# Part VII. Digital signatures

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Part VII Digital signatures

2 CHAPTER 7: Digital signatures Digital signatures are one of the most important inventions/applications of modern cryptography. The problem is how can a user sign a message such that everybody (or the intended addressee only) can verify the digital signature and the signature is good enough also for legal purposes. Example: Assume that each user A uses a public-key cryptosystem (e A,d A ). Signing a message w by a user A, so that any user can verify the signature; d A (w) Signing a message w by a user A so that only user B can verify the signature; e B (d A (w)) Sending a message w, and a signed message digest of w, obtained by using a hash function h: (w, d A (h(w))) Example Assume Alice succeeds to factor the integer Bob used, as modulus, to sign his will, using RSA, 20 years ago. Even the key has already expired, Alice can rewrite Bob s will, leaving fortune to her, and date it 20 years ago. Moral: It may pay off to factor a single integers using many years of many computers power. prof. Jozef Gruska IV Digital signatures 2/44

3 Digital signatures basic goals Digital sigantures should be such that each user should be able to verify signatures of other users, but that should give him/her no information how to sign a message on behind of other users. An important difference from a handwritten signature is that digital signature of a message is always intimately connected with the message, and for different messages is different, whereas the handwritten signature is adjoined to the message and always looks the same. Technically, a digital signature signing is performed by a signing algorithm and a digital signature is verified by a verification algorithm. A copy of a digital (classical) signature is identical (usually distinguishable) to (from) the origin. A care has therefore to be taken that a classical signature is not misused. This chapter contains some of the main techniques for design and verification of digital signatures (as well as some attacks to them). prof. Jozef Gruska IV Digital signatures 3/44

4 Digital signatures If only signature (but not the encryption of the message) are of importance, then it suffices that Alice sends to Bob Caution: Signing a message w by A for B by (w, d A (w)) e B (d A (w)) is O.K., but the symmetric solution, with encoding first: is not good. c = d A (e B (w)) An active enemy, the tamperer, can intercept the message, then can compute d T (e A (c)) = d T (e B (w)) and can send the outcome to Bob, pretending that it is from him/tamperer (without being able to decrypt/know the message). Any public-key cryptosystem in which the plaintext and cryptotext spaces are the same can be used for digital signature. prof. Jozef Gruska IV Digital signatures 4/44

5 Digital Signature Schemes I Digital signatures are basic tools for authentication and non-repudiation of messages. A digital signature scheme allows anyone to verify signature of any sender S without providing any information how to generate signatures of S. A Digital Signature Scheme (M, S, K s, K v ) is given by: M - a set of messages to be signed S - a set of possible signatures K s - a set of private keys for signing K v - a set of public keys for verification Moreover, it is required that: For each k from K s, there exists a single and easy to compute signing mapping sig k : {0, 1} M S For each k from K v there exists a single and easy to compute verification mapping ver k : M S {true, false} such that the following two conditions are satisfied: prof. Jozef Gruska IV Digital signatures 5/44

6 Digital Signature Schemes II Correctness: For each message m from M and public key k in K v, it holds if there is an r from {0, 1} such that ver k (m, s) = true s = sig l (r, m) for a private key l from K s corresponding to the public key k. Security: For any w from M and k in K v, it is computationally infeasible, without the knowledge of the private key corresponding to k, to find a signature s from S such that ver k (w, s) = true. prof. Jozef Gruska IV Digital signatures 6/44

7 Attacks on digital signatures Total break of a signature scheme: The adversary manages to recover the secret key from the public key. Universal forgery: The adversary can derive from the public key an algorithm which allows to forge the signature of any message. Selective forgery: The adversary can derive from the public key a method to forge signatures of selected messages (where selection was made prior the knowledge of the public key). Existential forgery: The adversary is able to create from the public key a valid signature of a message m (but has no control for which m). prof. Jozef Gruska IV Digital signatures 7/44

8 A digital signature of one bit Let us start with a very simple but much illustrating (though non-practical) example how to sign a single bit. Design of the signature scheme: A one-way function f(x) is chosen. Two integers k 0 and k 1 are chosen, by the signer, kept secret, and items are made public, where Signature of a bit b: Verification of such a signature f, (0, s 0), (1, s 1) s 0 = f (k 0), s 1 = f (k 1) (b, k b ). s b = f (k b ) SECURITY? prof. Jozef Gruska IV Digital signatures 8/44

9 RSA signatures and their attacks Let us have an RSA cryptosystem with encryption and decryption exponents e and d and modulus n. Signing of a message w s = (w, σ), where σ = w d mod n Verification of a signature s = (w, σ): w = σ e mod n? Attacks It might happen that Bob accepts a signature not produced by Alice. Indeed, let Eve, using Alice s public key, compute w e and say that (w e, w) is a message signed by Alice. Everybody verifying Alice s signature gets w e = w e. Some new signatures can be produced without knowing the secret key. Indeed, is σ 1 and σ 2 are signatures for w 1 and w 2, then σ 1σ 2 and σ 1 1 are signatures for w 1w 2 and w 1 1. prof. Jozef Gruska IV Digital signatures 9/44

10 ENCRYPTION versus SIGNATURE Let each user U uses a cryptosystem with encryption and decryption algorithms: e U, d U Let w be a message Encryption: Decryption: PUBLIC-KEY ENCRYPTIONS e U (w) d U (e U (w)) PUBLIC-KEY SIGNATURES Signing: Verification of the signature: d U (w) e U (d U (w)) prof. Jozef Gruska IV Digital signatures 10/44

11 DIGITAL SIGNATURE SYSTEMS simplified version A digital signature system (DSS) consists of: P - the space of possible plaintexts (messages). S - the space of possible signatures. K - the space of possible keys. For each k K there is a signing algorithm sig k S a and a corresponding verification algorithm ver k V such that and sig k : P S. ver k : P S {true, false} ver k (w, s) = ( true ifs = sig k (w);, false otherwise. Algorithms sig k and ver k should be computable in polynomial time. Verification algorithm can be publicly known; signing algorithm (actually only its key) should be kept secret prof. Jozef Gruska IV Digital signatures 11/44

12 FROM PKC to DSS - again Any public-key cryptosystem in which the plaintext and cryptotext space are the same, can be used for digital signature. Signing of a message w by a user A so that any user can verify the signature: d A (w). Signing of a message w by a user A so that only user B can verify the signature; e B (d A (w)). Sending a message w and a signed message digest of w obtained by using a (standard) hash function h: (w, d A (h(w))). If only signature (but not the encryption of the message) are of importance, then it suffices that Alice sends to Bob (w, d A (w)). prof. Jozef Gruska IV Digital signatures 12/44

13 ElGamal signatures Design of the ElGamal digital siganture system: choose: prime p, integers 1 q x p, where q is a primitive element of Z p ; Compute: y = q x mod p key K = (p, q, x, y) public key (p, q, y) - trapdoor: x Signature of a message w: Let r Z p 1 be randomly chosen and kept secret. sig(w, r) = (a, b), where a = q r mod p and b = (w xa)r 1 (mod (p 1)). Verification: accept a signature (a,b) of w as valid if y a a b q w (mod p) (Indeed: y a a b q ax q rb q ax+w ax+k(p 1) q w (mod p)) prof. Jozef Gruska IV Digital signatures 13/44

14 ElGamal signatures - example Example choose: p = 11, q = 2, x = 8 compute: y = 2 8 mod 11 = 3 w = 5 is signed as (a,b), where a = q r mod p, w = xa + rb mod (p 1) choose r = 9 (this choice is O.K. because gcd(9, 10) = 1) compute a = 2 9 mod 11 = 6 solve equation: b (mod 10) that is 7 9b (mod 10) b=3 signature: (6, 3) Note: equation that has to be solved: w = xa + rb mod (p 1). prof. Jozef Gruska IV Digital signatures 14/44

15 Security of ElGamal signatures Let us analyze several ways an eavesdropper Eve can try to forge ElGamal signature (with x - secret; p, q and y = q x mod p - public): sig(w, r) = (a, b); where r is random and a = q r mod p; b = (w xa)r 1 (mod p 1). 1 First suppose Eve tries to forge signature for a new message w, without knowing x. If Eve first chooses a value a and tries to find the corresponding b, it has to compute the discrete logarithm lg aq w y a, because a b q r(w xa)r 1 q w xa q w y a. If Eve first chooses b and then tries to find a, she has to solve the equation y a a b q xa q rb q w (mod p). It is not known whether this equation can be solved for any given b efficiently. 2 If Eve chooses a and b and tries to determine such w that (a,b) is signature of w, then she has to compute discrete logarithm lg qy a a b. Hence, Eve can not sign a random message this way. prof. Jozef Gruska IV Digital signatures 15/44

16 Forging and misusing of ElGamal signatures There are ways to produce, using ElGamal signature scheme, some valid forged signatures, but they do not allow an opponent to forge signatures on messages of his/her choice. For example, if 0 i, j p 2 and gcd(j, p - 1) = 1, then for the pair a = q i y j mod p; b = aj 1 mod (p 1); w = aij 1 mod (p 1) (a, b) is a valid signature of the message w. This can be easily shown by checking the verification condition. There are several ways ElGamal signatures can be broken if they are not used carefully enough. For example, the random r used in the signature should be kept secret. Otherwise the system can be broken and signatures forged. Indeed, if r is known, then x can be computed by x = (w rb)a 1 mod (p 1) and once x is known Eve can forge signatures at will. Another misuse of the ElGamal signature system is to use the same r to sign two messages. In such a case x can be computed and the system can be broken. prof. Jozef Gruska IV Digital signatures 16/44

17 Digital Signature Standard In December 1994, on the proposal of the National Institute of Standards and Technology, the following Digital Signature Algorithm (DSA) was accepted as a standard. Design of DSA 1 The following global public key components are chosen: p - a random l-bit prime, 512 l 1024, l = 64k. q - a random 160-bit prime dividing p -1. r = h (p 1)/q mod p, where h is a random primitive element of Z p, such that r > 1, r 1 (observe that r is a q-th root of 1 mod p). 2 The following user s private key components are chosen: x - a random integer (once), 0 < x < q, y = r x mod p. 3 Key is K = (p, q, r, x, y) prof. Jozef Gruska IV Digital signatures 17/44

18 Digital Signature Standard Signing and Verification Signing of a 160-bit plaintext w choose random 0 < k < q such that gcd(k, q) = 1 compute a = (r k mod p) mod q compute b = k 1 (w + xa) mod q where kk 1 1 (mod q) signature: sig(w, k) = (a, b) Verification of signature (a, b) compute z = b 1 mod q compute u 1 = wz mod q, u 2 = az mod q verification: ver K (w, a, b) = true (r u1 y u2 mod p) mod q = a prof. Jozef Gruska IV Digital signatures 18/44

19 From ElGamal to DSA DSA is a modification of ElGamal digital signature scheme. It was proposed in August 1991 and adopted in December Any proposal for digital signature standard has to go through a very careful scrutiny. Why? Encryption of a message is usually done only once and therefore it usually suffices to use a cryptosystem that is secure at the time of the encryption. On the other hand, a signed message could be a contract or a will and it can happen that it will be needed to verify a signature many years after the message is signed. Since ElGamal signature is no more secure than discrete logarithm, it is necessary to use large p, with at least 512 bits. However, with ElGamal this would lead to signatures with at least 1024 bits what is too much for such applications as smart cards. In DSA a 160 bit message is signed using 320-bit signature, but computation is done modulo with bits. Observe that y and a are also q-roots of 1. Hence any exponents of r,y and a can be reduced module q without affecting the verification condition. This allowed to change ElGamal verification condition: y a a b = q w. prof. Jozef Gruska IV Digital signatures 19/44

20 Fiat-Shamir signature scheme Choose primes p, q, compute q n = pq and choose: as public key v 1,..., v k and compute secret key s 1,..., s k, s i = v 1 i mod n. Protocol for Alice to sign a message w: 1 Alice chooses t random integers 1 r 1,..., r t < n, computes x i = ri 2 mod n, 1 i t. 2 Alice uses a publicly known hash function h to compute H = h(wx 1x 2... x t) and then uses first kt bits of H, denoted as b ij, 1 i t, 1 j k as follows. 3 Alice computes y 1,..., y t y i = r i ky j=1 s b ij j mod n 4 Alice sends to Bob w, all b ij all y i and also h {Bob already knows Alice s public key v 1,..., v k } 5 Bob computes z 1,..., z k ky Z i = yi 2 v b ij j mod n = ri 2 j=1 ky j=1 (v 1 j ) b ij ky j=1 v b ij j = ri 2 = x i and verifies that the first k t bits of h(wx 1x 2... x t) are the b ij values that Alice has sent to him. Security of this signature scheme is 2 kt. Advantage over the RSA-based signature scheme: only about 5% of modular multiplications are needed. prof. Jozef Gruska IV Digital signatures 20/44

21 Sad story Alice and Bob got to jail - and, unfortunately, to different jails. Walter, the warden, allows them to communicate by network, but he will not allow that their messages are encrypted. Problem: Can Alice and Bob set up a subliminal channel, a covert communication channel between them, in full view of Walter, even though the messages themselves that they exchange contain no secret information? prof. Jozef Gruska IV Digital signatures 21/44

22 Ong-Schnorr-Shamir subliminal channel scheme Story Alice and Bob are in different jails. Walter, the warden, allows them to communicate by network, but he will not allow messages to be encrypted. Can they set up a subliminal channel, a covert communication channel between them, in full view of Walter, even though the messages themselves contain no secret information? Yes. Alice and Bob create first the following communication scheme: They choose a large n and an integer k such that gcd(n, k) = 1. They calculate h = k 2 mod n = (k 1 ) 2 mod n. Public key: h, n Trapdoor information: k Let secret message Alice wants to send be w (it has to be such that gcd(w, n) =1) Denote a harmless message she uses by w (it has to be such that gcd(w,n) = 1) Signing by Alice: S 1 = 1 2 ( w w S 2 = k 2 ( w w + w) mod n w) mod n Signature: (S 1, S 2). Alice then sends to Bob (w, S 1, S 2) Signature verification by Walter: w = S1 2 hs2 2 ( mod n) w Decryption by Bob: w = mod n (S 1 + k 1 S 2) prof. Jozef Gruska IV Digital signatures 22/44

23 One-time signatures Lamport signature scheme shows how to construct a signature scheme for one use only from any one-way function. Let k be a positive integer and let P = {0, 1} k be the set of messages. Let f: Y Z be a one-way function where Y is a set of signatures. For 1 i k, j = 0,1 let y ij Y be chosen randomly and z ij = f (y ij ). The key K consists of 2k y s and z s. y s are secret, z s are public. Signing of a message x = x 1... x k {0, 1} k and sig(x 1... x k ) = (y 1,x1,..., y k,xk ) = (a 1,..., a k ) - notation ver K (x 1... x k, a 1,..., a k ) = true f (a i ) = z i,xi, 1 i k Eve cannot forge a signature because she is unable to invert one-way functions. Important note: Lampert signature scheme can be used to sign only one message. prof. Jozef Gruska IV Digital signatures 23/44

24 Undeniable signatures I Undeniable signatures are signatures that have two properties: A signature can be verified only in the cooperation with the signer by means of a challenge-and-response protocol. The signer cannot deny a correct signature. To achieve that, steps are a part of the protocol that force the signer to cooperate by means of a disavowal protocol this protocol makes possible to prove the invalidity of a signature and to show that it is a forgery. (If the signer refuses to take part in the disavowal protocol, then the signature is considered to be genuine.) Undeniable signature protocol of Chaum and van Antwerpen (1989), discussed next, is again based on infeasibility of the computation of the discrete logarithm. prof. Jozef Gruska IV Digital signatures 24/44

25 Undeniable signatures II Undeniable signatures consist of: Signing algorithm Verification protocol, that is a challenge-and-response protocol. In this case it is required that a signature cannot be verified without a cooperation of the signer (Bob). This protects Bob against the possibility that documents signed by him are duplicated and distributed without his approval. Disavowal protocol, by which Bob can prove that a signature is a forgery. This is to prevent Bob from disavowing a signature he made at an earlier time. Chaum-van Antwerpen undeniable signature schemes (CAUSS) p, r are primes p = 2r + 1 q Z p is of order r; 1 x r 1, y = q x mod p; G is a multiplicative subgroup of Zp of order q (G consists of quadratic residues modulo p). Key space: K = {p, q, x, y}; p, q, y are public, x G is secret. Signature: s = sig K (w) = w x mod p. prof. Jozef Gruska IV Digital signatures 25/44

26 Fooling and Disallowed protocol Since it holds: Theorem If s w x probability 1/r. mod p, then Alice will accept s as a valid signature for w with Bob cannot fool Alice except with very small probability and security is unconditional (that is, it does not depend on any computational assumption). Disallowed protocol Basic idea: After receiving a signature s Alice initiates two independent and unsuccessful runs of the verification protocol. Finally, she performs a consistency check to determine whether Bob has formed his responses according to the protocol. Alice chooses e 1, e 2 Z r. Alice computes c = s e1 y e2 Bob computes d = c x( 1) mod p and sends it to Bob. mod r Alice verifies that d w e1 q e2 (mod p). Alice chooses f 1, f 2 Z r. Alice computes C = s f 1 y f 2 Bob computes D = C x( 1) mod p and sends it to Alice. mod p and sends it to Bob. mod r mod p and sends it to Alice. prof. Jozef Gruska IV Digital signatures 26/44

27 Fooling and Disallowed protocol Alice verifies that D w f 1 q f 2 (mod p). Alice concludes that s is a forgery iff It can be shown: (dq e2 ) f 1 (Dq f 2 ) e1 (mod p). CONCLUSIONS Bob can convince Alice that an invalid signature is a forgery. In order to do that it is sufficient to show that if s w x, then (dq e2 ) f 1 (Dq f 2 ) e1 (mod p) what can be done using congruency relation from the design of the signature system and from the disallowed protocol. Bob cannot make Alice believe that a valid signature is a forgery, except with a very small probability. prof. Jozef Gruska IV Digital signatures 27/44

28 Signing of fingerprints Signature schemes presented so far allow to sign only short messages. For example, DSS is used to sign 160 bit messages (with 320-bit signatures). A naive solution is to break long message into a sequence of short ones and to sign each block separately. Disadvantages: signing is slow and for long signatures integrity is not protected. The solution is to use a fast public hash function h which maps a message of any length to a fixed length hash. The hash is then signed. Example: message message digest El Gamal signature w z = h (w) y = sig(z) arbitrary length 160bits 320bits If Bob wants to send a signed message w he sends (w, sig(h(w)). prof. Jozef Gruska IV Digital signatures 28/44

29 Collision-free hash functions revisited For a hash function it is necessary to be good enough for creating fingerprints that do not allow various forgeries of signatures. Example 1, Eve starts with a valid signature (w, sig(h(w))), computes h(w) and tries to find w such that h(w) = h(w ). Would she succeed, then would be a valid signature, a forgery. (w, sig(h(w))) In order to prevent the above type of attacks, and some other, it is required that a hash function h satisfies the following collision-free property. Definition A hash function h is strongly collision-free if it is computationally infeasible to find messages w and w such that h(w) = h(w ). Example 2: Eve computes a signature y on a random fingerprint z and then find an x such that z = h(x). Would she succeed (x,y) would be a valid signature. In order to prevent the above attack, it is required that in signatures we use one-way hash functions. It is not difficult to show that for hash-functions the (strong) collision-free property implies the one-way property. prof. Jozef Gruska IV Digital signatures 29/44

30 Timestamping There are various ways that a digital signature can be compromised. For example: if Eve determines the secret key of Bob, then she can forge signatures of any Bob s message she likes. If this happens, authenticity of all messages signed by Bob before Eve got the secret key is to be questioned. The key problem is that there is no way to determine when a message was signed. A timestamping should provide proof that a message was signed at a certain time. A method for timestamping of signatures: In the following pub denotes some publically known information that could not be predicted before the day of the signature (for example, stock-market data). Timestamping by Bob of a signature on a message w, using a hash function h. Bob computes z = h(w); Bob computes z = h(z pub); Bob computes y = sig(z ); Bob publishes (z, pub, y) in the next days s newspaper. It is now clear that signature was not be done after triple (z, pub, y) was published, but also not before the date pub was known. prof. Jozef Gruska IV Digital signatures 30/44

31 Blind signatures The basic idea is that Sender makes Signer to sign a message m without Signer knowing m, therefore blindly this is needed in e-commerce. Blind signing can be realized by a two party protocol, between the Sender and the Signer, that has the following properties. In order to sign (by a Signer) a message m, the Sender computes, using a blinding procedure, from m an m* from which m can not be obtained without knowing a secret, and sends m* to the Signer. The Signer signs m* to get a signature s m (of m*) and sends s m to the Sender. Signing is done in such a way that the Sender can afterwards compute, using an unblinding procedure, from Signer s signature s m of m* the signer signature s m of m. prof. Jozef Gruska IV Digital signatures 31/44

32 Chaum s blind signatures This blind signature protocol combines RSA with blinding/unblinding features. Bob s RSA public key is (n,e) and his private key is d. Let m be a message, 0 < m < n, PROTOCOL: Alice chooses a random 0 < k < n with gcd(n,k)=1. Alice computes m = mk e (mod n) and sends it to Bob (this way Alice blinds the message m). Bob computed s = (m ) d (mod n) and sends s* to Alice (this way Bob signs the blinded message m*). Alice computes s = k 1 s (mod n) to obtain Bob s signature m d of m (Alice performs unblinding of m*). Verification is equivalent to that of the RSA signature scheme. prof. Jozef Gruska IV Digital signatures 32/44

33 Fail-then-stop signatures They are signatures schemes that use a trusted authority and provide ways to prove, if it is the case, that a powerful enough adversary is around who could break the signature scheme and therefore its use should be stopped. The scheme is maintained by a trusted authority that chooses a secret key for each signer, keeps them secret, even from the signers themselves, and announces only the related public keys. An important idea is that signing and verification algorithms are enhanced by a so-called proof-of-forgery algorithm. When the signer sees a forged signature he is able to compute his secret key and by submitting it to the trusted authority to prove the existence of a forgery and this way to achieve that any further use of the signature scheme is stopped. So called Heyst-Pedersen Scheme is an example of a Fail-Then-Stop signature Scheme. prof. Jozef Gruska IV Digital signatures 33/44

34 Digital signatures with encryption and resending 1 Alice signs the message: s A (w). 2 Alice encrypts the signed message: e B (s A (w)). 3 Bob decrypt the signed message: d B (e B (s A (w))) = s A (w). 4 Bob verifies signature and recovers the message v A (s A (w)) = w. Resending the message as a receipt 5 Bob signs and encrypts the message and sends to Alice e A (s B (w)). 6 Alice decrypts the message and verifies the signature. Assume now: v x = e x, s x = d x for all users x. prof. Jozef Gruska IV Digital signatures 34/44

35 A surprising attack to the previous scheme 1 Mallot intercept e B (s A (w)). 2 Later Mallot sends e B (s A (w)) to Bob pretending it is from him (from Mallot). 3 Bob decrypts and verifies the message by computing e M (d B (e B (d A (w)))) = e M (d A (w)) a garbage. 4 Bob goes on with the protocol and reterns Mallot the receipt: e M (d B (e M (d A (w)))) 5 Mallot can then get w. Indeed, Mallot can compute e A (d M (e B (d M (e M (d B (e M (d A (w)))))))) = w. prof. Jozef Gruska IV Digital signatures 35/44

36 A MAN-IN-THE-MIDDLE attack Consider the following protocol: 1 Alice sends Bob the pair (e B (e B (w)a), B) to B. 2 Bob uses d B to get A and w, and acknowledges by sending the pair (e A (e A (w)b), A) to Alice. (Here the function e and d are assumed to operate on numbers, names A, B,... are sequences of digits and e B (w)a is a sequence of digits obtained by concatenating e B (w) and A.) What can an active eavesdropper C do? C can learn (e A (e A (w)b), A) and therefore e A (w ), w = e A (w)b. C can now send to Alice the pair (e A (e A (w )C), A). Alice, thinking that this is the step 1 of the protocol, acknowledges by sending the pair (e C (e C (w )A), C) to C. C is now able to learn w and therefore also e A (w). C now sends to Alice the pair (e A (e A (w)c), A). Alice acknowledges by sending the pair (e C (e C (w)a), C). C is now able to learn w. prof. Jozef Gruska IV Digital signatures 36/44

37 Probabilistic signature schemes - PSS Let us have integers k, l, n such that k + l < n, a permutation a pseudorandom bit generator and a hash function f : D D, D {0, 1} n, G : {0, 1} l {0, 1} k {0, 1} n (l+k), w (G 1(w), G 2(w)) h : {0, 1} {0, 1} l. The following PSS scheme is applicable to messages of arbitrary length. Signing: of a message w {0, 1}. 1 Choose random r {0, 1} k and compute m = h(w r). 2 Compute G(m) = (G 1(m), G 2(m)) and y = m (G 1(m) r) G 2(m). 3 Signature of w is σ = f 1 (y). Verification of a signed message (w, σ). Compute f (σ) and decompose f (σ) = m t u, where m = l, t = k and u = n (k + l). Compute r = t G 1(m). Accept signature σ if h(w r) = m and G 2(m) = u; otherwise reject it. prof. Jozef Gruska IV Digital signatures 37/44

38 Authenticated Diffie-Hellman key exchange Let each user U has a signature algorithm s U and a verification algorithm v U. The following protocol allows Alice and Bob to establish a key K to use with an encryption function e K and to avoid the man-in-the-middle attack. 1 Alice and Bob choose large prime p and a generator q Z p. 2 Alice chooses a random x and Bob chooses a random y. 3 Alice computes q x mod p, and Bob computes q y mod p. 4 Alice sends q x to Bob. 5 Bob computes K = q xy mod p. 6 Bob sends q y and e K (s B (q y, q x )) to Alice. 7 Alice computes K = q xy mod p. 8 Alice decrypts e K (s B (q y, q x )) to obtain s B (q y, q x ). 9 Alice verifies, using an authority, that v B is Bob s verification algorithm. 10 Alice uses v B to verify Bob s signature. 11 Alice sends e K (s A (q x, q y )) to Bob. 12 Bob decrypts, verifies v A, and verifies Alice s signature. An enhanced version of the above protocol is known as Station-to-Station protocol. prof. Jozef Gruska IV Digital signatures 38/44

39 Security of digital signatures It is very non-trivial to define security of a digital signature. Definition A chosen message attack is a process by which on an input of a verification key one can obtain a signature (corresponding to the given key) to a message of its choice. A chosen message attack is considered to be successful (in so called existential forgery) if it outputs a valid signature for a message for which it has not requested a signature during the attack. A signature scheme is secure (or unforgeable) if every feasible chosen message attack succeeds with at most negligible probability. prof. Jozef Gruska IV Digital signatures 39/44

40 Treshold Signature Schemes The idea of a (t+1, n) treshold signature scheme is to distribute the power of the signing operation to (t+1) parties out of n. A (t+1) treshold signature scheme should satisfy two conditions. Unforgeability means that even if an adversary corrupts t parties, he still cannot generate a valid signature. Robustness means that corrupted parties cannot prevent uncorrupted parties to generate signatures. Shoup (2000) presented an efficient, non-interactive, robust and unforgeable treshold RSA signature schemes. There is no proof yet whether Shoup s scheme is provably secure. prof. Jozef Gruska IV Digital signatures 40/44

41 Digital Signatures - Observation Can we make digital signatures by digitalizing our usual signature and attaching them to the messages (documents) that need to be signed? No, because such signatures could be easily removed and attached to some other documents or messages. Key observation: Digital signatures have to depend not only on the signer, but also on the message that is being signed. prof. Jozef Gruska IV Digital signatures 41/44

42 SPECIAL TYPES of DIGITAL SIGNATURES Append-Only Signatures (AOS) have the property that any party given an AOS signature sig[m 1] on message M 1 can compute sig[m 1 M 2] for any message M 2. (Such signatures are of importance in network applications, where users need to delegate their shares of resources to other users). Identity-Based signatures (IBS) at which the identity of the signer (i.e. her address) plays the role of her public key. (Such schemes assume the existence of a TA holding a master public-private key pair used to assign secret keys to users based on their identity.) Hierarchically Identity-Based Signatures are such IBS in which users are arranged in a hierarchy and a user at any level at the hierarchy can delegate secret keys to her descendants based on their identities and her own secret keys. prof. Jozef Gruska IV Digital signatures 42/44

43 GROUP SIGNATURES At Group Signatures (GS) a group member can compute a signature that reveals nothing about the signer s identity, except that he is a member of the group. On the other hand, the group manager can always reveal the identity of the signer. Hierarchical Group Signatures (HGS) are a generalization of GS that allow multiple group managers to be organized in a tree with the signers as leaves. When verifying a signature, a group manager only learns to which of its subtrees, if any, the signer belongs. prof. Jozef Gruska IV Digital signatures 43/44

44 Unconditionally secure digital signatures Any of the digital signature schemes introduced so far can be forged by anyone having enough computer power. Chaum and Roijakkers (2001) developed, for any fixed set of users, an unconditionally secure signature scheme with the following properties: Any participant can convince (except with exponentially small probability) any other participant that his signature is valid. A convinced partipant can convince any other participant of the signature s validity, without interaction with the original signer. prof. Jozef Gruska IV Digital signatures 44/44

### Digital signatures are one of the most important inventions/applications of modern cryptography.

CHAPTER 7: DIGITAL SIGNATURES Digital signatures are one of the most important inventions/applications of modern cryptography. Part VII Digital signatures The problem is how can a user sign (electronically)

### Outline. Computer Science 418. Digital Signatures: Observations. Digital Signatures: Definition. Definition 1 (Digital signature) Digital Signatures

Outline Computer Science 418 Digital Signatures Mike Jacobson Department of Computer Science University of Calgary Week 12 1 Digital Signatures 2 Signatures via Public Key Cryptosystems 3 Provable 4 Mike

### Signature Schemes. CSG 252 Fall 2006. Riccardo Pucella

Signature Schemes CSG 252 Fall 2006 Riccardo Pucella Signatures Signatures in real life have a number of properties They specify the person responsible for a document E.g. that it has been produced by

### Digital Signatures. Good properties of hand-written signatures:

Digital Signatures Good properties of hand-written signatures: 1. Signature is authentic. 2. Signature is unforgeable. 3. Signature is not reusable (it is a part of the document) 4. Signed document is

### Digital signatures. Informal properties

Digital signatures Informal properties Definition. A digital signature is a number dependent on some secret known only to the signer and, additionally, on the content of the message being signed Property.

### CSCE 465 Computer & Network Security

CSCE 465 Computer & Network Security Instructor: Dr. Guofei Gu http://courses.cse.tamu.edu/guofei/csce465/ Public Key Cryptogrophy 1 Roadmap Introduction RSA Diffie-Hellman Key Exchange Public key and

### Digital Signatures. (Note that authentication of sender is also achieved by MACs.) Scan your handwritten signature and append it to the document?

Cryptography Digital Signatures Professor: Marius Zimand Digital signatures are meant to realize authentication of the sender nonrepudiation (Note that authentication of sender is also achieved by MACs.)

### Introduction. Digital Signature

Introduction Electronic transactions and activities taken place over Internet need to be protected against all kinds of interference, accidental or malicious. The general task of the information technology

### Computer Science 308-547A Cryptography and Data Security. Claude Crépeau

Computer Science 308-547A Cryptography and Data Security Claude Crépeau These notes are, largely, transcriptions by Anton Stiglic of class notes from the former course Cryptography and Data Security (308-647A)

### CIS 6930 Emerging Topics in Network Security. Topic 2. Network Security Primitives

CIS 6930 Emerging Topics in Network Security Topic 2. Network Security Primitives 1 Outline Absolute basics Encryption/Decryption; Digital signatures; D-H key exchange; Hash functions; Application of hash

### Cryptographic hash functions and MACs Solved Exercises for Cryptographic Hash Functions and MACs

Cryptographic hash functions and MACs Solved Exercises for Cryptographic Hash Functions and MACs Enes Pasalic University of Primorska Koper, 2014 Contents 1 Preface 3 2 Problems 4 2 1 Preface This is a

### RSA Attacks. By Abdulaziz Alrasheed and Fatima

RSA Attacks By Abdulaziz Alrasheed and Fatima 1 Introduction Invented by Ron Rivest, Adi Shamir, and Len Adleman [1], the RSA cryptosystem was first revealed in the August 1977 issue of Scientific American.

### Digital Signatures. Murat Kantarcioglu. Based on Prof. Li s Slides. Digital Signatures: The Problem

Digital Signatures Murat Kantarcioglu Based on Prof. Li s Slides Digital Signatures: The Problem Consider the real-life example where a person pays by credit card and signs a bill; the seller verifies

### Textbook: Introduction to Cryptography 2nd ed. By J.A. Buchmann Chap 12 Digital Signatures

Textbook: Introduction to Cryptography 2nd ed. By J.A. Buchmann Chap 12 Digital Signatures Department of Computer Science and Information Engineering, Chaoyang University of Technology 朝 陽 科 技 大 學 資 工

### 1 Signatures vs. MACs

CS 120/ E-177: Introduction to Cryptography Salil Vadhan and Alon Rosen Nov. 22, 2006 Lecture Notes 17: Digital Signatures Recommended Reading. Katz-Lindell 10 1 Signatures vs. MACs Digital signatures

### Digital Signatures. Prof. Zeph Grunschlag

Digital Signatures Prof. Zeph Grunschlag (Public Key) Digital Signatures PROBLEM: Alice would like to prove to Bob, Carla, David,... that has really sent them a claimed message. E GOAL: Alice signs each

### Digital Signature. Raj Jain. Washington University in St. Louis

Digital Signature Raj Jain Washington University in Saint Louis Saint Louis, MO 63130 Jain@cse.wustl.edu Audio/Video recordings of this lecture are available at: http://www.cse.wustl.edu/~jain/cse571-11/

### Introduction to Cryptography CS 355

Introduction to Cryptography CS 355 Lecture 30 Digital Signatures CS 355 Fall 2005 / Lecture 30 1 Announcements Wednesday s lecture cancelled Friday will be guest lecture by Prof. Cristina Nita- Rotaru

### CIS 5371 Cryptography. 8. Encryption --

CIS 5371 Cryptography p y 8. Encryption -- Asymmetric Techniques Textbook encryption algorithms In this chapter, security (confidentiality) is considered in the following sense: All-or-nothing secrecy.

### Principles of Public Key Cryptography. Applications of Public Key Cryptography. Security in Public Key Algorithms

Principles of Public Key Cryptography Chapter : Security Techniques Background Secret Key Cryptography Public Key Cryptography Hash Functions Authentication Chapter : Security on Network and Transport

### Overview of Public-Key Cryptography

CS 361S Overview of Public-Key Cryptography Vitaly Shmatikov slide 1 Reading Assignment Kaufman 6.1-6 slide 2 Public-Key Cryptography public key public key? private key Alice Bob Given: Everybody knows

### 1 Digital Signatures. 1.1 The RSA Function: The eth Power Map on Z n. Crypto: Primitives and Protocols Lecture 6.

1 Digital Signatures A digital signature is a fundamental cryptographic primitive, technologically equivalent to a handwritten signature. In many applications, digital signatures are used as building blocks

### Public Key (asymmetric) Cryptography

Public-Key Cryptography UNIVERSITA DEGLI STUDI DI PARMA Dipartimento di Ingegneria dell Informazione Public Key (asymmetric) Cryptography Luca Veltri (mail.to: luca.veltri@unipr.it) Course of Network Security,

### Cryptography and Network Security

Cryptography and Network Security Spring 2012 http://users.abo.fi/ipetre/crypto/ Lecture 9: Authentication protocols, digital signatures Ion Petre Department of IT, Åbo Akademi University 1 Overview of

### Public Key Cryptography. c Eli Biham - March 30, 2011 258 Public Key Cryptography

Public Key Cryptography c Eli Biham - March 30, 2011 258 Public Key Cryptography Key Exchange All the ciphers mentioned previously require keys known a-priori to all the users, before they can encrypt

### Crittografia e sicurezza delle reti. Digital signatures- DSA

Crittografia e sicurezza delle reti Digital signatures- DSA Signatures vs. MACs Suppose parties A and B share the secret key K. Then M, MAC K (M) convinces A that indeed M originated with B. But in case

### Communications security

University of Roma Sapienza DIET Communications security Lecturer: Andrea Baiocchi DIET - University of Roma La Sapienza E-mail: andrea.baiocchi@uniroma1.it URL: http://net.infocom.uniroma1.it/corsi/index.htm

### Network Security. Computer Networking Lecture 08. March 19, 2012. HKU SPACE Community College. HKU SPACE CC CN Lecture 08 1/23

Network Security Computer Networking Lecture 08 HKU SPACE Community College March 19, 2012 HKU SPACE CC CN Lecture 08 1/23 Outline Introduction Cryptography Algorithms Secret Key Algorithm Message Digest

### Network Security. Gaurav Naik Gus Anderson. College of Engineering. Drexel University, Philadelphia, PA. Drexel University. College of Engineering

Network Security Gaurav Naik Gus Anderson, Philadelphia, PA Lectures on Network Security Feb 12 (Today!): Public Key Crypto, Hash Functions, Digital Signatures, and the Public Key Infrastructure Feb 14:

### CRYPTOGRAPHY IN NETWORK SECURITY

ELE548 Research Essays CRYPTOGRAPHY IN NETWORK SECURITY AUTHOR: SHENGLI LI INSTRUCTOR: DR. JIEN-CHUNG LO Date: March 5, 1999 Computer network brings lots of great benefits and convenience to us. We can

### Public-Key Cryptography. Oregon State University

Public-Key Cryptography Çetin Kaya Koç Oregon State University 1 Sender M Receiver Adversary Objective: Secure communication over an insecure channel 2 Solution: Secret-key cryptography Exchange the key

### Lecture Note 7 AUTHENTICATION REQUIREMENTS. Sourav Mukhopadhyay

Lecture Note 7 AUTHENTICATION REQUIREMENTS Sourav Mukhopadhyay Cryptography and Network Security - MA61027 In the context of communications across a network, the following attacks can be identified: 1.

### The Mathematics of the RSA Public-Key Cryptosystem

The Mathematics of the RSA Public-Key Cryptosystem Burt Kaliski RSA Laboratories ABOUT THE AUTHOR: Dr Burt Kaliski is a computer scientist whose involvement with the security industry has been through

### Lecture 15 - Digital Signatures

Lecture 15 - Digital Signatures Boaz Barak March 29, 2010 Reading KL Book Chapter 12. Review Trapdoor permutations - easy to compute, hard to invert, easy to invert with trapdoor. RSA and Rabin signatures.

### Overview of Cryptographic Tools for Data Security. Murat Kantarcioglu

UT DALLAS Erik Jonsson School of Engineering & Computer Science Overview of Cryptographic Tools for Data Security Murat Kantarcioglu Pag. 1 Purdue University Cryptographic Primitives We will discuss the

### Outline. CSc 466/566. Computer Security. 8 : Cryptography Digital Signatures. Digital Signatures. Digital Signatures... Christian Collberg

Outline CSc 466/566 Computer Security 8 : Cryptography Digital Signatures Version: 2012/02/27 16:07:05 Department of Computer Science University of Arizona collberg@gmail.com Copyright c 2012 Christian

### Digital Signature CHAPTER 13. Review Questions. (Solution to Odd-Numbered Problems)

CHAPTER 13 Digital Signature (Solution to Odd-Numbered Problems) Review Questions 1. We mentioned four areas in which there is a differences between a conventional and a digital signature: inclusion, verification,

### Title Goes Here An Introduction to Modern Cryptography. Mike Reiter

Title Goes Here An Introduction to Modern Cryptography Mike Reiter 1 Cryptography Study of techniques to communicate securely in the presence of an adversary Traditional scenario Goal: A dedicated, private

### CSC474/574 - Information Systems Security: Homework1 Solutions Sketch

CSC474/574 - Information Systems Security: Homework1 Solutions Sketch February 20, 2005 1. Consider slide 12 in the handout for topic 2.2. Prove that the decryption process of a one-round Feistel cipher

### Ch.9 Cryptography. The Graduate Center, CUNY.! CSc 75010 Theoretical Computer Science Konstantinos Vamvourellis

Ch.9 Cryptography The Graduate Center, CUNY! CSc 75010 Theoretical Computer Science Konstantinos Vamvourellis Why is Modern Cryptography part of a Complexity course? Short answer:! Because Modern Cryptography

### CRC Press has granted the following specific permissions for the electronic version of this book:

This is a Chapter from the Handbook of Applied Cryptography, by A. Menezes, P. van Oorschot, and S. Vanstone, CRC Press, 1996. For further information, see www.cacr.math.uwaterloo.ca/hac CRC Press has

### Lecture 3: One-Way Encryption, RSA Example

ICS 180: Introduction to Cryptography April 13, 2004 Lecturer: Stanislaw Jarecki Lecture 3: One-Way Encryption, RSA Example 1 LECTURE SUMMARY We look at a different security property one might require

### Study of algorithms for factoring integers and computing discrete logarithms

Study of algorithms for factoring integers and computing discrete logarithms First Indo-French Workshop on Cryptography and Related Topics (IFW 2007) June 11 13, 2007 Paris, France Dr. Abhijit Das Department

### Lukasz Pater CMMS Administrator and Developer

Lukasz Pater CMMS Administrator and Developer EDMS 1373428 Agenda Introduction Why do we need asymmetric ciphers? One-way functions RSA Cipher Message Integrity Examples Secure Socket Layer Single Sign

### 2. Cryptography 2.4 Digital Signatures

DI-FCT-UNL Computer and Network Systems Security Segurança de Sistemas e Redes de Computadores 2010-2011 2. Cryptography 2.4 Digital Signatures 2010, Henrique J. Domingos, DI/FCT/UNL 2.4 Digital Signatures

### Notes on Network Security Prof. Hemant K. Soni

Chapter 9 Public Key Cryptography and RSA Private-Key Cryptography traditional private/secret/single key cryptography uses one key shared by both sender and receiver if this key is disclosed communications

### Lecture Note 5 PUBLIC-KEY CRYPTOGRAPHY. Sourav Mukhopadhyay

Lecture Note 5 PUBLIC-KEY CRYPTOGRAPHY Sourav Mukhopadhyay Cryptography and Network Security - MA61027 Modern/Public-key cryptography started in 1976 with the publication of the following paper. W. Diffie

### Cryptography and Network Security

Cryptography and Network Security Fifth Edition by William Stallings Chapter 9 Public Key Cryptography and RSA Private-Key Cryptography traditional private/secret/single key cryptography uses one key shared

### DIGITAL SIGNATURES 1/1

DIGITAL SIGNATURES 1/1 Signing by hand COSMO ALICE ALICE Pay Bob \$100 Cosmo Alice Alice Bank =? no Don t yes pay Bob 2/1 Signing electronically Bank Internet SIGFILE } {{ } 101 1 ALICE Pay Bob \$100 scan

### QUANTUM COMPUTERS AND CRYPTOGRAPHY. Mark Zhandry Stanford University

QUANTUM COMPUTERS AND CRYPTOGRAPHY Mark Zhandry Stanford University Classical Encryption pk m c = E(pk,m) sk m = D(sk,c) m??? Quantum Computing Attack pk m aka Post-quantum Crypto c = E(pk,m) sk m = D(sk,c)

### CS 758: Cryptography / Network Security

CS 758: Cryptography / Network Security offered in the Fall Semester, 2003, by Doug Stinson my office: DC 3122 my email address: dstinson@uwaterloo.ca my web page: http://cacr.math.uwaterloo.ca/~dstinson/index.html

### Network Security. Abusayeed Saifullah. CS 5600 Computer Networks. These slides are adapted from Kurose and Ross 8-1

Network Security Abusayeed Saifullah CS 5600 Computer Networks These slides are adapted from Kurose and Ross 8-1 Public Key Cryptography symmetric key crypto v requires sender, receiver know shared secret

### UNIVERSITY OF MASSACHUSETTS Dept. of Electrical & Computer Engineering. Introduction to Cryptography ECE 597XX/697XX

UNIVERSITY OF MASSACHUSETTS Dept. of Electrical & Computer Engineering Introduction to Cryptography ECE 597XX/697XX Part 6 Introduction to Public-Key Cryptography Israel Koren ECE597/697 Koren Part.6.1

### Cryptography Lecture 8. Digital signatures, hash functions

Cryptography Lecture 8 Digital signatures, hash functions A Message Authentication Code is what you get from symmetric cryptography A MAC is used to prevent Eve from creating a new message and inserting

### Improved Online/Offline Signature Schemes

Improved Online/Offline Signature Schemes Adi Shamir and Yael Tauman Applied Math. Dept. The Weizmann Institute of Science Rehovot 76100, Israel {shamir,tauman}@wisdom.weizmann.ac.il Abstract. The notion

### Public Key Cryptography and RSA. Review: Number Theory Basics

Public Key Cryptography and RSA Murat Kantarcioglu Based on Prof. Ninghui Li s Slides Review: Number Theory Basics Definition An integer n > 1 is called a prime number if its positive divisors are 1 and

### MTAT.07.003 Cryptology II. Digital Signatures. Sven Laur University of Tartu

MTAT.07.003 Cryptology II Digital Signatures Sven Laur University of Tartu Formal Syntax Digital signature scheme pk (sk, pk) Gen (m, s) (m,s) m M 0 s Sign sk (m) Ver pk (m, s)? = 1 To establish electronic

### Digital Signatures. What are Signature Schemes?

Digital Signatures Debdeep Mukhopadhyay IIT Kharagpur What are Signature Schemes? Provides message integrity in the public key setting Counter-parts of the message authentication schemes in the public

### ACTA UNIVERSITATIS APULENSIS No 13/2007 MATHEMATICAL FOUNDATION OF DIGITAL SIGNATURES. Daniela Bojan and Sidonia Vultur

ACTA UNIVERSITATIS APULENSIS No 13/2007 MATHEMATICAL FOUNDATION OF DIGITAL SIGNATURES Daniela Bojan and Sidonia Vultur Abstract.The new services available on the Internet have born the necessity of a permanent

### Asymmetric Encryption. With material from Jonathan Katz, David Brumley, and Dave Levin

Asymmetric Encryption With material from Jonathan Katz, David Brumley, and Dave Levin Warmup activity Overview of asymmetric-key crypto Intuition for El Gamal and RSA And intuition for attacks Digital

### Software Implementation of Gong-Harn Public-key Cryptosystem and Analysis

Software Implementation of Gong-Harn Public-key Cryptosystem and Analysis by Susana Sin A thesis presented to the University of Waterloo in fulfilment of the thesis requirement for the degree of Master

### A New Generic Digital Signature Algorithm

Groups Complex. Cryptol.? (????), 1 16 DOI 10.1515/GCC.????.??? de Gruyter???? A New Generic Digital Signature Algorithm Jennifer Seberry, Vinhbuu To and Dongvu Tonien Abstract. In this paper, we study

### PUBLIC KEY ENCRYPTION

PUBLIC KEY ENCRYPTION http://www.tutorialspoint.com/cryptography/public_key_encryption.htm Copyright tutorialspoint.com Public Key Cryptography Unlike symmetric key cryptography, we do not find historical

### Introduction to Cryptography

Introduction to Cryptography Part 2: public-key cryptography Jean-Sébastien Coron January 2007 Public-key cryptography Invented by Diffie and Hellman in 1976. Revolutionized the field. Each user now has

### SECURITY IN NETWORKS

SECURITY IN NETWORKS GOALS Understand principles of network security: Cryptography and its many uses beyond confidentiality Authentication Message integrity Security in practice: Security in application,

### Introduction to Computer Security

Introduction to Computer Security Hash Functions and Digital Signatures Pavel Laskov Wilhelm Schickard Institute for Computer Science Integrity objective in a wide sense Reliability Transmission errors

### Final Exam. IT 4823 Information Security Administration. Rescheduling Final Exams. Kerberos. Idea. Ticket

IT 4823 Information Security Administration Public Key Encryption Revisited April 5 Notice: This session is being recorded. Lecture slides prepared by Dr Lawrie Brown for Computer Security: Principles

### Secure Network Communication Part II II Public Key Cryptography. Public Key Cryptography

Kommunikationssysteme (KSy) - Block 8 Secure Network Communication Part II II Public Key Cryptography Dr. Andreas Steffen 2000-2001 A. Steffen, 28.03.2001, KSy_RSA.ppt 1 Secure Key Distribution Problem

Family Name:... First Name:... Section:... Advanced Cryptography Final Exam July 18 th, 2006 Start at 9:15, End at 12:00 This document consists of 12 pages. Instructions Electronic devices are not allowed.

### Understanding Cryptography A Textbook for Students and Practitioners by Christof Paar and Jan Pelzl

Understanding Cryptography A Textbook for Students and Practitioners by Christof Paar and Jan Pelzl www.crypto-textbook.com Chapter 10 Digital Signatures ver. October 29, 2009 These slides were prepared

### Cryptography and Network Security Chapter 10

Cryptography and Network Security Chapter 10 Fifth Edition by William Stallings Lecture slides by Lawrie Brown (with edits by RHB) Chapter 10 Other Public Key Cryptosystems Amongst the tribes of Central

### Applied Cryptology. Ed Crowley

Applied Cryptology Ed Crowley 1 Basics Topics Basic Services and Operations Symmetric Cryptography Encryption and Symmetric Algorithms Asymmetric Cryptography Authentication, Nonrepudiation, and Asymmetric

### Network Security. Security Attacks. Normal flow: Interruption: 孫 宏 民 hmsun@cs.nthu.edu.tw Phone: 03-5742968 國 立 清 華 大 學 資 訊 工 程 系 資 訊 安 全 實 驗 室

Network Security 孫 宏 民 hmsun@cs.nthu.edu.tw Phone: 03-5742968 國 立 清 華 大 學 資 訊 工 程 系 資 訊 安 全 實 驗 室 Security Attacks Normal flow: sender receiver Interruption: Information source Information destination

### 1 Message Authentication

Theoretical Foundations of Cryptography Lecture Georgia Tech, Spring 200 Message Authentication Message Authentication Instructor: Chris Peikert Scribe: Daniel Dadush We start with some simple questions

### In this paper a new signature scheme and a public key cryptotsystem are proposed. They can be seen as a compromise between the RSA and ElGamal-type sc

Digital Signature and Public Key Cryptosystem in a Prime Order Subgroup of Z n Colin Boyd Information Security Research Centre, School of Data Communications Queensland University of Technology, Brisbane

### Module 8. Network Security. Version 2 CSE IIT, Kharagpur

Module 8 Network Security Lesson 2 Secured Communication Specific Instructional Objectives On completion of this lesson, the student will be able to: State various services needed for secured communication

### Public Key Cryptography Overview

Ch.20 Public-Key Cryptography and Message Authentication I will talk about it later in this class Final: Wen (5/13) 1630-1830 HOLM 248» give you a sample exam» Mostly similar to homeworks» no electronic

### Today ENCRYPTION. Cryptography example. Basic principles of cryptography

Today ENCRYPTION The last class described a number of problems in ensuring your security and privacy when using a computer on-line. This lecture discusses one of the main technological solutions. The use

### Public Key Cryptography. Basic Public Key Cryptography

Public Key Cryptography EJ Jung Basic Public Key Cryptography public key public key? private key Alice Bob Given: Everybody knows Bob s public key - How is this achieved in practice? Only Bob knows the

### Authentication Protocols Using Hoover-Kausik s Software Token *

JOURNAL OF INFORMATION SCIENCE AND ENGINEERING 22, 691-699 (2006) Short Paper Authentication Protocols Using Hoover-Kausik s Software Token * WEI-CHI KU AND HUI-LUNG LEE + Department of Computer Science

### Capture Resilient ElGamal Signature Protocols

Capture Resilient ElGamal Signature Protocols Hüseyin Acan 1, Kamer Kaya 2,, and Ali Aydın Selçuk 2 1 Bilkent University, Department of Mathematics acan@fen.bilkent.edu.tr 2 Bilkent University, Department

### CS 161 Computer Security

Song Spring 2015 CS 161 Computer Security Discussion 11 April 7 & April 8, 2015 Question 1 RSA (10 min) (a) Describe how to find a pair of public key and private key for RSA encryption system. Find two

### Midterm Exam Solutions CS161 Computer Security, Spring 2008

Midterm Exam Solutions CS161 Computer Security, Spring 2008 1. To encrypt a series of plaintext blocks p 1, p 2,... p n using a block cipher E operating in electronic code book (ECB) mode, each ciphertext

### SAMPLE EXAM QUESTIONS MODULE EE5552 NETWORK SECURITY AND ENCRYPTION ECE, SCHOOL OF ENGINEERING AND DESIGN BRUNEL UNIVERSITY UXBRIDGE MIDDLESEX, UK

SAMPLE EXAM QUESTIONS MODULE EE5552 NETWORK SECURITY AND ENCRYPTION September 2010 (reviewed September 2014) ECE, SCHOOL OF ENGINEERING AND DESIGN BRUNEL UNIVERSITY UXBRIDGE MIDDLESEX, UK NETWORK SECURITY

### Improvement of digital signature with message recovery using self-certified public keys and its variants

Applied Mathematics and Computation 159 (2004) 391 399 www.elsevier.com/locate/amc Improvement of digital signature with message recovery using self-certified public keys and its variants Zuhua Shao Department

### Lecture 9 - Message Authentication Codes

Lecture 9 - Message Authentication Codes Boaz Barak March 1, 2010 Reading: Boneh-Shoup chapter 6, Sections 9.1 9.3. Data integrity Until now we ve only been interested in protecting secrecy of data. However,

### Hybrid Signcryption Schemes with Insider Security (Extended Abstract)

Hybrid Signcryption Schemes with Insider Security (Extended Abstract) Alexander W. Dent Royal Holloway, University of London Egham Hill, Egham, Surrey, TW20 0EX, U.K. a.dent@rhul.ac.uk http://www.isg.rhul.ac.uk/~alex/

### CS549: Cryptography and Network Security

CS549: Cryptography and Network Security by Xiang-Yang Li Department of Computer Science, IIT Cryptography and Network Security 1 Notice This lecture note (Cryptography and Network Security) is prepared

### Discrete logarithms within computer and network security Prof Bill Buchanan, Edinburgh Napier

Discrete logarithms within computer and network security Prof Bill Buchanan, Edinburgh Napier http://asecuritysite.com @billatnapier Introduction. Encryption: Public/Private Key. Key Exchange. Authentication.

### Elements of Applied Cryptography Public key encryption

Network Security Elements of Applied Cryptography Public key encryption Public key cryptosystem RSA and the factorization problem RSA in practice Other asymmetric ciphers Asymmetric Encryption Scheme Let

### Module: Applied Cryptography. Professor Patrick McDaniel Fall 2010. CSE543 - Introduction to Computer and Network Security

CSE543 - Introduction to Computer and Network Security Module: Applied Cryptography Professor Patrick McDaniel Fall 2010 Page 1 Key Distribution/Agreement Key Distribution is the process where we assign

### APNIC elearning: Cryptography Basics. Contact: esec02_v1.0

APNIC elearning: Cryptography Basics Contact: training@apnic.net esec02_v1.0 Overview Cryptography Cryptographic Algorithms Encryption Symmetric-Key Algorithm Block and Stream Cipher Asymmetric Key Algorithm

### 7! Cryptographic Techniques! A Brief Introduction

7! Cryptographic Techniques! A Brief Introduction 7.1! Introduction to Cryptography! 7.2! Symmetric Encryption! 7.3! Asymmetric (Public-Key) Encryption! 7.4! Digital Signatures! 7.5! Public Key Infrastructures

### CUNSHENG DING HKUST, Hong Kong. Computer Security. Computer Security. Cunsheng DING, HKUST COMP4631

Cunsheng DING, HKUST Lecture 08: Key Management for One-key Ciphers Topics of this Lecture 1. The generation and distribution of secret keys. 2. A key distribution protocol with a key distribution center.

### Chapter 3. Network Domain Security

Communication System Security, Chapter 3, Draft, L.D. Chen and G. Gong, 2008 1 Chapter 3. Network Domain Security A network can be considered as the physical resource for a communication system. This chapter

### Chapter 10 Asymmetric-Key Cryptography

Chapter 10 Asymmetric-Key Cryptography Copyright The McGraw-Hill Companies, Inc. Permission required for reproduction or display. 10.1 Chapter 10 Objectives Present asymmetric-key cryptography. Distinguish

### Digital Signatures. Meka N.L.Sneha. Indiana State University. nmeka@sycamores.indstate.edu. October 2015

Digital Signatures Meka N.L.Sneha Indiana State University nmeka@sycamores.indstate.edu October 2015 1 Introduction Digital Signatures are the most trusted way to get documents signed online. A digital

### The application of prime numbers to RSA encryption

The application of prime numbers to RSA encryption Prime number definition: Let us begin with the definition of a prime number p The number p, which is a member of the set of natural numbers N, is considered