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An asymmetric key algorithm requires two keys: private (which is never shared) and public (which is openly available), so no key exchange is needed for communication. Messages are encrypted with the recipient's public key and decrypted with the recipient's private key. Mar 03, 2020 Run the following commands to generate an ES256 key with a self-signed X.509 certificate: openssl req -x509 -new -key ecprivate.pem -out eccert.pem -subj '/CN=unused' You can replace the -subj argument with an actual certificate subject and use that certificate, or you can omit -subj and supply the certificate information when prompted.
Lecturer: Tom Roeder
Lecture notes byTom Roeder
Table of Contents
- 1 Introduction
- 2 Definitions
- 3 Examples of Public-Key Cryptosystems
1 Introduction
We have now defined two functions that are hard to perform:computing the inverse of a one-way function and distinguishing theoutput of a pseudo-random function from a random function. We thengave high-level definitions of more useful operations: cryptographichash functions and encryption, which can be based on one-wayfunctions and pseudo-random functions, respectively. But shared keysare inherently limiting; these keys must be shared between each pairof principals and complicate the process of adding new principals tothe system. Similarly, shared key operations are not easilyapplicable to cases where one principal performs an operation thataffects many principals. An asymmetric key setup would solve bothof these problems: each principal has its own key information thatit does not need to share in secret with other principals.
For an example of how problems arise in symmetric-key settings,consider how we might perform some of our shared-key operations in acontext with, say, three principals, A, B, and C. Principal A wantsto send a message to B and C in such a way that both know that itcame from A. If A and B share key kAB and A and C share keykAC, then it's not obvious how to send a bit string thatguarantees this property (though such schemes exist); the naivesolution of computing a pair (MAC(m, kAB), MAC(m, kAC)) andsending it as an authenticator doesn't work if B and C don't trusteach other or don't trust A, since one element of the pair mightpass the check for one principal and the other not pass the checkfor the other principal. If A, B, and C all share a single key, thenB or C could create a MAC that appears to come from A.
So, shared keys between more than two principals lose some properties.First, they lose their binding to identities. Second,authentication for different principals cannot be guaranteed. Third,they complicate open systems, in which new principals can appear at anytime, since new principals must be given a key shared with each otherprincipal.
To get around this problem, recall the example of the stock broker.The client published a pair M1 and M2 of numbers. It happenedthat the stock broker was the principal that used these numbers andchecked them, but any principal could have performed the stockbroker's actions, since M1 and M2 were published by the client.We say that key information published like M1 and M2 is a public key and m1 and m2 are the corresponding private key.
1.1 Two-key/trapdoor functions
Two-key or asymmetric cryptography relies on the existence of acomputational primitive called trapdoor functions. A trapdoorfunction takes a domain to a range in such a way that it is easy togo from the domain to range and it is hard to go from the range tothe domain, but it is easy to go from the range to the domain givena special string called the trapdoor.
We parametrize families of trapdoor functions on keys. So, thepublic key K (written as a capital letter) is used to go from thedomain to the range and the private key k (written as a lower-caseletter) is used to go from the range to the domain. As in ourexample, the private key should only be known to one principal.Further, the private key and the public key are usually related bysome function f such that K = f(k); we would like f to be one-wayso that it is hard to find the private key given the public key.
The existence of trapdoor functions implies the existence ofone-way functions, so trapdoor functions are not known to exist,but there are many candidate functions that are believed to betrapdoor functions. For an intuition as to why such functionsexist, consider a related problem: finding and checking witnessesfor NP languages. The definition of NP requires that it be easy tocheck a witness, but it is not guaranteed to be easy to find awitness in the worst case.
Note further that unlike shared key operations, there can be nopublic-key security against adversaries with unlimitedcomputational resources! Since K = f(k) and f is public, any suchadversary can just search the space of keys to find k given K.
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Now let's return to our examples from symmetric cryptography andsee if we can generalize them to run in open systems usingasymmetric cryptography.
1.2 Example application: encryption
In an open system, given any two principals A and B, A should beable to encrypt a message that can only be decrypted by B. Ifthere is some binding established between principal identities andpublic keys, then these operations can easily be performed. Anaive scheme might function as follows: principal A looks up publickey KB for principal B and uses it to compute an encryption for Busing some trapdoor function c = fKB(m). Then B, on receipt ofthis message computes f-1kB(c) = m.
But there's a significant problem with this scheme given ourdefinitions of security for shared-key encryption: it doesn'tsatisfy Semantic Security, since it's trivial for an adversary tocompute fKB(m) and fKB(m') and compare them against givenciphertexts in the different attack models. Once again we see thatthere is no Semantic Security without probabilistic encryption.This is especially true in the public-key setting, since everyprincipal has access to an encryption function for every otherprincipal, by definition. Especially when the space of possiblemessages is small, it is easy to simply check all messages underthe encryption function to figure out what has been encrypted.
1.3 Example application: signatures
Handwritten signatures on physical documents are of great practicalvalue under the assumption (though rarely true) that only the givenindividual could have written the signature. By analogy withhandwritten signatures, digital signature schemes computesignatures that are appended to documents and used forauthentication. It is known that one-way functions are bothnecessary and sufficient for digital signatures, but practicalconstructions of digital signature schemes built directly onone-way functions are not known.
Given a trapdoor function and a setup giving a binding fromidentities to public keys, a simplistic digital signature schememight do exactly the opposite of encryption: principal B uses keykB on message m to compute s = f-1kB(m). Then, to check s,any other principal A would use KB to check that fKB(s) = m.
1.4 History of asymmetric cryptography
- US
- 1975: Diffie imagines asymmetric cryptographyWhitfield Diffie and Martie E. Hellman write a paper calledNew directions in cryptography, in which they describe theidea of asymmetric cryptography.
- 1976: Diffie-Hellman key exchange.This operation allows two principals to set up a shared keygiven a public-key system. It uses a computational assumptioncalled, unsurprisingly, the Computational Diffie-Hellman(CDH) assumption. CDH states that, given g, a generator for afinite field of size n and randomly chosen values a and b inthis field, it is hard for an adversary to construct ga bgiven only g, ga, and gb. If this assumption is true, thenit can be used to exchange shared keys between two principals Aand B as follows:
- A chooses a random a, sets M1 = ga mod n, and sends M1 toB
- B chooses a random b, sets M2 = gb mod n, and send M2 to A
- A computes K = M2a mod n = ga b mod n
- B computes K = M1b mod n = ga b mod n
- now A and B share a key K, but CDH implies that noeavesdropper can construct K given only the information thatwas transmitted between A and B.
- April 1977: (Rivest, Shamir, Adelman) RSA algorithm.RSA was the first publicly known instantiation of a public-keycryptosystem. Rivest, Shamir, and Adelman shared the 2002Turing Award for this development.
- 1975: Diffie imagines asymmetric cryptography
- UKIt turned out in 1997, however, that the RSA function andalgorithm had already been known elsewhere when it was invented.Of course, Rivest, Shamir, and Adelman did their work entirelyindependently.
- 1969: GCHQ asks James Ellis to look at key distribution problem
- 1973: Clifford Cocks thinks of RSA function for asymmetriccryptography. It appears, however, never to have beendeployed.
- January 1974: Malcolm Williams at GCHQ discovers Diffie-Hellmanwhile attempting to find a way to break Cocks' work.
2 Definitions
The definition of encryption in the public-key setting is verysimilar to the definition in the shared-key setting, but sincepublic keys allow encryption and are known to all principals byassumption, every principal has access to an encryption machine asin the CPA attack model. In shared key encryption we can talk aboutsecurity of schemes when an adversary has seen the encryption ofonly one message. But, since adversaries have access to encryptionfunctions by default in the public-key setting, public-keyencryption schemes must always be secure under CPA.
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2.1 Public-key encryption
A public-key encryption scheme (E, D) has two properties
- (Completeness) Given any message m and key pair (K, k), theencryption function and decryption function are inverses:EK(Dk(m)) = Dk(EK(m)) = m.
- (Semantic Security) For any pair of messages m and m', it iscomputationally hard to distinguish encryptions of m fromencryptions of m', even given the public key of a principal.
2.1.1 Attack Models
The same attack models apply here. Although an encryptionfunction is automatically given to each principal, nothingimmediately guarantees that adversaries have access to adecryption function. As before, Chosen Ciphertext Attack (CCA)and Chosen Ciphertext Attack 2 (CCA2) apply, and CCA2 impliesNon-Malleability, as before.
2.1.2 Modes of Encryption
Public-key encryption functions operate on fixed-size inputs andproduce fixed-size outputs, just like shared-key functions, so thesame comments on encryption modes apply here.
2.1.3 Speed
Public-key operations are significantly slower than correspondingshared-key operations. There are two factors at play here: First,shared key operations are often based on low-level bitmanipulation primitives, although these primitives might havealgebraic interpretations. Standard computer hardware isoptimized already for these operations, so they can be performedvery quickly. Public-key operations are often based onexponentiation of large integers; modern hardware is not optimizedfor these operations, though special hardware exists to computethem more quickly.
Second, public keys must have many more bits than shared keys thatachieve the same level of security. This is mostly because sharedkeys are kept secret between each pair of principals, so anadversary has little choice (barring clever attacks on thecryptosystem) but to search the space of keys to try to find theright one. A public key, on the other hand, uniquely determinesthe corresponding private key, so some structure can be exploitedby an adversary trying to find the private key.
The original RSA algorithm was specified with keys of length lessthan 500 bits, but recently it has been claimed that even 1024-bitkeys could be factored, so 1024-bit RSA should be deprecated infavor of larger key sizes like 2048-bit RSA. On fast modernprocessors as of 2007, it takes on the order of 1ms to perform a1024-bit RSA operation, whereas a hash function can be computed onthe order of 1 us. 2048-bit RSA takes on the order of 30 ms.
To contrast and compare the key sizes, although 1024-bit RSA isalready suspect and systems are moving to 2048-bit RSA, 128-bitAES keys are expected to remain secure long into the future(barring new attacks on AES).
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2.1.4 Hybrid Encryption
To get the speed of symmetric key operations in open systems, keyexchange protocols have been developed that initially usepublic-key operations to establish a shared key for a givencommunication session and then use that shared key (under, e.g.,AES) for the remainder of the session. A simplistic exampleinvolves encrypting a large amount of data x. Given a securepublic-key encryption scheme (E, D) with public key Kj forprincipal j, principal i can generate a new shared key k for AESand send AESk(x) || EKj(k). Then j can decrypt k and use kto decrypt x at high speed. Key k can then be used for a sessionof communication between i and j.
2.2 Signature algorithms
To generalize MACs to signatures requires definitions that defendagainst the equivalent of CPA. Instead of being able to seeencryptions of its choice, an adversary should be able to seesignatures of its choice, but still not be able to come up with asignature on a new message. A signature scheme consists of asigning algorithm S and a verification algorithm V. S takes amessage and a private key and outputs a signature. V takes amessage, a signature, and a public key and outputs 1 if thesignature is valid for the message given the key. Otherwise, Voutputs 0. A message m along with a signature s under key k iswritten <m>k, which we call a signed message.
2.2.1 CMA security:
This attack model is called Chosen Message Attack (CMA). Noadversary should be able to find a message m and signature s suchthat V(m, s, K) = 1, even given access to a machine that computesS, as long as the adversary never actually requested S(m, k).
It is instructive in this case to consider why there are nocorresponding attack models to CCA and CCA2. In public-keycryptosystems, verification function V is public, so allprincipals automatically have access to a verification functionand can perform arbitrary verification requests. These models ofsemantic security do not apply here in any case, since there is nonotion of secrecy with respect to signature schemes; a signatureattests to the fact that a message was signed by a givenprincipal, but does not hide any information.
2.2.2 Consistency
Note that given a binding between identities and public keys,signature algorithms satisfy a very useful sort of consistency:any principal that receives a message and signature pair and findsthat V(m, s, K) = 1 knows that it can send m and s to any otherprincipal and that principal will also find that V(m, s, K) = 1.
Note that a signature scheme is a fundamentally asymmetricoperation: one principal attests to many principals about a givendocument.
3 Examples of Public-Key Cryptosystems
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3.1 Merkle's Puzzles
Merkle's Puzzles was one of the first public key cryptographicsystems to be described. It allows principals A and B to agree ona secret key. Principal A invents a million keys and a millionpuzzles, where each puzzle encodes a different one of thekeys. Each puzzle is assumed to take at least two minutes to solveand to fit into 96 bits. A sends these puzzles to B. B then picksa puzzle at random and solves it. B encrypts a pre-arranged string(say 0000) with the key from the puzzle it solved. B sends thisencrypted string back to A. A tries each of the million keys onthe message it receives from B. The one that decrypts the messageand obtains the pre-arranged string is the shared key that A willuse henceforth to communicate with B.
A wiretapper C could steal the million puzzles. However, C wouldneed to crack all million of the puzzles in order to discover theshared key. (If the wiretapper didn't know the pre-arrangedstring, then it can't even use a known-plaintext attack.) Sincecracking each puzzle requires at least 2 minutes, the wiretapperwould need on average 330 days to find the key.
3.2 RSA
The RSA cryptosystem is based on the assumption that factoringlarge integers is computationally hard. This assumption is notknown to be true, but is widely believed. It is not even known iffactoring is an NP-complete problem. RSA keys are often of lengtha power of two, like 512, 1024, or 2048 bits.
The RSA algorithm operates as follows
- Setup:
- To setup a public-private key pair, principal A chooses twoprimes p and q and keeps them secret. A then computes n = p*q.
- A chooses e relatively prime to (p-1)*(q-1) (e can be small;often 3, 17, or 65537 are chosen). It must be less than(p-1)(q-1).
- public key is (e, n) and the private key is (d, n), where d isthe multiplicative inverse of e mod(p-1)(q-1). A publishes (e,n) to complete the setup (usually with a Certificate Authority (CA)).
- Operations:
- encryption of plaintext m: compute me mod n
- decryption of ciphertext c: compute m = cd mod n
RSA works because c = me, so cd = md e mod n = m mod n (needto apply Fermat's Little Theorem and the Chinese RemainderTheorem)
RSA as described above suffers from several problems given ourdefinitions. First, it is deterministic, since a given messagealways encrypts to the same ciphertext. Further, it does notsatisfy Non-Malleability, since two encryptions, can, for example,be multiplied to get a new encryption: me * (m')e mod n =(m*m')e mod n. In some contexts, this kind of malleability can beuseful, but it should be taken into account when designing systemsthat use RSA.
One simple way to solve malleability problems is to add some sortof Message Integrity Check (MIC) using hash functions. Forinstance, if instead of computing me mod n, we computed (m ||h(m))e mod n for some h chosen from a family H ofcollision-resistant hash functions, then the product would be((m || h(m)) * (m' || h(m')))e mod n, which is (m' || h(m'))emod n for some m', but h being a one-way function guarantees thatit is hard for the adversary to find an m' for which this holds.
3.2.1 RSA signatures
Sftp generate new host key. As noted in the discussion on trapdoor functions, signatures cansometimes be constructed from encryption functions. RSAsignatures function in a similar manner.
- signing message m: s = md mod n.
- checking message-signature pair (m, s): return true if m = se modn
RSA signatures suffer from similar problems to malleability, butnow these problems violate the fundamental CMA property ofsignature schemes. Since any two signatures can be combined toget a signature on a new message, it is trivial for an attacker inthe CMA model to violate security.
The common solution to this problem is to hash the message first.Compute signature s = h(m)d mod n. Then even though the aboveattacks can be applied, the adversary can't figure out whichmessage has been signed (to do so would require, given h(m)*h(m'),finding an m' such that h(m') = h(m)*h(m') mod n, which isprevented by pre-image resistance of the hash function).
3.3 ElGamal
ElGamal is an encryption scheme that, like RSA, depends oncomputational assumptions to guarantee security. Unlike the RSAassumption, however, ElGamal depends on an type of assumptioncalled the discrete-logarithm assumption. Roughly, thisassumption claims that it is hard in some groups to find x givengx mod n. The name comes from the fact that x log(g) mod n =log(gx) mod n and division is easy to compute in a group, so x iseasy to compute given log(gx) mod n. ElGamal operates as follows.
- Setup:
- generate safe prime p (this means that there exists a prime qsuch that p = 2q+1).
- Let H be the field of integers mod p and G be the field ofintegers mod q. Choose g as a generator of G, which means thatall elements of G are expressible as some power of g (it iseasy to find such generators)
- Choose a random x in G as a private key and compute y = gx modp. Public key is (p, g, y).
- Operations:
- encryption of plaintext m in G: encode m into M in H, choose random rin G, and output c = (gr, yr * M mod p)
- decryption of ciphertext (a, b): output m = b / ax mod p
Avira antivirus review. ElGamal works because ax = gr x and b = yr * M = gr x *M. Note that this uses probabilistic encryption, and has beenproven secure in some models.
ElGamal as described here also suffers from malleability attacks.Of course, these 'attacks' can be useful in some systems thatrequire manipulation of encrypted values without revealing thesevalues. There exist signature functions, called Schnorr signatures, that give non-malleable ElGamal construction.
3.4 Elliptic Curve Cryptography (ECC)
Most public-key cryptosystems are built over arithmetic in finitefields (algebraic structures that have addition and multiplicationoperations each with inverses). Elliptic Curve Cryptography (ECC)builds a finite field out of the set of solutions to an ellipticcurve equation y2 = x3 + ax + b along with an additive identityelement (that corresponds to the point at infinity).
We won't go into the details of constructing ECC versions of commoncryptosystems, but several such examples exist. It should be notedthat the addition operation in the group is not simply addingsolutions in the underlying field.
ECC is valuable because it is believed to be harder to compute,e.g., discrete logs over the finite fields of ECC than in theunderlying integer finite fields. This means that key sizes in ECCcan be smaller than the corresponding key sizes in cryptosystemsbased on other fields. ECC is not, however, known to be harderthan any other system.