Furtherance Of Elliptic Curve Cryptography Algorithm In .
International Journal Of Scientific & Engineering Research, Volume 3, Issue 10, October-2012ISSN 2229-55181Furtherance of Elliptic Curve CryptographyAlgorithm in the field of GSM securitySatarupa ChakrabortyAbstract-- Mobile Phones have totally changed the world; nowadays people can afford to forget their daily household needs but not theirown mobile phones. This increasing popularity has sensed a huge growth in the acceptance of modern mobile phones. With the increasingnumber of features in mobile phones security has become the chief area of concern as it is apposite to all the authoritative applicationsthroughout the world. Today as GSM accounts for 80% of the total mobile phone technologies in the market so lack of security measurescan crumple resulting in hampering of its service. Some of these security issues have been sort out using 3GPP. In this paper we will becentralizing our discussion on the operational methodology of RSA algorithm and elliptic curve cryptography algorithm and will be closelyexamining which of the above anticipates a secure method of encryption for GSM key generation.Index Terms-- elliptic curve cryptography, RSA, GSM, security, cryptography, finite field, discrete logarithmic problem—————————— ——————————‘burst period’ and it lasts 15/26 ms (or approx. 0.577 ms).Therefore the eight ‘time slots’ are actually ‘burst periods’,1 INTODUCTIONwhich are grouped into a TDMA frame, whichsubsequently form the basic unit for the definition of logicalThe last few years have witnessed an unprecedentedchannels. One physical channel is one burst period peremergence in the wireless industry. The ever increasingTDMA frame [3]. GSM was designed to grow and meet thedemands of users have triggered an increasing involvementneeds of new technologies. GSM is currently composed ofamong researchers and industries to come up with aGPRS, 3G, and EDGE. EDGE is a technology that allowscomprehensive manifestation for more upgraded mobileimproved data transmission rates as a backwardcommunication systems. Mobile phones are being used oncompatible extension of GSM. GPSR is designed for webdaily basis by millions of users so privacy of user’s phonebrowsing. 3GSM is the GSM running on third generationcalls and text messages (data) need to be ensured andstandards for multimedia services [4]. It allows fullunauthorized use of the service needs to be prevented. It isroaming from operator to operator if mutual bilateralhighly mandatory to take sensible technological securityagreements are in place [5]. For security of the transmit ionmeasures. The risk of encroachment and eavesdropping hasof sensitive information cryptography and securitybeen increasing with gadgets becoming wireless andauthentication protocols were devised to ensureubiquitous. As such it gives rise to an alarming issue ectra of hackers/crackers bulking large [1] [2].Thecommunications. Security protocols like SSL [6] and SET [7]successful deployment of GSM over the last two to threealready exist. Encryption can be broadly categorized intodecades has been noteworthy as it has been to corroboratetwo forms: Symmetric and asymmetric encryptionmost of world’s mobile phone networks. GSM has beentechniques. Most of symmetric encryption is based on RSAdubbed the “Wireless Revolution” and it doesn’t take muchpublickeycryptography.Butasymmetrickeyto realize why. GSM provides a secure and confidentialcryptography using elliptic curve cryptography (ECC) ismethod of communication. Most GSM systems operate indesigned which has been able to maintain the security levelthe 900 MHz and 1.8 GHz frequency bands.GSM divides upset by other protocols [8].the radio spectrum bandwidth by using a combination ofIn this paper Section 2 discusses about the importance ofTime- and Frequency Division Multiple AccessGSM and the requirements of GSM security. Section 3(TDMA/FDMA) schemes on its 25 MHz wide frequencydiscusses about RSA algorithm. Section 4 overview aboutspectrum, dividing it into 124 carrier frequencies (spacedECC while 5 discusses its implementations. Section 6200 KHz apart). Each frequency is then divided into eightdiscusses about the comparison of RSA and ECC.time slots using TDMA, and one or more carrierfrequencies are assigned to each base station. Thefundamental unit of time in this TDMA scheme is called a———————————————— Satarupa Chakraborty has completed her master’s degree (M.TECH) incomputer science and engineering from Institute of Engineering andManagement, India, E-mail: satarupa85@gmail.com2 GSM AND ITS SECURITYGSM technology was introduced in the early 1980s by theEuropean Telecommunications Standards Institute (ETSI).Global System for Mobile communications or GSM usesdigital modulation to improve voice quality but thenetwork offers limited data service. As demand drove
2International Journal Of Scientific & Engineering Research, Volume 3, Issue 10, October-2012ISSN 2229-5518uptake of cell phones, GSM continued to improvetransmission quality and coverage. GSM carriers also beganto offer additional services, such as paging, faxes, textmessages and voicemail. Developed as a replacement forfirst generation (1G) analog cellular networks, the GSMstandard originally described a digital, circuit switchednetwork optimized for full duplex voice telephony. Thestandard was expanded over time to include first circuitswitched data transport, then packet data transport viaGPRS (General Packet Radio Services). Packet datatransmission speeds were later increased via EDGE(Enhanced Data rates for GSM Evolution) referred asEGPRS. The GSM standard is more improved after thedevelopment of third generation (3G) UMTS standarddeveloped by the 3GPP. GSM networks will evolve furtheras they begin to incorporate fourth generation (4G) LTEAdvanced standards. GSM was designed to grow and meetthe needs of new technologies.The main advantages of GSM are:1) Better voice quality and low-cost alternatives tomaking calls.2) Ease for the network operators in deployingequipment from any vendors that implement thestandard.3) GSM allows network operators to offer roamingservices so that subscribers can use their phones onGSM networks all over the world.4) Delivering mobile data.5) Offering greater network capacity.6) .7) Enabling rich data applications such as VoIP, videotelephony, mobile multimedia, interactive gamingand more.Though GSM faces the disadvantages of having bandwidthlimitations as multiple users uses the same bandwidth andit even causes electronic interference.GSM was designed with security in mind. Older cellularsystems were analog based and therefore very susceptibleto security attacks. It was common for attackers toeavesdrop and intercept people’s conversations and data.Even worse yet, attackers were capable of stealing customerIDs to make fraudulent calls. GSM also beats out itscompetition by providing authentication, secure datatransfer, and subscriber data transfer. GSM has manybenefits over its predecessors in terms of security, capacity,clarity, and area coverage. A GSM network is composed ofseveral functional entities, whose functions and interfacesare specified. Figure 1 shows the layout of a generic GSMnetwork. The GSM network can be divided into three broadparts. The Mobile Station is carried by the subscriber. TheBase Station Subsystem controls the radio link with theMobile Station. The Network Subsystem, the main part ofwhich is the Mobile services Switching Center (MSC),performs the switching of calls between the mobile users,and between mobile and fixed network users. The MSCalso handles the mobility management operations. Notshown is the Operations and Maintenance Center, whichoversees the proper operation and setup of the network.The Mobile Station and the Base Station Subsystemcommunicate across the Um interface (represents the radiolink), also known as the air interface or radio link. The BaseStation Subsystem communicates with the Mobile servicesSwitching Center across the A interface [9].Figure 1: GSM architectureThe best way to appreciate security is by looking at howchaotic and dangerous a mobile communication systemwould be without security. At any given moment, anybodycould eavesdrop into your conversation. Ones bankaccount information, daily schedule, and any otherinformation that one may disclose on the phone would beat risk. Besides listening in, at any given moment, a hackercould impersonate user information to make calls thatwould later amount to thousands of dollars in servicecharges [4]. The security methods standardized for theGSM System make it the most secure cellulartelecommunications standard currently available. Althoughthe confidentiality of a call and anonymity of the GSMsubscriber is only guaranteed on the radio channel, this is amajor step in achieving end-to-end security. Thesubscriber’s anonymity is ensured through the use oftemporary identification numbers. The confidentiality ofthe communication itself on the radio link is performed bythe application of encryption algorithms and frequencyhopping which could only be realized using digital systemsand signaling. The security architecture of GSM wasoriginally intended to provide security services such asanonymity, authentication and confidentiality of user dataand signaling information [10].
International Journal Of Scientific & Engineering Research, Volume 3, Issue 10, October-2012ISSN 2229-5518This substantial loss incurred by the operator due to GSMfraud against a specific wireless carrier may include thefollowing:1) Indirect financial loss resulting from decrease inthe number of customers and increase in use of thesystem with no revenue.2) Direct financial loss, where money is paid out toothers, such as other networks, carriers andoperators of 'Value Added Networks' such asPremium Rate service lines.3) Potential embarrassment, where customers maymove to another service because of the lack ofsecurity.Failure to meet legal and regulatory requirements, such asLicense conditions, Companies Acts or Data ProtectionLegislation. [11]The security goals of GSM are as follows:1) Confidentiality and Anonymity on the radio path.2) Authentication of mobile users for the network.3) Confidentiality of user data and signalinginformation even in competition pressure oraccidentally.4) Anonymity of subscriber’s identity.5) Using SIM (Subscriber Identity Module) as asecurity module [10].6) Keys are securely stored [4] [17].GSM security design requirements must take into accountenvironment and security measures as:1) Must define security procedures for generationand distribution of keys.2) Exchange information between operators.3) Maintain confidentiality of algorithms.4) Must be a cost effective scheme [12].Whereas GSM security measures must not:1) Increase the bandwidth of the channel.2) Increase the error rate.3) Increase error propagation.4) Add expensive complexity to the system.5) Increase the error rate.6) Significantly add to the delay of initial call setup orsubsequent communication [12].The GSM security architecture:1) Each mobile subscriber is authenticated with aunique 128 bit secret key (Ki).2) Ki is stored in Subscriber Identity Module (SIM)which is inserted in the mobile phone.3) Ki of each subscriber also gets scored inAuthentication Center (AuC) associated with theHLR in the home network.4)3The SIM is designed as a tamper resistant smartcard to avoid extraction of the customer’s Ki (as ifthe Ki would have been extracted then thesubscription could be cloned and the subscriber’scalls could be eavesdropped and it would beimpossible even for the subscriber to obtain the Ki).The levels of GSM security:Level I:1)2)3)4)5)The subscription is authenticated in the SIM.The SIM is inserted in the phone.The key of the subscriber gets stored in the AuC.The owner gets authenticated and billed.GSM checks for the validity of the subscription.Level II:1) The caller makes the call.2) GSM identifies the location of the caller.3) The receiver identifies the caller before acceptingthe call.Level III:1) Digital encryption is made to avoid other partiesfrom listening to the conversation.But practically such levels cannot be bifurcated properlydue to their non-existence in vacuum.Now for the key generation for the GSM security weconsider the two algorithms: RSA algorithm and ECCalgorithm.3 OVERVIEW OF RSA ALGORITHMCryptography not only protects data from theft oralteration but can also be used for user authentication.Within the context of any application-to-applicationcommunication some specific security requirements areauthentication, privacy/confidentiality, integrity, nonrepudiation. In general there are three types ofcryptographic schemes typically used to accomplish thesegoals: secret key (or symmetric) cryptography, public-key(or asymmetric) cryptography, and hash functions. Publickey cryptography has been said to be the most significantnew development in cryptography in the last 300-400 years.RSA, the first PKC implementation, has been named afterthe three MIT mathematicians who developed it — RonaldRivest, Adi Shamir, and Leonard Adleman. RSA today isused in hundreds of software products and can be used forkey exchange, digital signatures, or encryption of smallblocks of data. RSA uses a variable size encryption blockand a variable size key [13]. Public key cryptography isbased on the creation of mathematical puzzles that aredifficult to solve without certain knowledge about how
International Journal Of Scientific & Engineering Research, Volume 3, Issue 10, October-2012ISSN 2229-5518they were created. The creator keeps that knowledge secret(the private key) and publishes the puzzle (the public key).The public key consists of the modulus n and the public (orencryption) exponent e. The private key consists of themodulus n and the private (or decryption) exponent dwhich must be kept secret [4]. Encryption and decryptionare performed by identical modular exponentiationoperations using a public and private key pair [10] [14]. Theprimary advantage of public-key cryptography is increasedsecurity and convenience: private keys never need to betransmitted or revealed to anyone. Another majoradvantage of public-key systems is that they can providedigital signatures that cannot be repudiated. Authenticationvia secret-key systems requires the sharing of some secretand sometimes requires trust of a third party as well. As aresult, a sender can repudiate a previously authenticatedmessage by claiming the shared secret was somehowcompromised by one of the parties sharing the secret [15].43.3 Decryption:For cipher text block C, its plaintext is P Cd mod n3.4 Working Example:1) Choose p 17 and q 23.2) Calculate n p q 17 * 23 391.3) Calculate ɸ (n) (p - 1) (q - 1) 16 * 22 352.4) Choose e 13 as e and n are co prime and1 e ɸ (n).5) Calculate d such that (d * e) mod ɸ (n) 1.One solution is d 325 since [(325 13) % 352 1].6) Public Key is (e, n) {13, 391}.7) Private Key is (d, n) {325, 391}.The encryption of plain text (P) 127 to cipher text C. Cipher text (C) Pe mod n 12713 mod 391 213.The decryption of cipher text (C) 213 to plain text P Figure 2: Working of RSA algorithm3.1 The RSA algorithm [16] [17]:The RSA algorithm uses two keys, d and e, which work inpairs, for decryption and encryption respectively.1)Choose two distinct prime numbers, say p and q,such that p q.2) Calculate n p q, n is used as modulus for bothpublic and private keys.3) Calculate ɸ (n) (p-1) (q-1), where ɸ (n) is Euler’stotient function.4) Consider an integer e, such that 1 e ɸ (n), so thatgcd (e, ɸ (n)) 1, i.e., e and ɸ (n) are co prime.5) Determine d, as d e mod ɸ (n) 1, i.e. d is themultiplicative inverse of (e mod ɸ (n)).6) Public Key consists of the modulus n and thepublic (or encryption) exponent e, i.e., KPU {n, e}.7) Private Key consists of the modulus n and theprivate (or decryption) exponent d which must bekept secret, i.e., KPR {d, n}.The set: (p, q, and ɸ (n) must also be kept secret becausethey can be used to calculate d.).3.2 Encryption:For plaintext block P n, its cipher text C P e mod nPlain text (P) Cd mod n 213325 mod 391 127.RSA is not secure if the same message is encrypted toseveral receivers, to completely break RSA one needs tofind the prime factors. A disadvantage of using public-keycryptography for encryption is speed. There are manysecret-key encryption methods that are significantly fasterthan any currently available public-key encryption method.In practice, RSA has proved to be quite slow, especially forkey generation algorithm. RSA is not well suited for limitedenvironments like mobile phones and smart cards withoutRSA co-processors because it is hard to implement largeinteger modular arithmetic on such environments [18]. RSAalgorithm encryption used in file encryption for small files,any file with asymmetric key encryption into its text can bemore convenient to communicate and manage, and it raphy may be vulnerable to impersonation, even ifusers' private keys are not available. Public-keycryptography is usually not necessary in a single-userenvironment. For example, if you want to keep yourpersonal files encrypted, you can do so with any secret keyencryption algorithm using, say, your personal passwordas the secret key. In general, public-key cryptography isbest suited for an open multi-user environment.4 OVERVIEW OF ELLIPTICCRYPTOGRAPHY ALGORITHMCURVEElliptic curve cryptosystems appear to offer newopportunities for public-key cryptography. Elliptic curvecryptography relies on the believed difficulty of the ellipticcurve discrete logarithm for its security. It was as an
5International Journal Of Scientific & Engineering Research, Volume 3, Issue 10, October-2012ISSN 2229-5518alternative mechanism for implementing public-keycryptography designed in 1985 by N. Koblitz (University ofWashington) and V. Miller (IBM) and is becoming acceptedas an alternative to cryptosystems such as RSA andELGamal over finite fields. In public key cryptography eachuser or the device taking part in the communicationgenerally have a pair of keys, a public key and a privatekey, and a set of operations associated with the keys to dothe cryptographic operations [7] [20] [21] [22]. ECC is basedon properties of a particular type of equation created fromthe mathematical group (a set of values for whichoperations can be performed on any two members of thegroup to produce a third member) derived from pointswhere the line intersects the axes. Multiplying a point onthe curve by a number will produce another point on thecurve, but it is very difficult to find what number was used,even if you know the original point and the result.Equations based on elliptic curves have a characteristic thatis very valuable for cryptography purposes: they arerelatively easy to perform, and extremely difficult toreverse. Elliptic curves are also used in several integerfactorization algorithms that have applications incryptography [16]. For the purpose of cryptography, anelliptic curve can be thought of as being given by an affineequationoftheform:23y x ax b(1)Where, 4a3 27b2 0(2)Where ‘a’ and ‘b’ comprises the elements of a finite fieldwith pn elements, where p is a prime larger than 3. (Theequation over binary and ternary fields looks slightlydifferent.) The set of points on the curve is the collection ofordered pairs (x, y) with coordinates in the field and suchthat x and y satisfy the relation given by the equationdefining the curve, plus an extra point that is said to be atinfinity. The set of points on an elliptic curve withcoordinates in a finite field also form a group and theoperation is as follows: to add two points on the curve Pand Q together, pass a straight line through them and lookfor the third point of i
cryptography using elliptic curve cryptography (ECC) is designed which has been able to maintain the security level set by other protocols [8]. In this paper Section 2 discusses about the importance of GSM and the requirements of GSM security
Zalka and indicate that, for current parameters at comparable classical security levels, the number of qubits required to tackle elliptic curves is less than for attacking RSA, suggesting that indeed ECC is an easier target than RSA. Keywords: Quantum cryptanalysis, elliptic curve cryptography, elliptic curve discrete log-arithm problem. 1 .
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Computer and Network Security by Avi Kak Lecture14 Back to TOC 14.1 WHY ELLIPTIC CURVE CRYPTOGRAPHY? As you saw in Section 12.12 of Lecture 12, the computational overhead of the RSA-based approach to public-key cryptography increases with the size of the keys. As algorithms for integer factorization have become more and more efficient, the RSA
SEC 1 Ver. 2.0 2 Mathematical Foundations This section gives an overview of the mathematical foundations necessary for elliptic curve cryp-tography. Use of each of the public-key cryptographic schemes described in this document involves arithmetic operations on an elliptic curve over a finite field. This section introduces the mathematical .
CCS Discrete Math I Professor: Padraic Bartlett Lecture 9: Elliptic Curves Week 9 UCSB 2014 It is possible to write endlessly on elliptic curves. (This is not a threat.) Serge Lang, Elliptic curves: Diophantine analysis. 1 Elliptic
applications. Smooth degree-3 curves, known as elliptic curves, were used in Andrew Wiles’s proof of Fermat’s Last Theorem [11]. The points on elliptic curves form a group with a nice geometric description. Hendrick Lenstra [5] exploited this group structure to show that elliptic curves can be used to factor large numbers with a relatively .
by Washington [83] on cryptography using elliptic curves is an excellent follow-up read; elliptic curve based cryptography is becoming the norm for the current gener-ation of public key cryptosystems. As we are writing for a mathematical
This study investigated microRNA and mRNA expression and protein function associated with DNA repair in human oocytes and embryos. MicroRNAs have been shown to down-regulate and in some cases to stabilise the expression of several genes including repair genes. The first aim of this study was to analyse the differences in the expression of microRNAs and their target mRNAs involved in repair .
