Secure Transmission of Sensitive data using multiple channels

Transcription

1 Secure Transmsson of Senstve data usng multple channels Ahmed A. Belal, Ph.D. Department of computer scence and automatc control Faculty of Engneerng Unversty of Alexandra Alexandra, Egypt. Abstract A new scheme for transmttng senstve data s proposed, the proposed scheme depends on parttonng the output of a block encrypton module usng the Chnese Remander Theorem among a set of channels. The purpose of usng the Chnese Remander Theorem s to hde the cpher text n order to ncrease the dffculty of attackng the cpher. The theory, mplementaton and the securty of ths scheme are descrbed n ths paper. Keywords Cryptography, Secure transmsson, Chnese Remander theorem, Block cpher. 1. Introducton As the need to protect senstve data aganst dfferent threats [2] ncreases the study of cryptography becomes of greater mportance [1]. Encrypton s the cryptographc prmtve mostly used n protectng the secrecy of the data. Block encrypton can be vewed as a mathematcal transformaton between the set of source blocks of sze S bts and the set of blocks of sze C bts through a mappng functon and a parameter K that s called the encrypton key. The form of any block Encrypton s C = EK (S) for some S {0,1 S and C {0,1 C. The nverse transform known as the decrypton s S = DK (D). The Chnese remander theorem CRT [5] states that f q0, q1... q k are k par wse relatvely prme postve ntegers and a0, a1... a k are postve ntegers then there exsts exactly one nteger a where 0 a < q k = 1 0 for q q such that a = a mod q for 0 < k. = The ntegers q 0, q1... q k are called the modul whle the ntegers a0, a1... a k are called the resdues. The CRT has well known applcatons n both secret sharng and error correctng codes [6]. In ths work, block encrypton and the Chnese remander theorem wll be combned to produce a scheme for transmttng senstve data over multple channels. Abdelhamd S. Abdelhamd, M.Sc. Department of computer scence and automatc control Faculty of Engneerng Unversty of Alexandra Alexandra, Egypt. asaladn@acm.org 2. Overvew of the proposed scheme The proposed scheme assumes the exstence of A transmsson channels between the sender and the recever partes from whch S Channels are chosen usng some selecton crtera. The orgnal message or plan-text s dvded nto unts that are referred to as super blocks of bts. The super blocks are encrypted usng an encrypton module that operates usng a block encrypton scheme E K ( B ) of block sze B bts where s equal to L for an B nteger L. The encrypton s performed usng an approprate mode of operaton such as CBC [2]. Each super block s treated as an bt nteger. A set of S relatvely prme modul q0, q1... q S are selected such that 2 q 0 q1.. qs then the bt nteger CRT remanders wth respect to the selected S modul r 0, r1... r S are calculated then sent over the selected S channels. The Selected S channels are used to transmt the remanders whle the rest of the channels carry some rrelevant data n order to decrease the ablty of the adversary to determne the used channels. At the recever sde, the nverse of the CRT s appled to get the orgnal -bt cpher super block, and then a decrypton module D K ( B ) s used to get the orgnal bt plan-text. The applcaton the CRT to the output of the encrypton module ams to hde the cpher from the adversary n order to prevent the adversary from takng beneft of any property of the cpher. 1

2 Fgure 1 Proposed Scheme Block Dagram 2.1 Selecton of the Channels The selecton of the S out of the A avalable channels should be done n a way that prevents the adversary from tellng whch channels carry relevant data and whch ones do not. The proposed selecton crteron s based on selectng pseudorandom numbers n the range [ 0, A ]. The generator s run on both sdes of communcaton S tmes to select the S channels. 2.2 Selecton of the modul It s mportant to determne the range of values for the modul used n the Chnese Remander transform. Assumng that, the sze of the super block s bts, and that t can be treated as an -bt nteger. The mod operaton can be consdered as a mappng between the -bt data space and the resdue space accordngly any super block b can be mapped to a remander r such that 0 r < q and r = b mod q where q s the selected modulus. Assumng a random varable B that denotes the possble super block nteger value and a random varable R that denotes the possble values of the resdue, the jont probablty between the two random varables can be represented as P( B = b / R = r) = q / 2. In order to decrease ths probablty we ether ncrease the super block sze or reduce the value of the modulus q. In order to decrease the value of all modul the sze of all of them can be chosen to be around the value / S. 2.3 umber of channels The maxmum number of channels used n the proposed scheme was estmated as S max = O( / log ). Ths estmaton s based on the fact that the maxmum number of relatvely prme numbers that can be selected less than or equal to a gven nteger x s O ( x / log x) and the assumpton that the modul wll be selected of length / S. 2.4 The super block Sze The super block sze s selected as a multple of the block sze B of the encrypton scheme used such that = L B. The selecton of the multpler L s governed by some factors 1. The loss resultng from representng the remanders of the Chnese remander theorem on fxed word length machnes. For example f AES s used as the encrypton method and 10 channels are used. If the encrypton module super block sze was taken to be 128 bts.e. L=1, each channel wll carry 12.8 bts or practcally 13 bts so 2 bytes are used for the representaton of that number whch leads to 18.75% loss n the bandwdth. If the super block sze was taken to be 512 bts.e. L=4, each channel wll carry 51.2 bts or practcally 52 bts so 7 bytes wll be used leadng to only 7.14% loss n bandwdth. Fgure 2 bandwdth Loss 2. The network algorthms used to manage the queung of data. For example n some mplementaton of the TCP/IP the agle algorthm s used n order to reduce the network overhead. The applcaton of that algorthm results n the bufferng of data tll a certan threshold s reached then ths data s sent as one packet. The use of larger remanders wll result n faster fllng of buffer and therefore ncreasng the throughput. 3. As the value ncreases the length of the modul selected ncreases. Ths wll ncrease the effort needed by the adversary to guess the modul 2

3 3. The detaled scheme The proposed scheme s composed of two phases: 1. The setup phase, durng ths phase the setup for the scheme wll be done. Ths phase s done once or every a relatvely long nterval. 2. The sesson phase, durng ths phase the actual transmsson s done between the two partes. The sesson phase starts as soon as the setup phase ends and s repeated whle the setup confguraton s vald. Scheme() { SetupPhase(); // IsVald represents outsde condtons whle( IsVald) Sesson(); Fgure 3 Proposed Scheme Flow 3.1 The setup phase In order to setup the envronment for the proposed scheme the followng steps should be done: 1. Selecton of the Encrypton method, ts parameters and mode of operaton. For example f we decde to use AES n the CBC mode, both the Encrypton key and the ntal vector should be selected. An mportant parameter to be set s the super block sze = L. B 2. Establshment of the set of avalable channels. 3. Selecton of the number of channels used n the communcatons S, ths number should be selected such that t s possble to select at least S relatvely prme ntegers to be used n the CRT. 4. Selecton of the S channels out of the A avalable channels. Ths s done through the use of a sutable pseudorandom number generator [3], [4]. The man condton of the random number generator s to produce a full cycle of values n the range [ 0, A ]. The output of the pseudorandom generator s taken to represent channels IDs [ 0, A ] so the generator s run S tmes to generate the S channels ds selected for the sesson. 5. Selecton of the modul. Two ways for usng these modul are proposed a. To generate a number of modul equal to the number of channels A, such that each channel s assgned a statc modulus. It should be kept n mnd that from the generated A modul any S modul can be used to represent the output of the encrypton module usng the CRT. Ths method wll be referred to as the statc assgnment of modul. b. To generate only S modul suffcent to represent the output of the Encrypton module usng CRT and on each sesson the S modul are assgned to the S selected channels. Ths method wll be referred to as the dynamc assgnment of modul. The output of the setup phase represents the key for the proposed scheme and so some sort of key exchange protocol should be used n order to exchange the output of the setup phase. 3.2 The sesson phase The sesson phase refers to the steps that should be done on each transmsson. It nvolves actvtes on both sdes of the transmsson. In the followng dscusson the two partes wll be referred to as the sender and the recever. Every sesson starts as soon as the prevous sesson ends. The operatons Input Wat, channels Input Wat, and channel Read wll be assumed to be blockng operatons. The sesson does not start as soon as the data arrves n order to keep the synchronzaton between the sender and the recever partes. It should be noted that the frst sesson starts as soon as the setup phase ends. In the followng, the two dfferent mplementatons for the Sender and the Recever wll be dscussed. Sender Sesson The Sender has a message M that should be delvered to the recever. The protocol steps used by the sender are: 1. Select channels step, ths step selects S channels from the Avalable Channels usng a pseudorandom number generator. After selectng the channels the modul are assgned to the correspondng channels. 2. Wat for nput step, ths step wats untl there are data avalable for sendng. Ths operaton s a blockng step. When the data s avalable steps three to sx wll be repeated untl the nput s consumed. 3. bts of data are read from the nput stream. If the amount of remanng data s less than, the data wll be padded as approprate by the applcaton. 4. The -bt super block wll be encrypted by the encrypton module. As mentoned earler, the encrypton module depends on a block cpher algorthm such as DES or AES. 5. The super block s represented as an bt nteger. For every channel the remander of the nteger resultng from the prevous step wll be calculated 3

4 wth respect to the modulus assocated wth the channel and 6. The calculated resdues wll be sent on the correspondng channels as packets of data. 4. The S ntegers read from the S channels are used n the nverse Chnese remander theorem operaton to produce an bt nteger. 5. The nteger s represented as an bt super block sutable to be nput to the decrypton module. 6. The resultng super block s decrypted usng the decrypton module. 7. The decrypted data s wrtten to the output stream. Fgure 5 Recever Flow chart Recever Sesson Fgure 4 Sender Flow Chart The recever party receves the data sent over the channels and reconstructs the Message usng the followng steps: 1. Select channels step, ths step s equvalent to the correspondng one on the sender sde. 2. Wat for channel nput step, ths step wats untl there are data avalable on the dfferent channels. Ths operaton s blockng. When the data s avalable steps three to seven wll be repeated untl end of transmsson. 3. For every channel, an nteger r s read that s the remander sent on the channel. 4. Securty of the proposed scheme In ths secton we analyze the securty of the proposed scheme. The analyss wll start by demonstratng some probablstc characterstcs of the scheme Theorem: The remanders of the CRT are ndependent. Proof: Suppose that the output of the encrypton module s an bt value x. Let p and q be two of the S modul and let Y, Z be random varables denotng the values of x mod p and x mod q respectvely. Let y = x mod p and z = x mod q. Accordng to the Chnese remander theorem the ordered par ( y, z) unquely dentfes x n the ranges of values [ 0, pq ], [ pq,2 pq ] and so on. 4

5 Ths mples that P ( Y = y Z = z) pq. It s clear that P ( Y = y) p and P ( Z = z) q. The result of the prevous steps mply that P ( Y = y Z = z) = P( Y = y) P( Z = z) so the two random varables are ndependent By generalzng the above proof, t s proved that the remanders of the Chnese Remander Theorem are ndependent. Based on ths a probablty model wll be bult to descrbe the output. Let the Random varable C denote the -Bts output of the Encrypton module and defne a set of random varables R for 0 < S such that R denotes the CRT remander wth respect to the modulus q. The probablty dstrbuton of such random varables for r cmodq can be defned as P( C = c) 0 c < 2 P( R0 = r0, R1 = r1.. RS = rs ) = 0 otherwse Based on the stated probablty relaton there are two sets of values that can never appear n the output of the proposed scheme 1. The set of remanders correspondng to the values c such that 2 c < q0 q1... qs. 2. For each channel the set of values defned l x : q x < 2 l = q. as log 2 Accordng to those results the proposed scheme seems to have two ponts that the adversary can use to compromse the system. Accordng to the concrete securty defnton [7], the adversary can compromse a securty system f he can dstngush t from an equvalent system wth truly random output. Based on the demonstrated ponts n the system we wll conclude that theoretcally the probablty of dstngushablty s L P D = 1 2 t s clear that P D But n order to use both of these ponts the adversary should guess the followng: 1. The number of channels S used from wthn the avalable channels. 2. The parameters of the pseudorandom generator used to select the S channels from the avalable A. 3. The relatvely prme numbers used n the CRT. In other words the adversary needs to guess a part of the system key to be able to dstngush t from a random system. If we assumed that the adversary has an algorthm called the CRT breaker that gven a set of channels carryng CRT remanders of encrypted blocks can recover nformaton about the transmtted plan text. The adversary can use CRT Breaker to develop a cpher breakng algorthm as n the followng fgure to break the underlyng encrypton scheme gven ts output. CpherBreaker( TC /* Cpher text */) { Select S relatvely prme ntegers Q[ S]; for(nt = 0; < S; + + ){ R[ ] = TC mod Q[ ]; Return CRTBreaker( Q); We can conclude from the above algorthm that the proposed scheme securty n terms of protectng the plan text s equvalent to the underlyng block cpher. 5. Concluson In ths paper a new scheme for transmttng senstve data s ntroduced. The proposed scheme was found to have some theoretcal ssues that prevent t from beng secure accordng to the concrete securty defnton but was proven to be equvalent to the underlyng block cpher. 6. References [1] Stnson D. R., Cryptography: Theory and Practce, CRC Press, [2] Menezes Alfred J., Van Oorschot Paul C. and Vanstone Scott A., Handbook of Appled Cryptography, CRC Press, [3] D. E. Knuth. The Art of Computer Programmng: Volume 2 / Sem numercal Algorthms. Addson- Wesley, 2nd edton, [4] Hellekalek Peter, Inversve Pseudorandom umber Generators: concepts, results and lnks. Department of mathematcs, Unversty of Salzburg, A-5020 Salzburg, Austra [5] Aho A., Hopcroft J., and Ullman J. The Desgn and Analyss of Computer Algorthms. Addson-Wesley, Readng, Mass., [6] Goldrech Oded, Ron Dana, Sudan Madhu, Chnese Remanderng wth errors. Proceedngs of the Thrty- Frst annual ACM Symposum on Theory of Computng [7] Bellare M., Desa A., Jokp E. and Pogaway P., A Concrete Securty Treatment of Symmetrc Encrypton: analyss of the DES modes of Operaton, Proceedngs of the 38 th Symposum on Foundatons of Computer Scence, IEEE,