A New Construction Method of Digital Signature Algorithms

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1 IJCSNS Iteratioal Joural o Computer Sciece ad Network Security VOL.6 No. December 06 A New Costructio Method o Diital Siature Alorithms Thuy Nuye Đuc ad Du Luu Ho Faculty o Iormatio Techoloy Ho Chi Mih City Techical ad coomic Collee Faculty o Iormatio Techoloy Military Techical Academy Summary The article presets a ew costructio method o diital siature alorithms based o diiculty o the discrete loarithm problem. From the proposed method the dieret siature schemes ca be deployed to choose suitably or applicatios i practice. Key words: Diital siature; Diital siature alorithm; Discrete loarithm problem.. Problem Curretly the diital siature has bee widely applied to the ields o e-govermet e-commerce... i the world ad has bee iitially deployed applicatios i Vietam to meet the autheticatio reuiremets or the orii ad the iterity o iormatio i electroic trasactios. However the iitiative research - developmet o ew diital siature schemes to meet the reuiremets or product saety euipmet desi - mauacture ad iormatio security i the coutry has always bee essetial problem arisi. I the coutry a umber o research results i this ield have bee published [] [] [] [] ad implemeted i practical applicatios. I this article the authors propose a ew costructio method o siature schemes based o diiculty o the discrete loarithm problem i the ield o iite elemets. As well as methods that have bee proposed i [] [] [] [] a advataes o the ewly proposed method here is that it ca be used or the purpose o developi dieret diital siature schemes to choose suitably to the reuiremets o applicatios i practice.. Costructio o diital siature alorithm. Costructio method This ewly proposed sheme is built up based o diiculty o the Discrete Loarithm Problem []. Discrete loarithm problem DLP(p ca be stated as ollows: Let p be prime umber r is the birth particle o the roup Zp*. For each positive iteer y Zp* id x satisyi the euatio: x = y Here the discrete loarithm problem is used as a oe-way uctio i ormatio o the key o etities i the same system with the commo parameter set {p}. It is easy to see that i x is a secret parameter calculatio o the public parameter y rom x ad systematic parameters {p} is absolutely easy. However the opposite is very diicult to implemet ie rom y ad {p} the calculatio o the secret parameter x is ueasible i practical applicatios. It should be oted that accordi to [6] ad [7] i order or discrete loarithm problem to be diicult p selected must be lare eouh with: p bit. Alorithm or the problem DLP(p ca be writte as a uctio calculati alorithm DLP(p(. with the iput variable y ad uctio value is the root x o the euatio: x = DLP ( p ( y This siature scheme built up based o the ewly proposed method allows etities sii i the same system to share the parameter set {p} where each member U o the system chooses oesel the secret key x satisyi: < x < ( p calculate ad disclosure the parameter: x It also should be oted that the secret parameter x must be chose so that the calculatio o DLP(p (y is diicult. With the above stated choice oly the sier U kows the value o x so the perso who kows x is eouh to autheticate is U. Assumi that the secret key o the sier x is radomized i the rae (p ad the correspodi public key y is ormed rom x i accordace with: x (. Here p is the chose prime umber so that solutio o the problem DLP(p (y is diicult is the birth particle o the roup Zp* has the deree o with (p-. Assume that (r is the siature o the messae M u is oe value i the rae ( ad r is calculated rom u by the ormula: r u (. Ad s calculated rom v by the ormula: s v (. Here: v is also oe value i the rae (. Also assume that the veriyi euatio o the scheme is Mauscript received December 06 Mauscript revised December 0 06

2 IJCSNS Iteratioal Joural o Computer Sciece ad Network Security VOL.6 No. December 06 s ( M ( r ( M ( r ( M ( r r y With ( r is the uctio o r ad s. Cosider the case: ( r = r s (. Where k is a radomly chose value i the rae (. Set: k = Z (. The the veriyi euatio ca be take to the orm: ( M Z ( M Z ( M Z s r y (.6 From (. (. (. ad (.6 we have: ( M Z u ( M Z x. ( M Z. (.7 From (.7 ier: v ( [ u ( So: x ( ]mod v = ( u ( ( (.8 x ( ( mod (.9 O the other had rom (. (. ad (. we have: ( v u mod = k (.0 From (.9 ad (.0 we have: [ u ( ( Or: x ( = k = k From (. ier: [ u ( ( x ( u = [( ( ( k x ( ( u]mod ( ( ]mod ( ( ]mod (. (. From (. the irst compoet o siature is calculated by (.: r u ad the secod compoet is calculated by (.: s v with v calculated by (.9: v = [ u ( ( x ( ( ]mod From here a orm o siature scheme correspodi to the case: ( r = r s k is show as Table Table ad Table below. Table. Alorithm or ormatio parameter ad key Iput: p x. Output: y. []. select h: <h<p ( p / []. h []. i ( = the o to [] y x []. retur {y} (i p: primes satisyi coditios: p = N N=. (ii xy: secret public keys o sii object U. Table. Alorithm or ormatio o siature Iput: p x M. Output: (r. []. select k: <k< []. []. []. w w mod w w mod u ( w ( k x w mod [8]. [9]. r u v u w x s v ( w mod [0]. []. []. retur (r (i M: the messae to be sied with: (ii (r: siature o U o M. M Table. Alorithm or veriyi siature Iput: p y {M(r}. []. Z ( r []. []. w ( w ( []. A s w B w w r y i (A=B the {retur true} else {retur alse} {0 } (i M (r: the messaes siature eed veriyi..

3 IJCSNS Iteratioal Joural o Computer Sciece ad Network Security VOL.6 No. December 06 (ii I the retur is true the iterity ad orii o M are coirmed. Coversely i the retur is alse M is deied the orii ad iterity. It should be oted that the siature created here is ot ecessarily the pair o (r. From the Table shows that the value v ca be selected as the secod compoet o the siature istead o s thus reduce oe calculatio step i the procedure or ormatio o siature. Ideed i the hypothesis o the veriyi euatio o the scheme is ( M ( r ( M ( r ( M ( r r y (. ad: v k ( r = r (. Set: k = Z The rom (. (. ad (. we also have: ( M ( r u ( M ( r x. ( M ( r. From here alorithms or ormatio ad veriyi siature o the orm o the scheme correspodi to ew assumptios ive i Table ad Table as ollows: Table. Alorithm or ormatio o siature Iput: p x M. Output: (r. []. select k: <k< []. k []. []. w ( w w w mod ( w w mod u ( w ( k x w mod [8]. [9]. r u v u w x []. retur (r ( w mod [0]. Table. Alorithm or veriyi siature Iput: p y {M(r}. []. Z ( r []. []. w []. A B w w r y i ( B A = the {retur true }. Several alorithms or siature built up uder the proposed method.. The irst scheme a Structure ad operatio The irst siature scheme proposed here - symbols LD is built up uder Table ad i sectio A with selectios: ( = H ( M ( = Z ( = H ( M. Alorithms or ormatio o parameter ad key alorithm or siature ad veriyi siature o the scheme are described i the Table 6 Table 7 ad Table 8 below. Table 6. Alorithm or ormatio o parameter ad key Iput: p x. Output: y H(.. []. select h: <h<p []. ( p / h []. i ( = the oto [] [] y. (. { 0} H : Z []. select retur {yh(.} - H(.: Hash uctio (SHA MD... < < Table 7. Alorithm or sii messaes Iput: p H(. x M. Output: (r. []. = H (M []. select k: <k< [] k u Z k x mod ( ( p (. []. r (. v ( u Z x mod (. s (. [8]. retur (r Table 8. Alorithm or veriyi siature Iput: p H(. y M (r. []. = H (M []. A s (.6 []. w r s (.7 w B r y (.8 []. i ( A = B the {retur true } b Correctess o the scheme LD The thi to be proved is: Let p are primes with (p- H : { 0} Z < < p < k x <

4 6 IJCSNS Iteratioal Joural o Computer Sciece ad Network Security VOL.6 No. December 06 x = H ( M ( Z ( k x mod ( u Z x mod Z k u = r u s =. I: w = r s A = s w B = r y the: A = B. Correctess o the ewly proposed scheme is proved as ollows: From (. (. ad (.6 we have: A = s (. Z x. = Z x. (.9 From (. (. (.7 ad (.8 we also have: w B = r y ( r. s Z x. x. Z x. (.0 From (.9 ad (.0 ier the thi to be proved: A = B Saety level o the scheme LD I orm o the ewly proposed scheme the public key is ormed rom the secret key based o diiculty o the discrete loarithm problem DLP(p. Thereore i the parameters {p} is selected or the problem DLP(p to be diicult the saety level o the ewly proposed scheme i terms o resistace to attacks disclosi secret key will be assessed by the level o diiculty o the problem DLP(p. It should be oted that i order or DLP(p to be diicult the parameters {p} ca be selected similarly to DSA [6] or GOST R.0-9 [7] with: p bit 60bit 60bit. The Alorithm or veriyi siature (Table 8 o the scheme LD shows ay pair o (r will be recoized as a valid siature o U o a messae M i it meets the coditio: ( s. r s r y (. Here: U is sii object owi a public key y ad = H (M are represetative value o the messae M to be veriied. To id (r rom (. the irst way is to select a value or r i advace the calculate s. The (. will be s b a s (. Or i the secod way is select s i advace the calculate r. The (. will be r r = b (. I both two cases a ad b costats. It is easy to see that solutios o (. ad (. to id s ad r is more diicult tha solutio o the discrete loarithm problem DLP(p... The secod scheme a Structure ad operatio The secod siature scheme - symbols LD is built up uder the method stated i Table ad i sectio A with selectios: (MZ = Z (MZ = H(M (MZ = H(M. The alorithm or ormatio o parameter ad key is similar to that i the scheme LD (Table 6 alorithms or siature ad veriyi siature o the scheme are described i Table 8 ad Table 0 below. Table 9. Alorithm or sii messaes Iput: p H(. x M. Output: (r. []. select k: <k< []. (. []. = H (M (. w = Z mod (. []. u ( w ( k x w mod (. r (. v w ( u x mod (.6 [8]. retur (r Table 0. Alorithm or veriyi siature Iput: p H(. y M (r. []. = H(M []. w r v w []. A []. B ( r y i ( A = B the {retur true } (.7 (.8 (.9 b Correctess o the scheme LD The thi to be proved is: Let p are primes with H : { 0} Z (p- < < p < k x < x Z k = = H ( M w = Z mod u = ( w ( k x w mod r u v = w ( u x mod. I: w r v w = A B = ( r y the: A = B. Correctess o the ewly proposed scheme is proved as ollows: From (.7 ad (.8 we have: w A v ( r. = Z Z.. x. Z x. (.0

IJCSNS Iteratioal Joural o Computer Sciece ad Network Security VOL.6 No. December 06 7 From (. (. ad (.9 we also have: B = r y ( x. (. From (.0 ad (. ier the thi to be proved: A = B.

9 0 shows that ay pair o (r will be recoized as a valid siature i the scheme eerated rom a messae M i it meets the coditio: ( v r. ( r y (. Similarly (. to id r ad v rom solutio o (.

5 IJCSNS Iteratioal Joural o Computer Sciece ad Network Security VOL.6 No. December 06 7 From (. (. ad (.9 we also have: B = r y ( x. (. From (.0 ad (. ier the thi to be proved: A = B. c Saety level o the scheme LD From the Alorithm or veriyi siature (Table 0 o the scheme LD shows that ay pair o (r will be recoized as a valid siature i the scheme eerated rom a messae M i it meets the coditio: ( v r. ( r y (. Similarly (. to id r ad v rom solutio o (. is more diicult tha solutio o the problem DLP (p.. Coclusio Produce ad check procedures o lectroic Diital Siature based o Asymmetric Cryptoraphic Alorithm. Govermet Committee o the Russia or Stadards (i Russia. Thuy N.D received the B.S rom HUFLIT Uiversity i 00 ad M.S deree rom Faculty o Iormatio Techoloy Military Techical Academy i 0. My research iterests iclude cryptoraphy commuicatio ad etwork security. Du L.H is a lecture at the Military Techical Academy (Ha Noi Viet Nam. He received the lectroics ieer deree (989 ad Ph.D (0 rom the Military Techical Academy. The article proposes a method o diital siature scheme desi based o diiculty discrete loarithm problem. A advataes o the ewly proposed method is that it ca be used or developi dieret diital siature schemes to choose suitably or applicatios i practice. Siature schemes o LD ad LD preseted here has somewhat showed the easibility o the ewly proposed method. Reereces [] Luu Ho Du Le Dih So Ho Nhat Qua Nuye Duc Thuy DVLOPING DIGITAL SIGNATUR SCHMS BASD ON DISCRT LOGARITHM PROBLM the ihth Natioal Scietiic Meeti o Basic Research ad Iormatio Techoloy Applicatios (FAIR 0 ISBN: [] Luu Ho Du Hoa Thi Mai Nuye Huu Mo A orm o siature scheme built up based o the diital aalysis problem the ihth Natioal Scietiic Meeti o Basic Research ad Iormatio Techoloy Applicatios (FAIR 0 ISBN: [] Luu Ho Du Ho Noc Duy Nuye Tie Gia Nuye Thi Thu Thuy Developmet o a ew orm o diital siature scheme the Proceedis o the Sixteeth Natioal Semiar: Some Selected Issues o Iormatio Techoloy ad Commuicatio - Da Na. [] Hoa Thi Mai Luu Ho Du A orm o siature scheme built up based o the diital aalysis problem ad the suare root problem Joural o Sciece ad ieeri - Military Techical Academy No. 7 (Joural o IT ad Commuicatio No.7-0/0 pae:. ISSN: [] T. lgamal A public key cryptosystem ad a siature scheme based o discrete loarithms I Trasactios o Iormatio Theory Vol. IT- No.. pp [6] Natioal Istitute o Stadards ad Techoloy. NIST FIPS PUB 86-(0. Diital Siature Stadard U.S. Departmet o Commerce. [7] GOST R.0-9. Russia Federatio Stadard. Iormatio Techoloy. Cryptoraphic data Security.

Growth of Functions. Chapter 3. CPTR 430 Algorithms Growth of Functions 1

Growth of Functions. Chapter 3. CPTR 430 Algorithms Growth of Functions 1 Growth o Fuctios Chapter 3 CPTR 430 Alorithms Growth o Fuctios 1 Asymptotic Eiciecy o Alorithms Idea: Look at iput sizes lare eouh to make rui time order o rowth relevat How does the rui time o a alorithm