 0,, 0) T The right side of (14) is the ier product of h i ad d(µ i ), ie, if u = (u 1, u 2,, u ) T,v = (v 1, v 2,, v ) T the h i d(µ i ) = (u 1 v 1, u 2 v 2,, u v ) T (16) It foows that the fmuae to compute the eemets i the upper triaguar matrix H are give by { 1 ψ h ij = j+1 (µi), if i j (17) 0, if i > j where is the matrix cotaiig cipher text This is equivaet to m groups of iear mappigs as foows y 11 = a 11 x a 1 x 1 + c 11 y 21 = a 21 x a 2 x 1 + c 21 (21) y 1 = a 1 x a 2 x 1 + c 1 y 12 = a 11 x a 1 x 2 + c 12 y 22 = a 21 x a 2 x 2 + c 22 (22) y 2 = a 1 x a 2 x 2 + c 2 y 1m = a 11 x 1m + + a 1 x m + c 1m y 2m = a 21 x 1m + + a 2 x m + c 2m (23) y m = a 1 x 1m + + a 2 x m + c m I the fied of character 2, the decryptio is expressed as foows X = V (Y C) = V (Y + C) This is equivaet to m groups of iear mappigs as foows ψ j+1(µ i ) = j 1 k=1,k i (µ i + µ k ) (18) 33 Improvemets of the iear mappigs x 11 = y y1 x 21 = µ 1 y µ y1 x 1 = µ 1 1 y µ 1 y 1 (24) I the fied GF (2 8 ), Let x 11 x 12 x 1m x X = 21 x 22 x 2m (19) x 1 x 2 x m be the pai text, ad c 11 c 12 c 1m c C = 21 c 22 c 2m (20) c 1 c 2 c m be the same size as XThe V 1 X + C x 12 = y y2 x 22 = µ 1 y µ y2 x 2 = µ 1 1 y µ 1 y 2 x 1m = y1m + + ym x 2m = µ 1 y1m + + µ ym x m = µ 1 1 y1m + + µ 1 where y ij = y ij c ij = y ij + c ij y m (25) (26) This cipher is obviousy me compex compared with the igia oe The security is ehaced Copyright (c) 2013 Iteratioa Joura of Computer Sciece Issues A Rights Reserved

4 IJCSI Iteratioa Joura of Computer Sciece Issues, Vo 10, Issue 1, No 1, Jauary 2013 wwwijcsig 34 Use of mobie phoe as a receiver I our igia scheme, we assumed that the user got the registratio strig vie web service emai Now we propose the registratio strig ca aso be received as a message via a mobie phoe Step1 The user pays f the software ad submits persoa ifmatio, eg hard id, ame, mobie phoe umber to the server via web Step2 The server checks the submissio Step3 The sever creates a registratio strig Step4 The server seds the registratio strig to the user s mobie phoe via wireess tue Step5 The user reads the message ad register the software The fow of the process is show by Fig Registratio steps 617 Agithm: Step1 Seects differet eemets from id, fms a ew id1 Step2 Creates matrices of ower triaguar L ad upper triaguar H from id1 as show i subsectio 32 Step3 Computes the iverse of Vadermode matrix V 1 = HL Step4 Appeds ecessary dots to the permissio cotro strig p1 to fit the dimesio, puts the resut i p2 Step5 Puts p2 ito matrix X ad creates a matrix C from ame which matches dimesios i step4, usig the eemets of ame circuary x11 x12 x1m x21 x22 x2m X= x1 x2 xm c11 c21 C= c12 c22 c1m c2m c1 c2 cm Step6 Computes V 1 X + C Fig Registratio steps Step7 Decomposes Y ad adds 0 A0 to each eemet respectivey to update Y Step8 Takes eemets from Y to create a registratio strig ike r = xxxxxx xxxxxx xxxxxx 4 Registratio scheme Set preimiary coditios o both sides of the ved ad user: A character strig as a permissio cotro strig p1 dimesio 41 The Creatio of registratio strig The user submits the message as foows Hard id(id f sht), ame, mobie phoe umber(mph f sht) O the ved s side, the server computes as foows: Iput: id, ame, mph Output:Registratio strig ike r = xxxxxx xxxxxx xxxxxx The the server seds the registratio strig above to the user s mobie phoe via wireess tue 42 The use of registratio strig Readig the registratio strig from mobie phoe, the user keys i to register the software Iput: Registratio strig ike r = xxxxxx xxxxxx xxxxxx Output: T RU E/F ALSE Agithm: Step1 Takes eemets from r to create a matrix Y Copyright (c) 2013 Iteratioa Joura of Computer Sciece Issues A Rights Reserved

5 IJCSI Iteratioa Joura of Computer Sciece Issues, Vo 10, Issue 1, No 1, Jauary 2013 wwwijcsig 618 Step2 Miuses A from each eemet of Y to update Y Step3 I the same way as step6 i ast subsectio, creates matrix C Step4 I the fied of character 2, computes Y C = Y + C Step5 Computes Vadermode matrix V from oca hardware id, id 1 Step6 Computes X = V (Y C) = V (Y + C) to get a possibe pai text Step7 Obtais p 2 from X Step8 Removes redudat dots if there are ay at the ed to get p 1 Step9 Compares p 1 with the preimiary oe Step10 The resut is T RUE F ALSE If the verificatio is successfu, the registratio wi be approved, otherwise it wi be defied A registered software picks up the registratio strig automaticay ad verifies it accdig to the above routie to decide which fuctios the software ca use 5 Experimeta resuts ad aaysis Embeds i the software o both sides of the ved ad user: p 1 = V IP versio as a permissio cotro strig Agree o dimesio = 6 51 The Creatio of registratio strig The user submits the messages as foowsid = 5V P 0567Q, ame = Aice, mph = O the ved s side, the server computes as foows Iput: id = 5V P 0567Q, ame = Aice Output: r = GEP CBJ LBDNF G P IMAMA AIHJN E Agithm: Step1 Seects differet eemets from id, fms a ew id 1 : id 1 = 5V P 067Q Step2 Creates ower triaguar ad upper triaguar matrices: L = ad H = BE A 2B B0 CC DF BE A2 8B B A Step3 ComputesV 1 = HL 07 A3 C0 A A5 10 BA F 5 D3 06 = B1 AA F 0 D7 F E 73 CD F D A F D BF 5C 7B 7B AD Step4 Appeds 2 dots to the permissio cotro strig to fit the dimesio, p 2 = V IP versio Step5 Puts p 2, ame ito matrices: V s I i X = P o v 50 6F 76 6E e 65 2E r 72 2E the A C = i c e A i c e A 6C which matche dimesio i step4 Step6 Computes 64 F 8 F 2 C0 19 C0 B1 08 3D D4 Copyright (c) 2013 Iteratioa Joura of Computer Sciece Issues A Rights Reserved

6 IJCSI Iteratioa Joura of Computer Sciece Issues, Vo 10, Issue 1, No 1, Jauary 2013 wwwijcsig 619 Step7 Decomposes Y ad adds A to each eemet respectivey to get G L P A E B I I P D M H C N A J B F M N J G A E Step8 Takes eemets from Y to create a registratio strig: r = GEP CBJ LBDNF G P IMAMA AIHJNE The the server seds the registratio strig to the user s mobie phoe via wireess tue 52 The use of registratio strig Readig the registratio strig from mobie phoe, the user keys i to register the software Iput: r = GEP CBJ LBDNF G P IMAMA AIHJN E Output: T RUE/F ALSE Agithm: Step1 Takes eemets from r = GEP CBJ LBDNF G P IMAMA AIHJN E to get a matrix G L P A E B I I P D M H C N A J B F M N J G A E Step2 Miuses A from each eemet of Y to get a ew matrix 64 F 8 F 2 C0 19 C0 B1 08 3D D4 Step3 I the same way as step5 i ast subsectio, creates A i 6C 69 C = i c c e e A A Step4 I the fied of character 2, computes E A9 Y C = Y + C = 70 A3 D2 6D B8 Step5 Computes Vadermode matrix V = 96 D9 CD D0 5F AF 0B DA F 0 CB CA 33 B4 6D CD 5D A1 from oca hardware id = 5V P 0567Q, id1 = 5V P 067Q Step6 Computes a possibe pai text X = V (Y C) = V (Y + C) V s I i = 50 6F 76 6E P o v 65 2E e 72 2E r Step7 Obtais p 2 = V IP versio from X Step8 Removes 2 redudat dots at the ed to get p 1 = V IP versio Step9 Compares p 1 with the preimiary oe Step10 The resut is T RUE 53 Aaysis of the resuts I our ove scheme, the computatios are perfmed i GF (2 8 ) to make the scheme suitabe f the whoe exteded ASCII tabe Meawhie, the key space expads from (37 + 1) f a give ( ) The uses of mobie phoe, user s ame, the decompositio ad compositio of etters aso ehace the cipher Experimeta resuts show the success of the scheme 6 Cocusios It foows from Experimeta resuts ad aaysis that the ove scheme is ehaced We get better resuts i our further research Copyright (c) 2013 Iteratioa Joura of Computer Sciece Issues A Rights Reserved

7 IJCSI Iteratioa Joura of Computer Sciece Issues, Vo 10, Issue 1, No 1, Jauary 2013 wwwijcsig 620 Ackowedgemets 1)The auths woud ike to express their thaks to the hard wk of the edits ad reviewers of this paper 2)This wk was suppted by the fud from Natura Sciece of Jiagxi Provice of Chia uder Grat No20114BAB The auths woud ike to express their thaks to the Committee of the fud Refereces [1] Copy protectio f computer software, protectio2012 [2] N Huag, Permissio cotro of software based o registratio system with Vadermode matrix i a Gaois fied,i Istrumetatio & Measuremet, Ses Netwk ad Automatio (IMSNA), 2012 Iteratioa Symposium o, 2012, Vo 2, pp [3] SH Hou, ad E Hou, Triaguar Facts of the Iverse of Vadermode Matrices, I Proceedigs of the Iteratioa MutiCoferece of Egieers ad Computer Scietists 2008 Vo 2, IMECS 2008, pp19-21 [4] Darre Hakerso, Afred Meezes ad Scott VastoeGuide to Eiptic Curve Cryptography, Beri:Spriger,2003 [5] Roberto M Avazi, Heri Cohe, Christophe Doche,Gerhard Frey, Taja Lage, Kim Nguye, ad Frederik Vercautere, Hadbook of eiptic ad hypereiptic curve cryptography, Lodo:Tay & Fracis Group,2006 [6] Wiiam JGibert ad WKeith Nichoso, Moder Aegbra with Appicatios,Secod EditioNew Jersy:Joh Wiey &Sos,Ic,2003 [7] SDS Moteiro ad RF Erbacher, Exempifyig Attack Idetificatio ad Aaysis i a Nove Fesicay Viabe Sysog Mode,I Washigto:IEEE Computer Society,Proceedigs of the Third Iteratioa Wkshop o Systematic Approaches to Digita Fesic Egieerig,2008,pp57-68 [8] L S Hi, Cryptography i a Agebraic Aphabet, The America Mathematica Mothy, Vo 36, No 6 (Ju - Ju, 1929), pp Nig Huag, b i 1958, received Master s degree i appied mathematics ad computer sciece from Jiagxi Uiversity, Chia i 1991, awarded sei egieer of the Idustria ad Commercia Bak of Chia i 2001 He is ow with Ceter of Moder Educatioa Techoogy, Gaa Nma Uiversity, Gazhou, Chia,as a associate profess His research iterests icude ifmatio security ad digita campus Xia-tog Huag, b i 1966, received Doct s degree i computatioa mathematics from Hua Uiversity, Chia i 2006 He is ow with Coege of Mathematics ad Computer Sciece, Gaa Nma Uiversity, Gazhou, Chia as a profess His research iterests icude computatioa mathematics ad ifmatio security Jig-i Re, b i 1980, received Master s degree i computer sciece from Wuha Uiversity of Techoogy, Chia i 2006 She is ow with Coege of Commuicatio ad Media, Gaa Nma Uiversity, Gazhou, Chia, as a ecturer Her research iterests icude Iteiget computatio ad simuatio Xia-we He, b i 1974, received Master s degree i computer sciece from Nachag Uiversity, Chia i 2007 He is ow with Coege of Mathematics ad Computer Sciece, Gaa Nma Uiversity, Gazhou,Chia, as a associate profess His research iterests icude etwk security ad image recogitio Yag Liu, b i 1972, received Master s degree i computer sciece from Nachag Uiversity, Chia i 2007 He is ow with Coege of Mathematics ad Computer Sciece, Gaa Nma Uiversity, Gazhou,Chia, as a ecturer His research iterests icude agithm optimizatio ad image recogitio Copyright (c) 2013 Iteratioa Joura of Computer Sciece Issues A Rights Reserved