DOCSLIB.ORG

On the Security of the Algebraic Eraser™Tag Authentication Protocol

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
On the Security of the Algebraic Eraser™Tag Authentication Protocol On the Security of the Algebraic Eraser™Tag Authentication Protocol Simon R. Blackburn1 M.J.B. Robshaw2 1Royal Holloway University of London, Egham, Surrey, UK 2Impinj, Seattle, WA, USA 20th June 2016 Simon R. Blackburn (RHUL) Security of Algebraic Eraser Authentication 1 / 8 Overview Anshel, Anshel, Goldfeld, Lemieux announce the Algebraic Eraser (AE) in 2002. SecureRF (trademark owners) marketing it for IoT. Nov 2015: AE-based RFID tag authentication proposal under ISO/IEC SC31: posted on SecureRF website. Attacks on underlying AE primitive: I Jan 2008: Myasnikov and Ushakov break proposed parameters. I May 2011: Gunnells recommends increasing parameter sizes. I Jan 2008: Kalka, Tsaban and Teicher break for generic parameters. I Feb 2012: Goldfeld and Gunnells avoid attack by careful choice of system parameters. I Nov 2015: Ben-Zvi, SRB, Tsaban break 128-bit parameters in 8 hours. I Jan 2016: Anshel, Atkins, Goldfeld and Gunnells claim this is not real time for proposed ISO standard. This work: very practical attacks on the proposed ISO standard. Simon R. Blackburn (RHUL) Security of Algebraic Eraser Authentication 2 / 8 AE Diffie Hellman Set N = 10, q = 256. Ω = f(M; σ): M 2 GLN (q) and σ 2 Sym(N): Two parties: Tag and Interrogator. 1 Tag generates private info. 2 Tag computes public key (MT ; σT ) 2 Ω and sends to Interrogator. 3 Interrogator generates private info. 4 Interrogator computes public key (MI ; σI ) 2 Ω and sends to Tag. 5 Parties compute shared value (M; σ) 2 Ω from private info and public keys. 6 M is the shared key. Simon R. Blackburn (RHUL) Security of Algebraic Eraser Authentication 3 / 8 Proposed ISO Protocol Aim: Authenticate Tag to Interrogator. Tag Interrogator Public key? −−−−−−−− (M ; σ ) −−−−−−−!T T Generate secret (M ; σ ), challenge Find (M; σ) −−−−−−−−−−−−−−I I Portion of M −−−−−−−−−−! Is this correct? 80-bit security Reasonable model: Adversary interacts with Tag; then interacts with Interrogator. Wins if Interrogator accepts. Simon R. Blackburn (RHUL) Security of Algebraic Eraser Authentication 4 / 8 A simple attack Observation 1 When (MT ; σT ) and σI are fixed, the shared key M is a linear function of MI . Eve challenges Tag with σI fixed, MI varying, to recover this linear function. Eve can then spoof the Tag whenever the Interrogator uses this σI . Observation 2 σI only takes 5!=120 values! So... After 273 interactions Eve can spoof with probability 120−1 ≈ 2−7. After2 7 × 273 ≈ 215 interactions, Eve can always spoof. Simon R. Blackburn (RHUL) Security of Algebraic Eraser Authentication 5 / 8 A more sophisticated attack The private key of the Tag has two parts: a secret matrix c 2 GLN (q) and a secret group element in the coloured Burau group. The matrix c can be recovered after 33 interactions with the Tag using a differential attack. Techniques due to Kalka, Tsaban and Teicher (KTT) can then be used to compute M from any challenge (MI ; σI ). KTT is heuristic in general. As parameters are small, we can replace heuristics by deterministic and efficient algorithms (see paper). So... After 33 interactions Eve can always spoof (in real time). Simon R. Blackburn (RHUL) Security of Algebraic Eraser Authentication 6 / 8 A key recovery attack There are about 280 possibilities for each part of the key. Eve can recover one part after 33 interactions with the Tag. There is a meet in the middle attack that recovers the remaining part (of an equivalent key). No more interactions. About 250 storage and operations. But why bother when you can spoof anyway? Simon R. Blackburn (RHUL) Security of Algebraic Eraser Authentication 7 / 8 Does the AE have a future? Atkins and Gunnells (Feb 2016) post a response: Change the protocol by adding a hash to the Tag's response. Disguises the linear nature of the AE DH protocol. Claims this thwarts all attacks. Does not address the combination of these attacks with the Ben-Zvi{SRB-Tsaban attack. A revised proposal has just emerged (5 May). Comments on AE: \Why Algebraic Eraser may be the riskiest cryptosystem you've never heard of", Dan Goodin, Ars Technica. There is a thread on Cryptography Stack Exchange. Twitter reaction overwhelmingly negative on AE security. I would currently not recommend using the Algebraic Eraser primitive in any applications. I would recommend independent security assessment of any future AE-based proposals before implementation. Simon R. Blackburn (RHUL) Security of Algebraic Eraser Authentication 8 / 8.
Load more

Recommended publications
  • A Study of Non-Abelian Public Key Cryptography
    International Journal of Network Security, Vol.20, No.2, PP.278-290, Mar. 2018 (DOI: 10.6633/IJNS.201803.20(2).09) 278 A Study of Non-Abelian Public Key Cryptography Tzu-Chun Lin Department of Applied Mathematics, Feng Chia University 100, Wenhwa Road, Taichung 40724, Taiwan, R.O.C. (Email: [email protected]) (Received Feb. 12, 2017; revised and accepted June 12, 2017) Abstract P.W. Shor pointed out that there are polynomial-time algorithms for solving the factorization and discrete log- Nonabelian group-based public key cryptography is a rel- arithmic problems based on abelian groups during the atively new and exciting research field. Rapidly increas- functions of a quantum computer. Research in new cryp- ing computing power and the futurity quantum comput- tographic methods is also imperative, as research on non- ers [52] that have since led to, the security of public key abelian group-based cryptosystems will be one of new re- cryptosystems in use today, will be questioned. Research search priorities. In fact, the pioneering work for non- in new cryptographic methods is also imperative. Re- abelian group-based public key cryptosystem was pro- search on nonabelian group-based cryptosystems will be- posed by N. R. Wagner and M. R. Magyarik [61] in 1985. come one of contemporary research priorities. Many inno- Their idea just is not suitable for practical applications. vative ideas for them have been presented for the past two For nearly two decades, numerous nonabelian groups have decades, and many corresponding problems remain to be been discussed to design efficient cryptographic systems.
    [Show full text]
  • The Algebraic Eraser: a Linear Asymmetric Protocol for Low-Resource Environments
    The Algebraic Eraser: a linear asymmetric protocol for low-resource environments Derek Atkins, Paul E. Gunnells SecureRF Corporation IETF92 (3/25/15) Algebraic Eraser I I. Anshel, M. Anshel, D. Goldfeld, and S. Lemieux, Key agreement, the Algebraic EraserTM, and lightweight cryptography, Algebraic methods in cryptography, Contemp. Math., vol. 418, Amer. Math. Soc., Providence, RI, 2006, pp. 1{34. I Asymmetric key agreement protocol I Designed for low-cost platforms with constrained computational resources I RFID I Bluetooth I NFC I \Internet of Things" I Complexity scales linearly with desired security level, unlike RSA, ECC. AE Performance vs ECC 2128 Security level (AES{128) ECC 283 AE B16, F256 Gain Cycles Gates Wtd. Perf. Cycles Gates Wtd. Perf. 164,823 29,458 4,855,355,934 71.7x 85,367 77,858 6,646,503,866 3,352 20,206 67,730,512 98.1x 70,469 195,382 13,768,374,158 203.3x Wtd. Perf. is Weighted Performance (clock cycles × gate count) and represents time and power usage. Gate counts are for 65nm CMOS. ECC data taken from A Flexible Soft IP Core for Standard Implementations of Elliptic Curve Cryptography in Hardware, B. Ferreira and N. Calazans, 2013 IEEE 20th International Conference on Electronics, Circuits, and Systems (ICECS), 12/2013. Overview of AE I The AE key exchange is a nonabelian Diffie–Hellman exchange. × I The underlying algebraic structure is not (Z=NZ) or E(Fq), but rather I Mn(Fq)(n × n matrices over Fq), I Bn (the braid group on n strands). I Private keys: a pair R = (m; µ) of a matrix and braid.
    [Show full text]
  • A Practical Cryptanalysis of the Algebraic Eraser
    A Practical Cryptanalysis of the Algebraic Eraser B Adi Ben-Zvi1, Simon R. Blackburn2( ), and Boaz Tsaban1 1 Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002, Israel 2 Department of Mathematics, Royal Holloway University of London, Egham TW20 0EX, Surrey, UK [email protected] Abstract. We present a novel cryptanalysis of the Algebraic Eraser prim- itive. This key agreement scheme, based on techniques from permutation groups, matrix groups and braid groups, is proposed as an underlying technology for ISO/IEC 29167-20, which is intended for authentication of RFID tags. SecureRF, the company owning the trademark Algebraic Eraser, markets it as suitable in general for lightweight environments such as RFID tags and other IoT applications. Our attack is practical on stan- dard hardware: for parameter sizes corresponding to claimed 128-bit secu- rity, our implementation recovers the shared key using less than 8 CPU hours, and less than 64 MB of memory. 1 Introduction The Algebraic EraserTM is a key agreement scheme using techniques from non- commutative group theory. It was announced by Anshel, Anshel, Goldfeld and Lemieaux in 2004; the corresponding paper [1] appeared in 2006. The Algebraic Eraser is defined in a very general fashion: various algebraic structures (monoids, groups and actions) need to be specified in order to be suitable for implementation. Anshel et al. provide most of this extra information, and name this concrete realisa- tion of the Algebraic Eraser the Colored Burau Key Agreement Protocol (CBKAP). This concrete representation involves a novel blend of finite matrix groups and per- mutation groups with infinite braid groups.
    [Show full text]
  • A Practical Cryptanalysis of the Algebraic Eraser
    A Practical Cryptanalysis of the Algebraic Eraser Adi Ben-Zvi∗ Simon R. Blackburn† Boaz Tsaban∗ June 3, 2016 Abstract We present a novel cryptanalysis of the Algebraic Eraser primitive. This key agreement scheme, based on techniques from permutation groups, matrix groups and braid groups, is proposed as an underlying technology for ISO/IEC 29167-20, which is intended for authentica- tion of RFID tags. SecureRF, the company owning the trademark Algebraic Eraser, markets it as suitable in general for lightweight en- vironments such as RFID tags and other IoT applications. Our attack is practical on standard hardware: for parameter sizes corresponding to claimed 128-bit security, our implementation recovers the shared key using less than 8 CPU hours, and less than 64MB of memory. 1 Introduction arXiv:1511.03870v2 [math.GR] 2 Jun 2016 The Algebraic Eraser™ is a key agreement scheme using techniques from non-commutative group theory. It was announced by Anshel, Anshel, Gold- feld and Lemieaux in 2004; the corresponding paper [1] appeared in 2006. ∗Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002, Israel †Department of Mathematics, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom © IACR 2016. This article is the final version submitted by the author(s) to the IACR and to Springer-Verlag on 1 June 2016. The version published by Springer-Verlag is available at [DOI to follow]. 1 The Algebraic Eraser is defined in a very general fashion: various algebraic structures (monoids, groups and actions) need to be specified in order to be suitable for implementation. Anshel et al.
    [Show full text]
  • On the Security of the Algebraic Eraser Tag Authentication Protocol
    On the Security of the Algebraic Eraser Tag Authentication Protocol Simon R. Blackburn Information Security Group Royal Holloway University of London Egham, TW20 0EX, United Kingdom and M.J.B. Robshaw Impinj 400 Fairview Ave. N., Suite 1200 Seattle, WA 98109, USA October 31, 2018 Abstract The Algebraic Eraser has been gaining prominence as SecureRF, the company commercializing the algorithm, increases its marketing reach. The scheme is claimed to be well-suited to IoT applications but a lack of detail in available documentation has hampered peer- review. Recently more details of the system have emerged after a tag arXiv:1602.00860v2 [cs.CR] 2 Jun 2016 authentication protocol built using the Algebraic Eraser was proposed for standardization in ISO/IEC SC31 and SecureRF provided an open public description of the protocol. In this paper we describe a range of attacks on this protocol that include very efficient and practical tag impersonation as well as partial, and total, tag secret key recovery. Most of these results have been practically verified, they contrast with the 80-bit security that is claimed for the protocol, and they emphasize the importance of independent public review for any cryptographic proposal. Keywords: Algebraic Eraser, cryptanalysis, tag authentication, IoT. 1 1 Introduction Extending security features to RAIN RFID tags1 and other severely con- strained devices in the Internet of Things is not easy. However the different pieces of the deployment puzzle are falling into place. Over-the-air (OTA) commands supporting security features have now been standardized [11] and both tag and reader manufacturers can build to these specifications knowing that interoperability will follow.
    [Show full text]
  • Algebraic Eraser™ and Lightweight Cryptography
    Walnut Digital Signature AlgorithmTM: A lightweight, quantum-resistant signature scheme for use in passive, low-power, and IoT devices Derek Atkins, SecureRF Corporation Historically “Lightweight Cryptography” has focused on symmetric schemes, yet asymmetric methods can also work effectively in these environments. Specifically, the Walnut Digital Signature AlgorithmTM (WalnutDSATM) provides a public-key signature scheme that verifies signatures significantly faster than ECC in both software and hardware, even in small, constrained environments and is resistant to all known attacks, both conventional and quantum. Specifically, WalnutDSA can validate a signature anywhere from 32 to 178 times faster than ECC using less ROM/RAM or fewer gates. This presentation previews the Walnut Digital Signature Algorithm, performance results, and shows how it can be applied to real-world uses. SecureRF Corporation 100 Beard Sawmill Road Suite 350 Shelton, CT 06484 203-227-3151 [email protected] www.SecureRF.com Walnut Digital Signature AlgorithmTM: A lightweight, quantum-resistant signature scheme for use in passive, low-power, and IoT devices 1. Introduction As Moore's Law continues, smaller and smaller devices are gaining more and more computational power. Where in the previous decades end systems were home personal computers, laptops, and even cloud servers, these days systems are getting smaller and more constrained. Growing into the Internet of Things, no longer are systems as powerful as before, as companies spend pennies to add computational resources to devices as small as light bulbs. However as these devices get smaller, the need to secure them grows exponentially. As more devices get connected the need to identify, authenticate, and protect those devices grows.
    [Show full text]
  • Special Signature Schemes and Key Agreement Protocols
    Special Signature Schemes and Key Agreement Protocols Caroline J. Kudla Thesis submitted to the University of London for the degree of Doctor of Philosophy Information Security Group Department of Mathematics Royal Holloway, University of London 2006 Declaration These doctoral studies were conducted under the supervision of Kenneth G. Paterson. The work presented in this thesis is the result of original research carried out by myself, in collaboration with others, whilst enrolled in the Department of Mathematics as a candidate for the degree of Doctor of Philosophy. This work has not been submitted for any other degree or award in any other university or educational establishment. Caroline J. Kudla January, 2006 2 Acknowledgements My sincere thanks go to my supervisor, Kenny Paterson, whose dedication has been out- standing, and who has given generously of his time to guide and encourage me in my work over the last 3 years. His invaluable comments, constructive criticism and unwavering support have been crucial in helping me to achieve my goals and develop as a researcher. I would also like to thank my advisor, Chris Mitchell, for his encouragement in my various endeavors. I would like to express my deepest gratitude to HP Labs in Bristol, whose generous financial support has enabled me to pursue my studies. HP also permitted me to work from their labs, allowing me insight into the world of industrial research and providing me with invaluable experience. In particular, I would like to thank Keith Harrison for all his encouragement and advice during my time at HP, and Liqun Chen for her guidance and support.
    [Show full text]
  • A Practical Cryptanalysis of the Algebraic Eraser
    A Practical Cryptanalysis of the Algebraic Eraser Adi Ben-Zvi1, Simon R. Blackburn2, and Boaz Tsaban1 1 Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002, Israel 2 Department of Mathematics, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom Abstract. We present a novel cryptanalysis of the Algebraic Eraser primitive. This key agreement scheme, based on techniques from per- mutation groups, matrix groups and braid groups, is proposed as an underlying technology for ISO/IEC 29167-20, which is intended for au- thentication of RFID tags. SecureRF, the company owning the trade- mark Algebraic Eraser, markets it as suitable in general for lightweight environments such as RFID tags and other IoT applications. Our attack is practical on standard hardware: for parameter sizes corresponding to claimed 128-bit security, our implementation recovers the shared key us- ing less than 8 CPU hours, and less than 64MB of memory. 1 Introduction The Algebraic EraserTM is a key agreement scheme using techniques from non- commutative group theory. It was announced by Anshel, Anshel, Goldfeld and Lemieaux in 2004; the corresponding paper [1] appeared in 2006. The Alge- braic Eraser is defined in a very general fashion: various algebraic structures (monoids, groups and actions) need to be specified in order to be suitable for implementation. Anshel et al. provide most of this extra information, and name this concrete realisation of the Algebraic Eraser the Colored Burau Key Agree- ment Protocol (CBKAP). This concrete representation involves a novel blend of finite matrix groups and permutation groups with infinite braid groups. A com- pany, SecureRF, owns the trademark to the Algebraic Eraser, and is marketing this primitive as suitable for low resource environments such as RFID tags and Internet of Things (IoT) applications.
    [Show full text]
  • A Practical Cryptanalysis of the Algebraic Eraser
    View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Royal Holloway - Pure A Practical Cryptanalysis of the Algebraic Eraser Adi Ben-Zvi∗ Simon R. Blackburn† Boaz Tsaban∗ June 1, 2016 Abstract We present a novel cryptanalysis of the Algebraic Eraser primitive. This key agreement scheme, based on techniques from permutation groups, matrix groups and braid groups, is proposed as an underlying technology for ISO/IEC 29167-20, which is intended for authentica- tion of RFID tags. SecureRF, the company owning the trademark Algebraic Eraser, markets it as suitable in general for lightweight en- vironments such as RFID tags and other IoT applications. Our attack is practical on standard hardware: for parameter sizes corresponding to claimed 128-bit security, our implementation recovers the shared key using less than 8 CPU hours, and less than 64MB of memory. 1 Introduction The Algebraic Eraser™ is a key agreement scheme using techniques from non-commutative group theory. It was announced by Anshel, Anshel, Gold- feld and Lemieaux in 2004; the corresponding paper [1] appeared in 2006. ∗Department of Mathematics, Bar-Ilan University, Ramat Gan 5290002, Israel †Department of Mathematics, Royal Holloway University of London, Egham, Surrey TW20 0EX, United Kingdom © IACR 2016. This article is the final version submitted by the author(s) to the IACR and to Springer-Verlag on 1 June 2016. The version published by Springer-Verlag is available at [DOI to follow]. 1 The Algebraic Eraser is defined in a very general fashion: various algebraic structures (monoids, groups and actions) need to be specified in order to be suitable for implementation.
    [Show full text]
  • PGP 7.0 Mac Users Guide
    PGP Freeware for MacOS User’s Guide Version 7.0 Copyright©1990-2001NetworkAssociates,Inc.anditsAffiliatedCompanies.AllRights Reserved. PGP*,Version7.0.3 01-2001.PrintedintheUnitedStatesofAmerica. PGP,PrettyGood,andPrettyGoodPrivacyareregisteredtrademarksofNetworkAssociates, Inc.and/oritsAffiliatedCompaniesintheUSandothercountries.Allotherregisteredand unregisteredtrademarksinthisdocumentarethesolepropertyoftheirrespectiveowners. PortionsofthissoftwaremayusepublickeyalgorithmsdescribedinU.S.Patentnumbers 4,200,770,4,218,582,4,405,829,and4,424,414,licensedexclusivelybyPublicKeyPartners;the IDEA(tm)cryptographiccipherdescribedinU.S.patentnumber5,214,703,licensedfrom AscomTechAG;andtheNorthernTelecomLtd.,CASTEncryptionAlgorithm,licensedfrom NorthernTelecom,Ltd.IDEAisatrademarkofAscomTechAG.NetworkAssociatesInc.may havepatentsand/orpendingpatentapplicationscoveringsubjectmatterinthissoftwareor itsdocumentation;thefurnishingofthissoftwareordocumentationdoesnotgiveyouany licensetothesepatents.ThecompressioncodeinPGPisbyMarkAdlerandJean-LoupGailly, usedwithpermissionfromthefreeInfo-ZIPimplementation.LDAPsoftwareprovided courtesyUniversityofMichiganatAnnArbor,Copyright©1992-1996Regentsofthe UniversityofMichigan.Allrightsreserved.Thisproductincludessoftwaredevelopedbythe ApacheGroupforuseintheApacheHTTPserverproject(http://www.apache.org/).Balloon helpsupportcourtesyofJamesW.Walker.Copyright©1995-1999TheApacheGroup.All rightsreserved.SeetextfilesincludedwiththesoftwareorthePGPwebsiteforfurther information.ThissoftwareisbasedinpartontheworkoftheIndependentJPEGGroup.Soft
    [Show full text]
  • Group Theoretic Cryptography and the Algebraic Eraser™
    Group Theoretic Cryptography and the Algebraic Eraser™ Simon R. Blackburn Royal Holloway University of London 9th March 2016 Simon R. Blackburn (RHUL) The Algebraic Eraser 1 / 16 Overview 1 The Discrete Log Problem and Diffie-Hellman 2 Group-theoretic analogues of DLP 3 The Algebraic Eraser 4 Attacks Simon R. Blackburn (RHUL) The Algebraic Eraser 2 / 16 Discrete logarithm problem (DLP) ∗ Let p be a prime and let g; h 2 Zp. Find an integer x (if it exists) such that h ≡ g x mod p. In general, this is a hard computational problem (for large p). Example: Let p = 11, g = 2 and h = 9. Solve the DLP. x 0 1 2 3 4 5 6 7 8 9 g x 1 2 4 8 5 10 9 7 3 6 Simon R. Blackburn (RHUL) The Algebraic Eraser 3 / 16 Diffie-Hellman key exchange Alice and Bob want to agree on a random key K. ∗ They decide upon a large prime p and some g 2 Zp, then: a Alice chooses a random integer 1 ≤ a < p − 1 and sends c1 = g mod p to Bob. b Bob chooses a random integer 1 ≤ b < p − 1 and sends c2 = g mod p to Alice. a On receiving c2 Alice computes K = c2 mod p. b On receiving c1 Bob computes K = c1 mod p. Alice and Bob both share the same key K = g ab mod p. It works because( g a)b = (g b)a. Simon R. Blackburn (RHUL) The Algebraic Eraser 4 / 16 What does security mean? a b An adversary Eve knows p and g, and sees c1 = g and c2 = g .
    [Show full text]
  • Mathematical Introduction to Coding Theory and Cryptography Yi Ouyang
    Mathematical Introduction to Coding Theory and Cryptography Yi Ouyang School of Mathematical Sciences, University of Science and Technology of China Email address: [email protected] Contents Chapter 1. Preliminaries 1 1. Theory of Integers 1 2. Polynomials over a field 4 3. Finite fields 6 Notations 9 Part 1. Coding Theory 11 Chapter 2. Introduction to Coding Theory 13 1. Background 13 2. Basic Definitions 15 Chapter 3. Linear Codes 19 1. Basic definitions 19 2. Hamming codes 23 3. Equivalence of codes 23 4. Encoding and decoding Algorithms 24 Chapter 4. Bounds of codes and codes with good bounds 29 1. Bounds of codes 29 2. Golay Codes 33 3. Other examples of optimal codes 36 Chapter 5. Constructing Codes from Other Codes 39 1. General Rules for Construction 39 2. Reed Muller Codes 41 Chapter 6. Weight Enumerators and the MacWilliams Theorem 47 1. MacWilliams Identity 47 Chapter 7. Sequences over finite fields 53 1. Sequences and Power Series over Finite Fields 53 2. Linear Feedback Shift Registers(LFSR) 57 3. Berlekamp-Massey Algorithm 60 Chapter 8. Cyclic codes and BCH codes 63 1. Cyclic codes 63 2. Trace Expression of Cyclic codes 67 iii iv CONTENTS 3. The BCH codes 69 4. Goppa Code 75 Chapter 9. Generalized GRS codes 79 1. Generalized GRS codes 79 2. Decoding GRS codes 80 Part 2. Cryptography 83 Chapter 10. History and Basic knowledge about Cryptography 85 1. Cryptography from early age 85 Chapter 11. Hard Computational Problems 91 1. Trapdoor function and One-way function 91 2. Factoring and RSA 92 Chapter 12.
    [Show full text]