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© 2007 Levente Buttyán and Jean-Pierre Hubaux Security and Cooperation in Wireless Networks http://secowinet.epfl.ch/ Appendix A: Introduction to cryptographic algorithms and protocols symmetric and asymmetric key encryption; hash functions; MAC functions; digital signatures; key establishment protocols; advanced authentication techniques;

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 2/80 Chapter outline A.1 Introduction A.2 Encryption A.3 Hash functions A.4 Message authentication codes A.5 Digital signatures A.6 Session key establishment protocols A.7 Pseudo-random number generators A.8 Advanced authentication techniques

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 3/80 Introduction security is about how to prevent attacks, or -- if prevention is not possible -- how to detect attacks and recover from them an attack is a a deliberate attempt to compromise a system; it usually exploits weaknesses in the systems design, implementation, operation, or management attacks can be –passive attempts to learn or make use of information from the system but does not affect system resources examples: eavesdropping message contents, traffic analysis difficult to detect, should be prevented –active attempts to alter system resources or affect their operation examples: masquerade (spoofing), replay, modification (substitution, insertion, destruction), denial of service difficult to prevent, should be detected A.1 Introduction

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 4/80 Main security services authentication –aims to detect masquerade –provides assurance that a communicating entity is the one that it claims to be access control –aims to prevent unauthorized access to resources confidentiality –aims to protect data from unauthorized disclosure –usually based on encryption integrity –aims to detect modification and replay –provides assurance that data received are exactly as sent by the sender non-repudiation –provides protection against denial by one entity involved in a communication of having participated in all or part of the communication –two basic types: non-repudiation of origin and non-repudiation of delivery A.1 Introduction

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 5/80 Some security mechanisms encryption –symmetric key, asymmetric (public) key digital signature access control schemes –access control lists, capabilities, security labels,... data integrity mechanisms –message authentication codes, sequence numbering, time stamping, cryptographic chaining authentication protocols –passwords, cryptographic challenge-response protocols, biometrics traffic padding, route control, … A.1 Introduction

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 6/80 Chapter outline A.1 Introduction A.2 Encryption A.3 Hash functions A.4 Message authentication codes A.5 Digital signatures A.6 Session key establishment protocols A.7 Pseudo-random number generators A.8 Advanced authentication techniques

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 7/80 E E D D m plaintext K encryption key K decryption key E K (m) ciphertext D K (E K (m)) = m eavesdropping adversary Classical model of encryption goal of the adversary: –to systematically recover plaintexts from ciphertexts –to deduce the (decryption) key Kerckhoffs principle: –we must assume that the adversary knows all details of E and D –security of the system should be based on the protection of the decryption key A.2 Encryption

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 8/80 Adversary models ciphertext-only attack –the adversary can only observe ciphertexts produced by the same encryption key known-plaintext attack –the adversary can obtain corresponding plaintext-ciphertext pairs produced with the same encryption key (adaptive) chosen-plaintext attack –the adversary can choose plaintexts and obtain the corresponding ciphertexts (adaptive) chosen-ciphertext attack –the adversary can choose ciphertexts and obtain the corresponding plaintexts related-key attack –the adversary can obtain ciphertexts, or plaintext-ciphertext pairs that are produced with different encryption keys that are related in a known way to a specific encryption key A.2 Encryption

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 9/80 Security of encryption schemes an encryption scheme is secure in a given adversary model if it is computationally infeasible for the adversary to determine the target decryption key under the assumptions of the given model for many encryption schemes used in practice, no proof of security exists –these schemes are used, nevertheless, because they are efficient and they resist all known attacks some encryption schemes are provably secure, however these schemes are often inefficient A.2 Encryption

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 10/80 Basic classification encryption schemes symmetric-key encryption –it is easy to compute K from K (and vice versa) –usually K = K –two main types: stream ciphers – operate on individual characters of the plaintext block ciphers – process the plaintext in larger blocks of characters asymmetric-key encryption –it is hard (computationally infeasible) to compute K from K –K can be made public ( public-key cryptography) A.2 Encryption

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 11/80 Block ciphers an n bit block cipher is a function E: {0, 1} n x {0, 1} k {0, 1} n, such that for each K {0, 1} k, E(x, K) = E K (x) is an invertible mapping from {0, 1} n to {0, 1} n E E … … … n bit input n bit output k bit key A.2 Encryption Block ciphers

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 12/80 Block cipher design criteria completeness –each bit of the output block should depend on each bit of the input block and on each bit of the key avalanche effect –changing one bit in the input block should change approximately half of the bits in the output block –similarly, changing one key bit should result in the change of approximately half of the bits in the output block statistical independence –input and output should appear to be statistically independent A.2 Encryption Block ciphers

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 13/80 How to satisfy the design criteria? complex encryption function can be built by composing several simple operations which offer complementary – but individually insufficient – protection simple operations: –elementary arithmetic operations –logical operations (e.g., XOR) –modular multiplication –transpositions –substitutions –... combine two or more transformations in a manner that the resulting cipher is more secure than the individual components A.2 Encryption Block ciphers

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 14/80 Example: DES (Data Encryption Standard) input size: 64 output size: 64 key size: 56 16 rounds Feistel structure Initial Permutation F F + F F + F F + F F + … Initial Permutation -1 (64) (32) (48) Key Scheduler (56) K K1K1 K2K2 K 16 K3K3 X Y A.2 Encryption Block ciphers

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 15/80 Example: the DES round function F Si – substitution box (S-box) P – permutation box (P-box ) A.2 Encryption Block ciphers

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 16/80 Example: the DES key scheduler each key bit is used in around 14 out of 16 rounds Permuted Choice 1 Permuted Choice 2 Left shift(s) Permuted Choice 2 Left shift(s) … (28) (56) K (28) (48) K1K1 K2K2 A.2 Encryption Block ciphers

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 17/80 Exhaustive key search attack given a small number of plaintext-ciphertext pairs encrypted under a key K, K can be recovered by exhaustive key search with 2 k-1 processing complexity (expected number of operations) –input: (X, Y), (X, Y), … –progress through the entire key space, and for each candidate key K, do the following: decrypt Y with K if the result is not X, then throw away K if the result is X, then check the other pairs (X, Y), … if K does not work for at least one pair, then throw away K and take another key if K worked for all pairs (X, Y), (X, Y), …, then output K as the target key –on average, the target key is found after searching half of the key space if the plaintexts are known to contain redundancy, then ciphertext-only exhaustive key search is possible with a relatively small number of ciphertexts in practice, key size should be at least 128 bits A.2 Encryption Block ciphers

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 18/80 Algebraic attacks having a large key size is only a necessary condition for the security of a block cipher –a block cipher can be broken due to the weaknesses in its internal (algebraic) structure, even if it uses large keys example: –naïve exhaustive key search against DES: 2 55 –attack using the complementation property of DES: 2 54 Y = DES K (X) implies Y* = DES K* (X*), where X* denotes the bitwise complement of X –differential cryptanalysis of DES: 2 47 –linear cryptanalysis of DES: 2 43 A.2 Encryption Block ciphers

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 19/80 Block cipher modes of operation ECB – Electronic Codebook –used to encipher a single plaintext block (e.g., a DES key) CBC – Cipher Block Chaining –repeated use of the encryption algorithm to encipher a message consisting of many blocks CFB – Cipher Feedback –used to encipher a stream of characters, dealing with each character as it comes OFB – Output Feedback –another method of stream encryption, used on noisy channels CTR – Counter –simplified OFB with certain advantages A.2 Encryption Block cipher modes of operation

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 20/80 ECB mode encrypt decrypt E E P1P1 C1C1 K E E P2P2 C2C2 K E E PNPN CNCN K … D D C1C1 P1P1 K D D C2C2 P2P2 K D D CNCN PNPN K … A.2 Encryption Block cipher modes of operation

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 21/80 Properties of the ECB mode identical plaintext blocks result in identical ciphertext blocks (under the same key, of course) –messages to be encrypted often have very regular formats –repeating fragments, special headers, string of 0s, etc. are quite common blocks are encrypted independently of other blocks –reordering ciphertext blocks result in correspondingly reordered plaintext blocks –ciphertext blocks can be cut from one message and pasted in another, possibly without detection error propagation: one bit error in a ciphertext block affects only the corresponding plaintext block (results in garbage) overall: not recommended for messages longer than one block, or if keys are reused for more than one block A.2 Encryption Block cipher modes of operation

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 22/80 CBC mode encrypt decrypt E E P1P1 C1C1 K + E E P2P2 C2C2 K + E E P3P3 C3C3 K + E E PNPN CNCN K + IV C N-1 … D D C1C1 P1P1 K + IV D D C2C2 P2P2 K + D D C3C3 P3P3 K + D D CNCN PNPN K + C N-1 A.2 Encryption Block cipher modes of operation

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 23/80 Properties of the CBC mode encrypting the same plaintexts under the same key, but different IVs result in different ciphertexts ciphertext block C j depends on P j and all preceding plaintext blocks –rearranging ciphertext blocks affects decryption –however, dependency on the preceding plaintext blocks is only via the previous ciphertext block C j-1 –proper decryption of a correct ciphertext block needs a correct preceding ciphertext block only error propagation: –one bit error in a ciphertext block C j has an effect on the j-th and (j+1)-st plaintext block P j is complete garbage and P j+1 has bit errors where C j had an attacker may cause predictable bit changes in the (j+1)-st plaintext block error recovery: –recovers from bit errors (self-synchronizing) –cannot, however, recover from frame errors (lost bits) A.2 Encryption Block cipher modes of operation

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 24/80 Integrity of the IV in the CBC mode the IV need not be secret, but its integrity should be protected –malicious modification of the IV allows an attacker to make predictable changes to the first plaintext block recovered one solution is to send the IV in an encrypted form at the beginning of the CBC encrypted message A.2 Encryption Block cipher modes of operation

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 25/80 Padding the length of the message may not be a multiple of the block size of the cipher one can add some extra bytes to the short end block until it reaches the correct size – this is called padding usually the last byte indicates the number of padding bytes added – this allows the receiver to remove the padding 05 short end blockpadding A.2 Encryption Block cipher modes of operation

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 26/80 CFB mode encrypt decrypt E E PiPi CiCi K + shift register (n) (n) select s bits (n) (s) initialized with IV E E CiCi PiPi K + shift register (n) (n) select s bits (n) (s) initialized with IV A.2 Encryption Block cipher modes of operation

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 27/80 Properties of the CFB mode encrypting the same plaintexts under the same key, but different IVs result in different ciphertexts the IV can be sent in clear ciphertext block C j depends on P j and all preceding plaintext blocks –rearranging ciphertext blocks affects decryption –proper decryption of a correct ciphertext block needs the preceding n/s ciphertext blocks to be correct error propagation: –one bit error in a ciphertext block C j has an effect on the decryption of that and the next n/s ciphertext blocks (the error remains in the shift register for n/s steps) P j has bit errors where C j had, all the other erroneous plaintext blocks are garbage an attacker may cause predictable bit changes in the j-th plaintext block error recovery: –self synchronizing, but requires n/s blocks to recover A.2 Encryption Block cipher modes of operation

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 28/80 OFB mode encrypt decrypt E E PiPi CiCi K + shift register (n) (n) select s bits (n) (s) initialized with IV E E CiCi PiPi K + shift register (n) (n) select s bits (n) (s) initialized with IV A.2 Encryption Block cipher modes of operation

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 29/80 Properties of the OFB mode a different IV should be used for every new message, otherwise messages will be encrypted with the same key stream the IV can be sent in clear –however, if the IV is modified by the attacker, then the cipher will never recover (unlike CFB) ciphertext block C j depends on P j only (does not depend on the preceding plaintext blocks) –however, rearranging ciphertext blocks affects decryption error propagation: –one bit error in a ciphertext block C j has an effect on the decryption of only that ciphertext block P j has bit errors where C j had an attacker may cause predictable bit changes in the j-th plaintext block error recovery: –recovers from bit errors –never recovers if bits are lost or the IV is modified A.2 Encryption Block cipher modes of operation

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 30/80 CTR mode encrypt decrypt E E PiPi CiCi K + (n) counter + i (n) E E CiCi PiPi K + counter + i (n) A.2 Encryption Block cipher modes of operation

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 31/80 Properties of the CTR mode similar to OFB cycle length depends on the size of the counter (typically 2 n ) the i-th block can be decrypted independently of the others –parallelizable (unlike OFB) –random access the values to be XORed with the plaintext can be pre-computed at least as secure as the other modes note1: in CFB, OFB, and CTR mode only the encryption algorithm is used (decryption is not needed), that is why some ciphers (e.g., AES) is optimized for encryption note2: the OFB and CTR modes essentially make a synchronous stream cipher out of a block cipher, whereas the CFB mode converts a block cipher into a self-synchronizing stream-cipher A.2 Encryption Block cipher modes of operation

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 32/80 Stream ciphers while block ciphers simultaneously encrypt groups of characters, stream ciphers encrypt individual characters –may be better suited for real time applications stream ciphers are usually faster than block ciphers in hardware (but not necessarily in software) limited or no error propagation –may be advantageous when transmission errors are probable note: the distinction between stream ciphers and block ciphers is not definitive –stream ciphers can be built out of block ciphers using CFB, OFB, or CTR modes –a block cipher in ECB or CBC mode can be viewed as a stream cipher that operates on large characters A.2 Encryption Stream ciphers

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 33/80 Synchronous stream ciphers the key stream is generated independently of the plaintext and of the ciphertext needs synchronization between the sender and the receiver –if a character is inserted into or deleted from the ciphertext stream then synchronization is lost and the plaintext cannot be recovered –additional techniques must be used to recover from loss of synch. no error propagation –a ciphertext character that is modified during transmission affects only the decryption of that character an attacker can make changes to selected ciphertext characters and know exactly what effect these changes have on the plaintext (if h = XOR) i i gkgk gkgk h h fkfk fkfk i+1 zizi pipi cici A.2 Encryption Stream ciphers

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 34/80 Self-synchronizing stream ciphers the key stream is generated as a function of a fixed number of previous ciphertext characters self-synchronizing –since the size t of the register is fixed, a lost ciphertext character affects only the decryption of the next t ciphertext characters limited error propagation –if a ciphertext character is modified, then decryption of the next t ciphertext characters may be incorrect ciphertext characters depend on all previous plaintext characters –better diffusion of plaintext statistics gkgk gkgk h h zizi pipi cici … register A.2 Encryption Stream ciphers

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 35/80 The Vernam cipher and the one-time pad Vernam cipher –c i = p i k i for i = 1, 2, … where p i are the plaintext digits, k i are the key stream digits, c i are the ciphertext digits, and is the bitwise XOR operation one-time pad –a Vernam cipher where the key stream digits are generated independently and uniformly at random –the one-time pad is unconditionally secure [Shannon, 1949] I(P; C) = H(P) - H(P|C) = 0 –a necessary condition for a symmetric key cipher to be unconditionally secure is that H(K) H(P) [Shannon, 1949] practically, the key must have as many bits as the compressed plaintext impractical because of key management problems A.2 Encryption Stream ciphers

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 36/80 Asymmetric-key encryption asymmetric-key encryption –it is hard (computationally infeasible) to compute K from K K can be made public (public-key cryptography) –no need for key setup before communication public-keys are not confidential but they must be authentic ! the security of asymmetric-key encryption schemes is usually based on some well-known or widely believed hard problems E E D D m plaintext K (public) encryption key K (private) decryption key E K (m) ciphertext D K (E K (m)) = m attacker A.2 Encryption A.2.2 Asymmetric-key encryption

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 37/80 Examples for hard problems factoring problem –given a positive integer n, find its prime factors true complexity is unknown it is believed that it does not belong to P discrete logarithm problem –given a prime p, a generator g of Z p *, and an element y in Z p *, find the integer x, 0 x p-2, such that g x mod p = y true complexity is unknown it is believed that it does not belong to P Diffie-Hellman problem –given a prime p, a generator g of Z p *, and elements g x mod p and g y mod p, find g xy mod p true complexity is unknown it is believed that it does not belong to P A.2 Encryption A.2.2 Asymmetric-key encryption

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 38/80 The RSA encryption scheme key generation –select p, q large primes (about 500 bits each) –n = pq, (n) = (p-1)(q-1) –select e such that 1 < e < (n) and gcd(e, (n)) = 1 –compute d such that ed mod (n) = 1 (this is easy if (n) is known) –the public key is (e, n) –the private key is d encryption –represent the message as an integer m in [0, n-1] –compute c = m e mod n decryption –compute m = c d mod n A.2 Encryption A.2.2 Asymmetric-key encryption

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 39/80 Relation to factoring the problem of computing d from (e, n) is computationally equivalent to the problem of factoring n –if one can factor n, then one can easily compute d –if one can compute d, then one can efficiently factor n the problem of computing m from c and (e, n) (called the RSA problem) is believed to be computationally equivalent to factoring –if one can factor n, then one can easily compute m from c and (e, n) –theres no formal proof for the other direction A.2 Encryption A.2.2 Asymmetric-key encryption

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 40/80 The need for salting let us assume that the adversary observes a ciphertext c = E K (m) let the set of possible plaintexts be M if M is small, then the adversary can try to encrypt every message in M with the publicly known key K until she finds the message m that maps into c the usual way to prevent this attack is to randomize the encryption –some random bytes are added to the plaintext message before encryption through the application of the PKCS #1 formatting rules –when the message is decrypted, the recipient can recognize and discard these random bytes A.2 Encryption A.2.2 Asymmetric-key encryption

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 41/80 The ElGamal encryption scheme key generation –generate a large random prime p and choose generator g of the multiplicative group Z p * = {1, 2, …, p-1} –select a random integer a, 1 a p-2, and compute A = g a mod p –the public key is (p, g, A) –the private key is a encryption –represent the message as an integer m in [0, p-1] –select a random integer r, 1 r p-2, and compute R = g r mod p –compute C = m A r mod p –the ciphertext is the pair (R, C) decryption –compute m = C R p-1-a mod p proof of decryption C R p-1-a m A r R p-1-a m g ar g r(p-1-a) m g p-1 ) r m (mod p) A.2 Encryption A.2.2 Asymmetric-key encryption

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 42/80 Relation to hard problems security of the ElGamal scheme is said to be based on the discrete logarithm problem in Z p *, although equivalence has not been proven yet recovering m given p, g, A, R, and C is equivalent to solving the Diffie-Hellman problem A.2 Encryption A.2.2 Asymmetric-key encryption

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 43/80 Digital enveloping plaintext message symmetric-key cipher (e.g., in CBC mode) symmetric-key cipher (e.g., in CBC mode) public key of the receiver asymmetric-key cipher asymmetric-key cipher digital envelop generate random symmetric key generate random symmetric key bulk encryption key most popular public-key encryption methods are several orders of magnitude slower than the best known symmetric key schemes public-key encryption is used together with symmetric-key encryption; the technique is called digital enveloping A.2 Encryption A.2.2 Asymmetric-key encryption

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 44/80 Chapter outline A.1 Introduction A.2 Encryption A.3 Hash functions A.4 Message authentication codes A.5 Digital signatures A.6 Session key establishment protocols A.7 Pseudo-random number generators A.8 Advanced authentication techniques

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 45/80 Hash functions a hash function maps bit strings of arbitrary finite length to bit strings of fixed length (n bits) many-to-one mapping collisions are unavoidable however, finding collisions are difficult the hash value of a message can serve as a compact representative image of the message (similar to fingerprints) message of arbitrary length fix length hash value / message digest / fingerprint hash function hash function A.3 Hash functions

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 46/80 Desirable properties of hash functions ease of computation –given an input x, the hash value h(x) of x is easy to compute weak collision resistance (2 nd preimage resistance) –given an input x, it is computationally infeasible to find a second input x such that h(x) = h(x) strong collision resistance (collision resistance) –it is computationally infeasible to find any two distinct inputs x and x such that h(x) = h(x) one-way property (preimage resistance) –given a hash value y (for which no preimage is known), it is computationally infeasible to find any input x s.t. h(x) = y A.3 Hash functions

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 47/80 The Birthday Paradox Given a set of N elements, from which we draw k elements randomly (with replacement). What is the probability of encountering at least one repeating element? first, compute the probability of no repetition: –the first element x 1 can be anything –when choosing the second element x 2, the probability of x 2 x 1 is 1-1/N –when choosing x 3, the probability of x 3 x 2 and x 3 x 1 is 1-2/N –… –when choosing the k-th element, the probability of no repetition is 1- (k-1)/N –the probability of no repetition is (1 - 1/N)(1 - 2/N)…(1 – (k-1)/N) –when x is small, (1-x) e -x –(1 - 1/N)(1 - 2/N)…(1 – (k-1)/N) = e -1/N e -2/N … e -(k-1)/N = e -k(k-1)/2N the probability of at least one repetition after k drawing is 1 – e -k(k-1)/2N A.3 Hash functions

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 48/80 The Birthday Paradox (contd) How many drawings do you need, if you want the probability of at least one repetition to be ? solve the following for k: = 1 – e -k(k-1)/2N k(k-1) = 2N ln(1/1- ) k sqrt(2N ln(1/1- )) examples: = ½ k 1.177 sqrt(N) = ¾ k 1.665 sqrt(N) = 0.9 k 2.146 sqrt(N) origin of the name Birthday Paradox: –elements are dates in a year (N = 365) –among 1.177 sqrt(365) 23 randomly selected people, there will be at least two that have the same birthday with probability ½ A.3 Hash functions

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 49/80 Choosing the output size of a hash function the Birthday Paradox have a profound impact on the design of hash functions (and other cryptographic algorithms and protocols)! –let n be the output size of a hash function –among ~sqrt(2 n ) = 2 n/2 randomly chosen messages, with high probability, there will be a collision pair –it is easier to find collisions than to find preimages or 2 nd preimages for a given hash value in order to resist birthday attacks, 2 n/2 should be sufficiently large (e.g., n = 160 bits) A.3 Hash functions

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 50/80 Iterated hash functions input is divided into fixed length blocks x 1, x 2, …, x L last block is padded if necessary –Merkle-Damgard strengthening: padding contains the length of the message each input block is processed according to the following scheme f is called the compression function –can be based on a block cipher, or –can be a dedicated compression function x1x1 CV 0 (b) (n) CV 1 f f x2x2 (b) (n) CV 2 f f x3x3 (b) (n) CV 3 f f xLxL (b) (n) h(x) = CV L f f CV L-1 … A.3 Hash functions

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 51/80 Chapter outline A.1 Introduction A.2 Encryption A.3 Hash functions A.4 Message authentication codes A.5 Digital signatures A.6 Session key establishment protocols A.7 Pseudo-random number generators A.8 Advanced authentication techniques

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 52/80 Message authentication codes (MACs) MAC functions can be viewed as hash functions with two functionally distinct inputs: a message and a secret key they produce a fixed size output (say n bits) called the MAC practically it should be infeasible to produce a correct MAC for a message without the knowledge of the secret key MAC functions can be used to implement data integrity and message origin authentication services message of arbitrary length fix length MAC function MAC function secret key A.4 Message authentication codes

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 53/80 MAC generation and verification MAC message MAC generation secret key MAC message MAC verification secret key compare yes/no A.4 Message authentication codes

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 54/80 Desirable properties of MAC functions ease of computation –given an input x and a secret key k, it is easy to compute MAC k (x) key non-recovery –it is computationally infeasible to recover the secret key k, given one or more text-MAC pairs (x i, MAC k (x i )) for that k computation resistance –given zero or more text-MAC pairs (x i, MAC k (x i )), it is computationally infeasible to find a text-MAC pair (x, MAC k (x)) for any new input x x i –computation resistance implies key non-recovery but the reverse is not true in general A.4 Message authentication codes

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 55/80 CBC MAC CBC MAC is secure for messages of a fixed number of blocks (adaptive chosen-text existential) forgery is possible if variable length messages are allowed it is recommended to involve the length of the message in the CBC MAC computation E E x1x1 k + E E x2x2 k + E E x3x3 k + E E xNxN cNcN k + 0 c N-1 … c1c1 c2c2 c3c3 E -1 E E k k MAC optional A.4 Message authentication codes

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 56/80 HMAC k + ipad CV 0 f f x1x1 f f x L |padding 1 f f k + opad CV 0 f f M|padding 2 f f M CV 1 inner CV 1 outer HMAC k (x) … hash fn HMAC k (X) = H( k|H( k|X )) A.4 Message authentication codes

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 57/80 Chapter outline A.1 Introduction A.2 Encryption A.3 Hash functions A.4 Message authentication codes A.5 Digital signatures A.6 Session key establishment protocols A.7 Pseudo-random number generators A.8 Advanced authentication techniques

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 58/80 Digital signatures similar to MACs but –unforgeable by the receiver –verifiable by a third party used for message authentication and non-repudiation (of message origin) based on public-key cryptography –private key defines a signing transformation S A S A (m) = –public key defines a verification transformation V A V A (m, ) = true if S A (m) = V A (m, ) = false otherwise A.5 Digital signatures

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 59/80 Attacks on digital signature schemes key-only attack –only the public key is available to the adversary known-message attack –the adversary has signatures for a set of messages known to her but not chosen by her chosen-message attack –the adversary obtains signatures for messages chosen by her before attempting to break the signature scheme adaptive chosen-message attack –the adversary is allowed to use the signer as an oracle –she may request signatures for messages which depend on previously obtained signatures A.5 Digital signatures

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 60/80 Hash-and-sign paradigm public/private key operations are slow hash the message first and apply public/private key operations to the hash value only h h enc private key of sender message hash signature h h message hash dec public key of sender signature compare yes/no generation verification A.5 Digital signatures

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 61/80 Examples of digital signature scheme RSA –essentially identical to the RSA encryption scheme –signature = decryption with private key –typical signature length is 1024 bits DSA (Digital Signature Algorithm) –based on the ElGamal signature scheme –typical signature length is 1024 bits ECDSA (Elliptic Curve DSA) –same as DSA but works over elliptic curves –reduced signature length (typically 320 bits) A.5 Digital signatures

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 62/80 Chapter outline A.1 Introduction A.2 Encryption A.3 Hash functions A.4 Message authentication codes A.5 Digital signatures A.6 Session key establishment protocols A.7 Pseudo-random number generators A.8 Advanced authentication techniques

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 63/80 Session key establishment protocols goal of session key establishment protocols –to setup a shared secret between two (or more) parties –it is desired that the secret established by a fixed pair of parties varies on subsequent executions of the protocol (dynamicity) –established shared secret is used as a session key to protect communication between the parties motivation for use of session keys –to limit available ciphertext for cryptanalysis –to limit exposure caused by the compromise of a session key –to avoid long-term storage of a large number of secret keys (keys are created on-demand when actually required) –to create independence across communication sessions or applications A.6 Session key establishment protocols

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 64/80 Basic classification key transport protocols –one party creates or otherwise obtains a secret value, and securely transfers it to the other party key agreement protocols –a shared secret is derived by the parties as a function of information contributed by each, such that no party can predetermine the resulting value A.6 Session key establishment protocols

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 65/80 Further services entity authentication implicit key authentication –one party is assured that no other party aside from a specifically identified second party (and possibly some trusted third parties) may gain access to the established session key key confirmation –one party is assured that a second (possibly unidentified) party actually possesses the session key –possession of a key can be demonstrated by producing a one-way hash value of the key or encryption of known data with the key explicit key authentication –implicit key authentication + key confirmation key freshness –one party is assured that the key is new (never used before) A.6 Session key establishment protocols

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 66/80 Further protocol characteristics reciprocity –guarantees are provided unilaterally –guarantees are provided mutually efficiency –number of message exchanges (passes) required –total number of bits transmitted (i.e., bandwidth used) –complexity of computations by each party –possibility of precomputations to reduce on-line computational complexity third party requirements –on-line, off-line, or no third party at all –degree and type of trust required in the third party system setup –distribution of initial keying material A.6 Session key establishment protocols

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 67/80 The Wide-Mouth-Frog protocol AliceBob Server generate k A, E Kas ( B, k, T a ) E Kbs ( A, k, T s ) summary: a simple key transport protocol that uses a trusted third party Alice generates the session key and sends it to Bob via the trusted third party characteristics: implicit key authentication for Alice explicit key authentication for Bob key freshness for Bob with timestamps (flawed) unilateral entity authentication of Alice on-line third party (Server) trusted for secure relaying of keys and verification of freshness, in addition A is trusted for generating good keys initial long-term keys between the parties and the server are required A.6 Session key establishment protocols

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 68/80 A flaw in the Wide-Mouth-Frog protocol summary: after observing one run of the protocol, Trudy can continuously use the Server as an oracle until she wants to bring about re-authentication between Alice and Bob B, E K bs ( A, k, T s ) E K as ( B, k, T s (1) ) A, E K as ( B, k, T s (1) ) E K bs ( A, k, T s (2) )... E K bs ( A, k, T s (n) ) BobTrudy Server A.6 Session key establishment protocols

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 69/80 Signed encrypted keys BobAlice A, E Kb (k), T a, S Ka (B,E Kb (k),T a ) generate k summary: Alice generates a session key, encrypts it with Bobs public key, then sings it, and sends it to Bob characteristics: unilateral entity authentication (of Alice), mutual implicit key authentication, no key confirmation, key freshness with timestamp, clock synchronization, off-line third party for issuing public key certificates may be required, initial exchange of public keys between the parties may be required, Alice is trusted to generate keys, non-repudiation guarantee for Bob A.6 Session key establishment protocols

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 70/80 The Diffie-Hellman protocol BobAlice select random x compute g x mod p select random y compute g y mod p g x mod p g y mod p compute k = (g y ) x mod pcompute k = (g x ) y mod p summary: a key agreement protocol based on one-way functions; in particular, security of the protocol is based on the hardness of the discrete logarithm problem and that of the Diffie-Hellman problem characteristics: NO AUTHENTICATION, key freshness with randomly selected exponents, no party can control the key, no need for a trusted third party assumptions: p is a large prime, g is a generator of Z p *, both are publicly known system parameters A.6 Session key establishment protocols

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 71/80 Chapter outline A.1 Introduction A.2 Encryption A.3 Hash functions A.4 Message authentication codes A.5 Digital signatures A.6 Session key establishment protocols A.7 Pseudo-random number generators A.8 Advanced authentication techniques

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 72/80 Pseudo-random number generators (PRNGs) a random number is a number that cannot be predicted by an observer before it is generated –if the number is generated within the range [0, N-1], then its value cannot be predicted with any better probability than 1/N –the above is true even if the observer is given all previously generated numbers a cryptographic pseudo-random number generator (PRNG) is a mechanism that processes somewhat unpredictable inputs and generates pseudo-random outputs –if designed, implemented, and used properly, then even an adversary with enormous computational power should not be able to distinguish the PRNG output from a real random sequence A.7 Pseudo-random number generators

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 73/80 General operation of PRNGs A.7 Pseudo-random number generators internal state internal state unpredictable input samples (from physical processes) pseudo-random bits indistinguishable from real random bits … one-way fn entropy pool entropy pool re-keying state update

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 74/80 Desirable properties of PRNGs the adversary cannot compute the internal state of the PRNG, even if she has observed many outputs of the PRNG the adversary cannot compute the next output of the PRNG, even if she has observed many previous outputs of the PRNG if the adversary can observe or even manipulate the input samples that are fed in the PRNG, but she does not know the internal state of the PRNG, then the adversary cannot compute the next output and the next internal state of the PRNG if the adversary has somehow learned the internal state of the PRNG, but she cannot observe the input samples that are fed in the PRNG, then the adversary cannot figure out the internal state of the PRNG after the re-keying operation A.7 Pseudo-random number generators

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 75/80 Chapter outline A.1 Introduction A.2 Encryption A.3 Hash functions A.4 Message authentication codes A.5 Digital signatures A.6 Session key establishment protocols A.7 Pseudo-random number generators A.8 Advanced authentication techniques

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 76/80 Hash chains a hash chain is a sequence of hash values that are computed by iteratively calling a one-way hash function on an initial value v 0 properties: –given v i, it is easy to compute any v j for j > i (v j = h (j-i) (v i )) –but it is difficult to compute v k for k < i (one-way property of h) hash chains can be used for repeated authentications at the cost of a single digital signature –Alice computes a hash chain and commits to it by signing v n and distributing it to potential verifiers –later on, Alice can authenticate herself repeatedly (at most n times) by revealing the elements of the hash chain in reverse order –when v n-i is revealed, verifiers can check if h (i) (v n-i ) = v n (or h(v n-i ) = v n-i+1 if they remember the last revealed element) –each hash chain element can be used only once for authenticating Alice –verifiers are assured that only Alice could have released the next hash chain element hash chains can be stored efficiently with a storage complexity that is logarithmic in the length of the hash chain A.8 Advanced authentication techniques v0v0 v1v1 v2v2 v3v3 v n-1 vnvn hhhhh …

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 77/80 Merkle-trees the limitation of hash chains is that elements can only be revealed sequentially Merkle-trees overcome this problem by allowing for the pre-authentication of a set of values with a single digital signature (on the root u 0 of the tree) and for the revelation of those values in any order when revealing a value v i, Alice must also reveal all the values assigned to the sibling vertices on the path from v i to the root (e.g., v 3 is revealed together with v 4, u 12, u 5678 ) verifiers hash the revealed values appropriately and check if the result is u 0 A.8 Advanced authentication techniques

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 78/80 TESLA a broadcast authentication mechanism based on symmetric key cryptographic primitives main idea: asymmetry through delayed disclosure of authentication keys –Alice wants to broadcast a message m –Alice computes a MAC on m with a key unknown to the verifiers –verifiers receive message m with the MAC, but they cannot immediately verify authenticity –later, Alice discloses the key used to compute the MAC –verifiers can now verify the MAC; if it is correct, they know that the message was sent by Alice, because at the time of reception nobody else knew the key assumptions: –loose time synchronization between the participants –each party knows an upper bound on the maximum synchronization error –initial secret between the parties to bootstrap the whole mechanism A.8 Advanced authentication techniques

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 79/80 TESLA (contd) MAC keys are consecutive elements in a one-way key chain: –K 0 K 1 … K n –K i = h(K i-1 ) protocol operation: –setup: Alice sends K n to each verifier in an authentic manner –time is divided into epochs –each message sent in epoch i is authenticated with key K n-i –K n-i is disclosed in epoch i+d, where d is a system parameter –K n-i is verified by checking h(K n-i ) = K n-i+1 example: K n-1 K n-2 K n-3 K n-4 P1P1 P2P2 P3P3 P4P4 P5P5 P6P6 P7P7 time K n-1 K n-2 K n-3 key disclosure schedule KnKn A.8 Advanced authentication techniques

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Security and Cooperation in Wireless Networks Appendix A: Intro to cryptographic algorithms and protocols 80/80 Conclusions security services are implemented by using security mechanisms many security mechanisms are based on cryptography (e.g., encryption, digital signature, message authentication codes, …) but be cautious: If you think cryptography is going to solve your problem, you don't understand cryptography and you don't understand your problem. -- Roger Needham other important aspects are –physical protection –procedural rules –education

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