Novel pseudo random number generation using variant logic framework

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1 Edith Cowa Uiversity Research Olie Iteratioal Cyber Resiliece coferece Cofereces, Symposia ad Campus Evets 011 Novel pseudo radom umber geeratio usig variat logic framework Jeffrey Zheg Yua Uiversity, Chia Origially published i the Proceedigs of the d Iteratioal Cyber Resiliece Coferece, Edith Cowa Uiversity, Perth Wester Australia, 1st - d August 011 This Article is posted at Research Olie.

2 Proceedigs of the d Iteratioal Cyber Resiliece Coferece Abstract NOVEL PSEUDO-RANDOM NUMBER GENERATION USING VARIANT LOGIC FRAMEWORK Jeffrey Zheg Departmet of Iformatio Security, School of Software, Yua Uiversity, Chia cojugatesys@gmail.com Cyber Security requires cryptology for the basic protectio. Amog differet ECRYPT techologies, stream cipher plays a cetral role i advaced etwork security applicatios; i additio, pseudo-radom umber geerators are placed i the core positio of the mechaism. I this paper, a ovel method of pseudo-radom umber geeratio is proposed to take advatage of the large fuctioal space described usig variat logic, a ew framework for biary logic. Usig permutatio ad complemetary operatios o classical truth table to form relevat variat table, umbers ca be selected from table etries havig pseudo-radom properties. A simple geeratio mechaism is described ad show ad pseudo-radom sequeces are aalyzed for their cycle property ad complexity. Applyig this ovel method, it ca play a useful role i future applicatios for higher performace of cyber security eviromets. Keywords Pseudo Radom Number Geeratio, Variat Logic, Cryptology INTRODUCTION I advaced cyber eviromet, cyber security mechaism plays a guider role to protect secure iformatio commuicated ad stored i etwork facilities (Robshaw, 1995 & Xiao, Li, Choi, 004). To achieve adequate etwork security effects, cryptology has to be placed i the essetial positio (Robshaw, 1995). Differet from block ciphers operate with a fixed trasformatio o a large blocks of plaitext; stream ciphers operate with a time-varyig trasformatio o idividual plaitext digits. Uder the stream cipher methodology, Pseudo- Radom Number Geerator PRNG is placed i the cetral part of the mechaism. From , New Europea Schemes for Sigatures, Itegrity ad Ecryptio NESSIE were started (Nessie). Durig , aother Europea stream cipher project: estream selected four software ad three hardware schemes for ECRYPT Stream Ciphers (The estream Project). Such extesive iteratioal activities o ECRYPT methodologies are showig the ultra-importace of Stream Cipher techologies i cyber eviromets for wider security applicatios. From a cyber resiliece viewpoit (Stadaert, Malki, Yug 009 & Stadaert, Pereira, Yug, 010), a set of researchers are focussig attetio o leakage resiliet pseudoradom geerator. This directio has show iterestig results to protect valuable iformatio from side-chael attack aspects. Sice PRNG plays a key role i stream cipher applicatios ad is the heart of cryptology (Agew, 1988, Atkiso, 1979, NIST, 010). May mathematical methodologies are applied to this field such as liear automata, cellular automata, Galois fields ad other algebraic costructios (Atkiso, 1979, Matsumoto ad Nishimura, 000, Robshaw, 1995). I cryptology, Boolea logic operatios are essetial to create highly effective cryptology systems (Atkiso, 1979, Park ad Miller, 1988, Satha ad Vazirai, 1986) Robshaw, 1995) as biary logic geerates the greatest efficiecy through maipulatio of oly 1 s ad 0 s. Therefore, it is advatageous to ivestigate potetial mechaisms i biary logic due to the follow-o effect it has i cryptology. CLASSICAL LOGIC FUNCTION TABLE A classic logic fuctio i variables ca be represeted as a truth table (Agew, 1988, Atkiso, 1979). For a classic sequece i a ordiary umber sequece, each table cotais colums ad rows with a total of bits respectively. A example of the stadard truth table ca be see i Figure 1a. 100

Proceedigs of the d Iteratioal Cyber Resiliece Coferece VARIANT LOGIC FUNCTION TABLE Variat Logic costructio is a ew proposed theoretical structure (Zheg, Zheg, Kuii, 011) to exted classical logic

3 Proceedigs of the d Iteratioal Cyber Resiliece Coferece VARIANT LOGIC FUNCTION TABLE Variat Logic costructio is a ew proposed theoretical structure (Zheg, Zheg, Kuii, 011) to exted classical logic from the three basic operators:,,. Two additioal vector-operators: Permutatio P ad Complemetary are icluded with the origial three to form the five basic operators withi the ovel framework. Let S (N) deote a permutatio group with N elemets, the (N) permutatio operators. Let N N N complemetary operators. B 0, 1 deote a biary group with N elemets, the S cotais a total of N! N B cotais a total of The Permutatio operator (P) ad Complemetary ( ) are two vector operators performed o each colum vector of bits. For a give P ad, two operators trasforms the truth table ito a variat table. Permutatio operators chages positios of relevat colums but do ot chage their values. Complemetary operators do ot chage the positio for each colum, but may chage etire values of the colum. Two give operators ca be performed together to geerate a variat table for further usages. There are colums i the table as permutatio elemets, so this permutatio group S( ) cotais a total of!permutatio operators; ad its complemetary group B icludes a total of complemetary operators. A example of the Variat Table ca be see i Figure 1b. (a) Truth Table Example (b) Variat Table Example Fig 1. variable Truth Table ad Variat Table uder P ad operators VARIANT METHOD OF PSEUDO-RANDOM NUMBER GENERATION Iput: Output: K, P,, m, L variables, N, P S ( ), m, K m 1,..., K m L 1 L bits sequeces, L, m B Method: The process for pseudo-radom umber geeratio ca be see i Figure : is the iput variable umber. Usig variables, a stadard truth table ca be costructed i rows. P is a give permutatio operator ), ( ) ( 1 I 0 P P P P P S B, ) colums ad, where PI correspods to the I-th colum. A give complemetary operator ( I 0, B I that the operator is performed o the I-th colum, where I 0, all values of the colum are reversed ad 1, all values are 0 m is a iitial positio for output sequeces, from m ivariat. output geerated 0-1 bit sequeces. I L K, L coditios 1 K m i i 0 are 101

Proceedigs of the d Iteratioal Cyber Resiliece Coferece SEQUENCE GENERATION EXAMPLE Fig.

Parameters are iitialized to arbitrary values: =, P=(103), =(0110) After the table is geerated, the pseudo-radom sequece ca read off the table.

Example for Geeratio of Pseudo-Radom Sequece From a applicatio viewpoit, it is importat to have the exact complexity evaluatio for the method.

4 Proceedigs of the d Iteratioal Cyber Resiliece Coferece SEQUENCE GENERATION EXAMPLE Fig. Variat Method of Radom Number Geeratio For coveiet uderstadig procedure, a example is selected to show i the = case show i Figure 3. Parameters are iitialized to arbitrary values: =, P=(103), =(0110) After the table is geerated, the pseudo-radom sequece ca read off the table. For m=4, L=6 coditios, a radom umber startig at positio 4 of the variat table cotaiig 6elemets ca be foud: COMPLEXITY ANALYSIS Fig 3. Example for Geeratio of Pseudo-Radom Sequece From a applicatio viewpoit, it is importat to have the exact complexity evaluatio for the method. I the iitial stage, it is ecessary to maipulate colums ad each colum with rows; the total umbers of bits are required. The total complexity is of order ( ) O. To geerate Variat Table values, P operatios eed at least to maipulate bits oce ad operatios to maipulate the same umber of bits. i.e. ( ) O. 10

5 Proceedigs of the d Iteratioal Cyber Resiliece Coferece Selectig L bits from the variat table, it is ecessary to perform O( L ) operatios. If a full table eeds to be geerated ad keep the full table as a radom resource, ( ) complexity is required. I geeral, their computatioal complexity is O( L ) - ( O ) 0 L. O computatioal Maximal cycle legth: uder this costructio, the maximal legth of the pseudo-radom umber sequece is bits. For ay short sequeces, the output sequece has a legth less tha this umber. No clear cycle effects ca be directly observed. CONCLUSION It is importat to desig this ew PRNG method to use variat logic costructio. Sice P ad potetially have a huge cofiguratio space! times larger tha classical Logic fuctio spaces. Explorig how difficulties for this mechaism to be decoded will be the mai issue for comig cryptologist s theoretical targets. I additio, it is importat to uderstad what type of distributio will be relevat to this geeratio mechaism. Owig to itrisic complexity of variat logic costructio, this provides potetial barriers to protect this type of sequeces decoded directly. Cosiderig PRNG placed i the cetral part of stream cipher mechaism, ad stream cipher techologies are more ad more importat i advaced etwork security eviromet, higher performace methodology ad relevat implemetatio will be useful i this fields. Ogoig approaches will be focus o whether this mechaism to provide better PRNG methods to help differet protectios o side-chael attacks (Robshaw, 1995, Nessie, the estream Project, gog, 00, Xiao, Li, Choi, 004, Aissa, Nouredie, 009, Stadaert, Malki, Yug, 009, Dwivedi, Tebbbe, Harshavardhaa, 010, Yu, Stadaert, Pereira, Yuk, 010) i wider etwork applicatios to resolve practical leakage-resiliet issues i the future. REFERENCES Agew, G.B., (1988) Radom Source for Cryptographic Systems," Advaces i Cryptology EUROCRYPT '87 Proceedigs, Spriger-Verlag, pp Aissa, B., ad Nouredie, D., (009) Desigig resiliet fuctios ad bet fuctio for stream ciphers. Georgia Electroic Scietific Joural: Computer Sciece ad Telecommuicatio, No.1(18), 7-33 Atkiso, C., (1979) "A Family of Switchig Algorithms for the Computer Geeratio of Beta Radom Variables." Biometrika 66, o. 1: Davies, R., (000) Hardware radom umber geerators. It. 15th Australia Statistical Coferece, Jul.. Dwivedi, A., Tebbe, D., ad P. Harshavardhaa, P., (010) Characterizig Cyber-Resiliecy. The 010 Military Commuicatio Coferece-Uclassified Program Cyber Security ad Network Maagemet, IEEE press Eastlake, D., Crocker, S.D. ad Schiller, J.I., (1994) Radomess Requiremets for Security," RFC 1750, Iteret Egieerig Task Force, Dec. Gog, G., (00) Cryptographic Properties of the Welch-Gog Trasformatio Sequece Geerators, IEEE Tras. O Iformatio Theory, Vol 48, N0.11, Kachitvichyaukul, V., ad Schmeiser, V.W. (1988) "Biomial Radom Variate Geeratio." Commuicatios of the ACM 31, o. : Matsumoto, M., ad T. Nishimura, T., (000) "Dyamic Creatio of Pseudoradom Number Geerators." I Proceedigs of the Third Iteratioal Coferece o Mote Carlo ad Quasi-Mote Carlo Methods i Scietific Computig: Mote Carlo ad Quasi-Mote Carlo Methods 1998, NESSIE New Europea Schemes for Sigatures, Itegrity ad Ecryptio NIST, (010) A statistical test suite for radom ad pseudoradom umber geerators for cryptographic applicatios, NIST Special Publicatio, Park, S.K., ad Miller, K.W., (1988) " Radom Number Geerators: Good Oes Are Hard To Fid", Commuicatios of the ACM, October, pp

6 Proceedigs of the d Iteratioal Cyber Resiliece Coferece Robshaw, M., (1995) Stream Ciphers. RSA Laboratories Techical Report TR-701. Satha, M. ad Vazirai, U.V., (1986) Geeratig Quasi-Radom Sequeces from Slightly Radom Sources, Joural of Computer ad System Scieces, v. 33, pp Stadaert, F.X., Malki, T., ad Yug, M., (009) A uified framework for the aalysis of side-chael key recovery attacks. EUROCRYPT, The estream Project Xiao, Y., Li, H., ad Choi, S., (004) Protectio ad Guaratee for Voice ad Video Traffic i IEEE 80.11e Wireless LANs, 11pages, IEEE INFOCOM Yu, Y., Stadaert, F.X., Pereira, O., ad Yug, M., (010) Practical Leakage-Resiliet Pseudoradom Geerator. CCS 010, , ACM. Zheg, J., Zheg, C. & Kuii, T.L., (011) A Framework Of Variat Logic Costructio For Cellular Automata, ITech - Ope Access Publisher, ISBN Zheg, J., & Zheg, C., (010) A Framework to express variat ad ivariat fuctioal spaces for biary logic, Frotiers of Electrical ad Electroic Egieerig i Chia, Higher Educatio Press & Spriger- Verlag. Vol.5 No., , 104

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