Post-quantum Key Exchange Protocol Using High Dimensional Matrix

Transcription

1 Post-quantum Key Exchange Protocol Usng Hgh Dmensonal Matrx Rchard Megrelshvl I. J. Tbls State Unversty Melksadeg Jnkhadze Akak Tseretel State Unversty Kutas, Georga Maksm Iavch Caucasus Unversty Avtandl Gagndze Geora Unversty Gorg Iashvl Geora Unversty Abstract Actve work s beng done to create and develop quantum computers. Google Corporaton, NASA and the Unverstes Space Research Assocaton USRA have teamed up wth DWAFE, the manufacturer of quantum processors. D-Wave X s a quantum processor that contans,8 physcal qubts. qubts from the whole number of qubts are used to perform the calculatons. As we see, quantum computers can easly solve the problem of calculatng the dscrete logarthm used n Dffe- Hellman algorthm. So t can break Dffe-Hellman algorthm. When quantum computers are released all exstng crypto systems wll be useless, because there wll be no way to transfer the key securely. In the artcle s proposed the new key exchange method usng hgh dmensonal matrx, ths method s safe aganst attacks mplemented usng quantum computers. The case concerns the matrx functon and algorthm for cryptographc keys exchange wth open channel. For the algorthm s offered the method of buldng a hgh dmensonal matrx multplcatve group. The arsng of ths goal s that tradtonal key exchange methods are vulnerable to quantum computer attacks. Keywords post-quantum cryptography, attacks, a matrx oneway functon, Abelan multplcatve group, asymmetrc cryptography, hgh dmensonal matrx fnte feld I. INTRODUCTION One of the fundamental problems of cryptography s the safe communcaton over the lstenng channel. Messages need to be encrypted and decrypted, but for ths, both partes need to have a common key. If ths key s transmtted va the same channel, then the lstenng sde wll also receve t, and the meanng of the encrypton wll dsappear. Dffe-Hellman algorthm allows the two partes to obtan a common secret key usng an unprotected, but spoofed, communcaton channel. The receved key can be used to exchange messages usng symmetrc encrypton. The securty of formng a common key n the Dffe-Hellman algorthm follows from the fact that, although t s relatvely easy to calculate exponents modulo a prme number, t s very dffcult to calculate dscrete logarthms. For large prme numbers of hundreds and thousands of bts, the task s consdered unsolvable, snce t requres a tremendous amount of computatonal resources. But ths problem can easly be solved by quantum computers usng Shor algorthm [,]. The securty of RSA algorthm reles on factorzaton problem, but ths problem can be easly solved usng quantum computers []. Actve work s beng done to create and develop quantum computers. Google Corporaton, NASA and the Unverstes Space Research Assocaton USRA have teamed up wth DWAFE, the manufacturer of quantum processors. D-Wave X s a quantum processor that contans,8 physcal qubts. qubts from the whole number of qubts are used to perform the calculatons. As we see, quantum computers can easly solve the problem of calculatng the dscrete logarthm used n Dffe- Hellman algorthm. So t can break Dffe-Hellman algorthm. When quantum computers are released all exstng crypto systems wll be useless, because there wll be no way to transfer the key securely [,]. In the artcle s proposed the new key exchange algorthm usng hgh dmensonal matrx. Ths algorthm s safe aganst quantum computer attacks. The case concerns the matrx functon and algorthm for cryptographc keys exchange wth open channel. For ths s offered the method buldng a hgh dmensonal matrx multplcatve group. The arsng of ths goal s that tradtonal key exchange methods are vulnerable to quantum computer attacks. One-way functon OWF s a functon whose value s easy to calculate for any argument, but t s dffcult to fnd an argument for the gven value of the functon. The word "dffcult" s to understand the complexty of the computaton. In other words, fndng the relevant argument of the gven functon n real tme s dffcult even wth the modern computng technques. The rreversblty of functon does not mean that the functon s one-way [,7]. The exstence of one-way functons s the bass for the dea of asymmetrc cryptography. It one-way functon s the foundaton of asymmetrc cryptography, personal dentfcaton, authentcaton, and other felds of nformaton protecton. Although there s no theoretcal proof of the exstence of one-way functons n general, there are several Copyrght held by the authors. 8

2 possble pretendents eg, multplcaton and factorzaton, squarng and module rootng, dscreet exponent and logarthmzaton, whose one-wayty e the dffculty of fndng the argument for the value of functon at ths tme real and s actvely used n nformaton exchange protocols. As we have mentoned, one-way functons are actvely used n the algorthms for developng a cryptographc open key. The ntal dea 97 belongs to Whtfeld Duffe and Martn Helman. Based of ther dea was establshed the frst practcal wel-known Dffe-Helman-Merkel method, whch enabled the development of a common cryptographc key usng the open unprotected channel. A year later, the frst RSA algorthm of asymmetrc encrypton was formed. The RSA n fact, resolved the problem of exchange nformaton wth open channel. Both algorthms are not safe aganst quantum computers attacks. Are proposed quantum key exchange protocols, but quantum computers are needed to mplement them [8,9]. II. ONE-WAY MATRIX FUNCTION The new one-way functon for the development of common cryptographc keys s based on hgh order cyclc matrx groups, wth the power e = n, where the n s row dmenson of the square matrx. Let's assume that "A" s the above matrx group, whle A s the ntal n n matrx, then "A" = A={A, A, A,, A n = I} where I represents an dentty matrx. One-way functon and algorthm for common key development are as follows: The sender chooses A A secret matrx to send to the recevng party va open channel the uu = va vector where v V n vector s known V n s a vector space on GF feld; The recevng party shall, on the other hand, choose A A secret matrx and send to the sender u = va vector; Sender calculates k = u A vector; Recever calculates k = u A where k and k are secret keys; Obvously, k = k = k, because k = va A = va A because of the commutatveness of the "A" group. The va = u 7 s one-way fast functon. Let v = v, v, v,, v n V n 8 and u = u, u, u,, u n V n are non-secret vectors from the above algorthm and a a n A = A 9 a n a nn s a secret matrx. Then, accordng to algorthm the followng system s formed: v a + v a + + v n a n u v va = a + v a + + v n a n u = v a n + v a n + + v a n u n The number of unknowns n the system of lnear equatons s the square of number of equatons. Obvously, the system can not be solved n lmted tme, f the sze of the matrx s large enough. Sze of the matrx must be chosen consderng Grover s algorthm. Classcally, searchng requres a lnear search, whch s ON n tme. Grover's algorthm needs ON / tme, t s consdered as fastest quantum algorthm for searchng an unsorted nformaton. Ths algorthm provdes a quadratc speedup [,]. One fact must be taken nto consderaton f the A matrx contans the nternal recurrence, or f each of ts rows are n a certan recurrence wth the prevous row, then the task of solvng the system wll be replaced by a smpler task that s easy to solve. It s so mportant that t puts tself n doubt the oneway character of our functon and requres the exstence of Abelan multplcatve matrx group wth a hgh order, that s free from the recurrence of the nsde. III. FINITE MATRIX GROUPS CONSTRUCTION Let's consder + α, where =,,,, and α represents the root of prmtve polynomal n the GF n feld odule wth the module p x. + α = + α = + α + α = + α + α = + α + α + α + α = + α + α = + α + α + α The polynomal coeffcents generated the structure known as Serpnsky Trangle. The derved structure contans a number of sub-structures that can be used as a generator generatng matrx for multplcatve groups, e prmtve elements. Such s, for example, = 9, P = And many more. Ther natural powers create Abelan multplcatve cyclc group. For example: P =, P =, =, P =, P =, P =, P 7 = P = It s easy to confrm that P, P, P,, P, P, 8

3 s an Abelan multplcatve group. Lets keep the structure of matrx and extend t by elements of set as follows:, =, where,=... Fore example, when = and =, we have, = = When = and =, we have pc. : Pc.:, and,, = = Consder P = [,] = = If we take nto consderaton that the set,,,,,,, s a feld, t s easy to assure that each sub-matrx of the P matrx s n the the same set: P, = + + =, P, = + + =, P, = + + =, P, = + + =, P, = + + =, P, = + + =,, = + + =,, = P P + P + P = P,, = P P + P + = P. or P = see pc.. pc.: P = [,] Usng the software package we have developed t has been confrmed that the, matrx s a prmtve element. Its natural powers generate Abelan multlplcatve group, whose power s. The elements of [,] k when k = 7,, 9, 9,, 8, are dagonal matrces see pc. :,,,,,, Pc. : [,] k, k = 7,, 9, 9,, 8, Also, matrx s a prmtve element, and elements of the set [,] k, when k=7,, 9, 9,, 8,, are dagonal matrcespc :,,,,, pc.: [,] k, k = 7,, 9, 9,, 8, 8

4 Set of non-zero elements of the dagonal matrces represents the perturbaton of the group P, P, P,, P, P, called as prmary group and one of the elements s P. Fnally, we can conclude that emprcally we proved the followng fact: The second order, + expanson, +, =.., of the matrx s a prmtve element and creates the Abelan multplcatve fnte group F, +, wth the power. Below we can see other prmtve elements that are results of expanson of matrx:, =,, =,, =, =, =, =, = 7 In order to get hgher order prmtve elements, we stll retan the structure of matrx and put nto the elements of the group F, +. We get order matrx call t a thrd order expanson. For example, f we use the elements of group F, for the frst and second expansons of the matrx of, respectvely [,] [,] matrces, we get the followng matrx pc. : pc.., P P P, = = = 8 Let's consder [,] k set. It has the same basc structure as the prmary group, as well as the frst and second expanson matrces taken from the prmary group. It s expected that ths set s characterzed by the same propertes as the prmary group has. Indeed, expermentally, t also has dagonal matrces, whose dagonal elements represent one of the perturbatons of the prmary group. For the set [,] k dagonal matrces are [,] + +, =,,,,. When =, we get the fnal element of the set [,] k : [,] + + = [,] = [,] = [,] 9 [,] We see that ths s an Identty matrx. Therefore, s a prmtve element and creates the Abelan multplcatve fnte group wth power. Defnton: We call the followng matrx k, + = k + k + k + k + k + k where k F k, +, as the k th order, + expanson of the matrx. Theorem: k, + s a prmtve element and creates the abelan multplcatve fnte group F k, + wth power k. In general matrces [ k, + ] k + k +, where =,,,, k are dagonal matrces and dagonal elements are one of the permutatons of the elements of the prmary group. 8

5 When = k, we get [ k, + ] k + k += = [ k, + ] k k + k += = [ k, + ] k = [ k, + ] [ k, + ] [ k, + ] Ths means that the structure s a prmtve matrx. The prmtve matrces obtaned have an nterestng fractal structure see pc.. Abelan multplcatve groups adopted by the above mentoned method represent suffcent sets for realzng our one-way matrx functons pc.., IV. CONCLUSION Basc matrx k, + expansons are prmtve matrces they generate abelan multplcatve matrx groups. An nterestng trend of research results n the dea: use the elements of the prmary feld as the frst and second expandng matrces wth the same characterstc polynom. It s also mportant the use of other baselne matrces, whch enlarges a new type of prmtve structures. Elements of abelan multplcatve matrx groups can be used n mplementaton of one way functon, that we offer. So the key exchange method s got and t s secure aganst quantum computers attacks. ACKNOWLEDGEMENT The Work Was Conducted as a Part of Research Grant of Jont Proect of Shota Rustavel Natonal Scence Foundaton and Scence & Technology Center n Ukrane [ STCU--8] REFERENCES []. Shor, P. Polynomal-tme algorthms for prme factorzaton and dscrete logarthms on a quantum computer. SIAM J. Comput., []. Jones, J. A. NMR quantum computaton. Prog. NMR Spectrosc. 8, []. Ekert, A. & Jozsa, R. Quantum computaton and Shor's factorng algorthm. Rev. Mod. Phys. 8, []. Avtandl Gagndze, Maksm Iavch, Gorg Iashvl// Novel Verson of Merkle Cryptosystem// BULLETIN OF THE GEORGIAN NATIONAL ACADEMY OF SCIENCES, vol., no., 7, p. 8- []. Avtandl Gagndze & Maksm Iavch & Gorg Iashvl, 7. "Some Aspects Of Post-Quantum Cryptosystems," Eurasan Journal of Busness and Management, Eurasan Publcatons, vol., pages - []. Werner Alex, Benny Chor, Oded Goldrech, Claus P. Schnorr, RSA and Rabn functons: certan parts are as hard as the whole, SIAM Journal on Computng, v.7 n., p.9-9, Aprl 988 [do>.7/7] [7]. R. P. Megrelshvl, Analyss of the matrx one-way functon and two varants of ts mplementaton, Internatonal J. of Multdscplnary Research And Advances In Engneerng IJMRAE, v., n. IV October, pp [8]. G. L. Long, X. S. Lu, Theoretcally effcent hgh-capacty quantum-key-dstrbuton scheme, Phys. Rev. A, [9]. W. Shor, John Preskll, Smple Proof of Securty of the BB8 Quantum Key Dstrbuton Protocol Peter Phys. Rev. Lett. 8,, []. Shu-Shen L, Gu-Lu Long, Feng-Shan Ba, Song-Ln Feng and Hou-Zh Zheng, Quantum computng, PNAS October, []. Marlan O. Scully and M. S. Zubary, Quantum optcal mplementaton of Grover's algorthm, PNAS August,