In , Bernstein published ChaCha , a new family of ciphers related to Salsa SEED : A block cipher using bit blocks and bit keys. Serpent : Serpent is another of the AES finalist algorithms. Serpent supports , , or bit keys and a block size of bits, and is a round substitution—permutation network operating on a block of four bit words.
The Serpent developers opted for a high security margin in the design of the algorithm; they determined that 16 rounds would be sufficient against known attacks but require 32 rounds in an attempt to future-proof the algorithm. As a hash function, SHA repeatedly calls on a compression scheme to alter the state of the data blocks. While SHA like other hash functions is irreversible, the compression function can be used for encryption by maintaining appropriate state information.
Simon and Speck : Simon and Speck are a pair of lightweight block ciphers proposed by the NSA in , designed for highly constrained software or hardware environments. While both cipher families perform well in both hardware and software, Simon has been optimized for high performance on hardware devices and Speck for performance in software.
Both are Feistel ciphers and support ten combinations of block and key size:. Skipjack : SKC scheme proposed, along with the Clipper chip , as part of the never-implemented Capstone project. Although the details of the algorithm were never made public, Skipjack was a block cipher using an bit key and 32 iteration cycles per bit block. Capstone, proposed by NIST and the NSA as a standard for public and government use, met with great resistance by the crypto community largely because the design of Skipjack was classified coupled with the key escrow requirement of the Clipper chip.
TEA was originally developed in , and employed a bit key, bit block, and 64 rounds of operation. TWINE's design goals included maintaining a small footprint in a hardware implementation i. Although not an SKC scheme, check out Section 5. There are several other references that describe interesting algorithms and even SKC codes dating back decades.
Savard's albeit old A Cryptographic Compendium page. Public key cryptography has been said to be the most significant new development in cryptography in the last years. Their paper described a two-key crypto system in which two parties could engage in a secure communication over a non-secure communications channel without having to share a secret key.
PKC depends upon the existence of so-called one-way functions , or mathematical functions that are easy to compute whereas their inverse function is relatively difficult to compute. Let me give you two simple examples:. While the examples above are trivial, they do represent two of the functional pairs that are used with PKC; namely, the ease of multiplication and exponentiation versus the relative difficulty of factoring and calculating logarithms, respectively.
The mathematical "trick" in PKC is to find a trap door in the one-way function so that the inverse calculation becomes easy given knowledge of some item of information. Generic PKC employs two keys that are mathematically related although knowledge of one key does not allow someone to easily determine the other key. One key is used to encrypt the plaintext and the other key is used to decrypt the ciphertext.
The important point here is that it does not matter which key is applied first , but that both keys are required for the process to work Figure 1B. Because a pair of keys are required, this approach is also called asymmetric cryptography. In PKC, one of the keys is designated the public key and may be advertised as widely as the owner wants. The other key is designated the private key and is never revealed to another party. It is straight-forward to send messages under this scheme.
Suppose Alice wants to send Bob a message. Alice encrypts some information using Bob's public key; Bob decrypts the ciphertext using his private key. This method could be also used to prove who sent a message; Alice, for example, could encrypt some plaintext with her private key; when Bob decrypts using Alice's public key, he knows that Alice sent the message authentication and Alice cannot deny having sent the message non-repudiation.
Public key cryptography algorithms that are in use today for key exchange or digital signatures include:. RSA today is used in hundreds of software products and can be used for key exchange, digital signatures, or encryption of small blocks of data.
RSA uses a variable size encryption block and a variable size key. The key-pair is derived from a very large number, n , that is the product of two prime numbers chosen according to special rules; these primes may be or more digits in length each, yielding an n with roughly twice as many digits as the prime factors.
The public key information includes n and a derivative of one of the factors of n ; an attacker cannot determine the prime factors of n and, therefore, the private key from this information alone and that is what makes the RSA algorithm so secure. Some descriptions of PKC erroneously state that RSA's safety is due to the difficulty in factoring large prime numbers.
In fact, large prime numbers, like small prime numbers, only have two factors! The ability for computers to factor large numbers, and therefore attack schemes such as RSA, is rapidly improving and systems today can find the prime factors of numbers with more than digits. Nevertheless, if a large number is created from two prime factors that are roughly the same size, there is no known factorization algorithm that will solve the problem in a reasonable amount of time; a test to factor a digit number took 1.
In , Kleinjung et al. Even so, they suggested that bit RSA be phased out by See the Wikipedia article on integer factorization. Regardless, one presumed protection of RSA is that users can easily increase the key size to always stay ahead of the computer processing curve.
As an aside, the patent for RSA expired in September which does not appear to have affected RSA's popularity one way or the other. A detailed example of RSA is presented below in Section 5. Diffie-Hellman is used for secret-key key exchange only, and not for authentication or digital signatures. More detail about Diffie-Hellman can be found below in Section 5. More detail about ECC can be found below in Section 5.
It is a PKC-based key authentication system where the public key can be derived from some unique information based upon the user's identity, allowing two users to exchange encrypted messages without having an a priori relationship. More detail about Identity-Based Encryption can be found below in Section 5.
These documents are no longer easily available; all links in this section are from archive. Cramer and V. Shoup of IBM in LUC : A public key cryptosystem designed by P. Smith and based on Lucas sequences. Can be used for encryption and signatures, using integer factoring. McEliece : A public key cryptosystem based on algebraic coding theory. Menezes, P. Vanstone CRC Press, A digression: Who invented PKC?
I tried to be careful in the first paragraph of this section to state that Diffie and Hellman "first described publicly" a PKC scheme. Although I have categorized PKC as a two-key system, that has been merely for convenience; the real criteria for a PKC scheme is that it allows two parties to exchange a secret even though the communication with the shared secret might be overheard. As shown in Section 5. And, indeed, it is the precursor to modern PKC which does employ two keys.
Their method, of course, is based upon the relative ease of finding the product of two large prime numbers compared to finding the prime factors of a large number. Diffie and Hellman and other sources credit Ralph Merkle with first describing a public key distribution system that allows two parties to share a secret, although it was not a two-key system, per se.
A Merkle Puzzle works where Alice creates a large number of encrypted keys, sends them all to Bob so that Bob chooses one at random and then lets Alice know which he has selected. An eavesdropper Eve will see all of the keys but can't learn which key Bob has selected because he has encrypted the response with the chosen key. In this case, Eve's effort to break in is the square of the effort of Bob to choose a key. While this difference may be small it is often sufficient.
Merkle apparently took a computer science course at UC Berkeley in and described his method, but had difficulty making people understand it; frustrated, he dropped the course. Merkle's method certainly wasn't published first, but he is often credited to have had the idea first.
An interesting question, maybe, but who really knows? Because of the nature of the work, GCHQ kept the original memos classified. In , however, the GCHQ changed their posture when they realized that there was nothing to gain by continued silence. Documents show that a GCHQ mathematician named James Ellis started research into the key distribution problem in and that by , James Ellis, Clifford Cocks, and Malcolm Williamson had worked out all of the fundamental details of PKC, yet couldn't talk about their work.
They were, of course, barred from challenging the RSA patent! Hash functions, also called message digests and one-way encryption , are algorithms that, in essence, use no key Figure 1C. Instead, a fixed-length hash value is computed based upon the plaintext that makes it impossible for either the contents or length of the plaintext to be recovered. Hash algorithms are typically used to provide a digital fingerprint of a file's contents, often used to ensure that the file has not been altered by an intruder or virus.
Hash functions are also commonly employed by many operating systems to encrypt passwords. Hash functions, then, provide a mechanism to ensure the integrity of a file. Hash functions are also designed so that small changes in the input produce significant differences in the hash value, for example:. This is an important distinction. Suppose that you want to crack someone's password, where the hash of the password is stored on the server.
Indeed, all you then need is a string that produces the correct hash and you're in! However, you cannot prove that you have discovered the user's password, only a "duplicate key. Message Digest MD algorithms: A series of byte-oriented algorithms that produce a bit hash value from an arbitrary-length message.
MD2 has been relegated to historical status, per RFC MD4 has been relegated to historical status, per RFC MD5 RFC : Also developed by Rivest after potential weaknesses were reported in MD4; this scheme is similar to MD4 but is slower because more manipulation is made to the original data. MD5 has been implemented in a large number of products although several weaknesses in the algorithm were demonstrated by German cryptographer Hans Dobbertin in "Cryptanalysis of MD5 Compress".
In , NIST announced that after reviewing 64 submissions, the winner was Keccak pronounced "catch-ack" , a family of hash algorithms based on sponge functions. The NIST version can support hash output sizes of and bits.
A root hash is used on peer-to-peer file transfer networks, where a file is broken into chunks; each chunk has its own MD4 hash associated with it and the server maintains a file that contains the hash list of all of the chunks. The root hash is the hash of the hash list file.
Zheng, J. Pieprzyk and J. Seberry, a hash algorithm with many levels of security. HAVAL can create hash values that are , , , , or bits in length. Skein supports internal state sizes of , and bits, and arbitrary output lengths. SM3 : SM3 is a bit hash function operating on bit input blocks.
More information can also be found at the SM3 hash function page. Whirlpool : Designed by V. Rijmen co-inventor of Rijndael and P. Whirlpool operates on messages less than 2 bits in length and produces a message digest of bits.
The design of this hash function is very different than that of MD5 and SHA-1, making it immune to the types of attacks that succeeded on those hashes. A digression on hash collisions. Hash functions are sometimes misunderstood and some sources claim that no two files can have the same hash value. This is in theory, if not in fact, incorrect.
Consider a hash function that provides a bit hash value. There are, then, 2 possible hash values. Now, while even this is theoretically correct, it is not true in practice because hash algorithms are designed to work with a limited message size, as mentioned above.
The difficulty is not necessarily in finding two files with the same hash, but in finding a second file that has the same hash value as a given first file. Consider this example. Since there are more than 7 billion people on earth, we know that there are a lot of people with the same number of hairs on their head. Finding two people with the same number of hairs, then, would be relatively simple.
The harder problem is choosing one person say, you, the reader and then finding another person who has the same number of hairs on their head as you have on yours. This is somewhat similar to the Birthday Problem. Alas, researchers as far back as found that practical collision attacks could be launched on MD5, SHA-1, and other hash algorithms and, today, it is generally recognized that MD5 and SHA-1 are pretty much broken. Readers interested in this problem should read the following:.
For historical purposes, take a look at the situation with hash collisions, circa , in RFC In October , the SHA-1 Freestart Collision was announced; see a report by Bruce Schneier and the developers of the attack as well as the paper above by Stevens et al. See also the paper by Stevens et al. Stevens, A. Lenstra, and B. Finally, note that certain extensions of hash functions are used for a variety of information security and digital forensics applications, such as:.
So, why are there so many different types of cryptographic schemes? Why can't we do everything we need with just one? The answer is that each scheme is optimized for some specific cryptographic application s. Hash functions, for example, are well-suited for ensuring data integrity because any change made to the contents of a message will result in the receiver calculating a different hash value than the one placed in the transmission by the sender.
Since it is highly unlikely that two different messages will yield the same hash value, data integrity is ensured to a high degree of confidence. Secret key cryptography, on the other hand, is ideally suited to encrypting messages, thus providing privacy and confidentiality. The sender can generate a session key on a per-message basis to encrypt the message; the receiver, of course, needs the same session key in order to decrypt the message.
Key exchange, of course, is a key application of public key cryptography no pun intended. Asymmetric schemes can also be used for non-repudiation and user authentication; if the receiver can obtain the session key encrypted with the sender's private key, then only this sender could have sent the message. Public key cryptography could, theoretically, also be used to encrypt messages although this is rarely done because secret key cryptography values can generally be computed about times faster than public key cryptography values.
Figure 4 puts all of this together and shows how a hybrid cryptographic scheme combines all of these functions to form a secure transmission comprising a digital signature and digital envelope. In this example, the sender of the message is Alice and the receiver is Bob.
A digital envelope comprises an encrypted message and an encrypted session key. Alice uses secret key cryptography to encrypt her message using the session key , which she generates at random with each session. Alice then encrypts the session key using Bob's public key. The encrypted message and encrypted session key together form the digital envelope.
Upon receipt, Bob recovers the session secret key using his private key and then decrypts the encrypted message. The digital signature is formed in two steps. First, Alice computes the hash value of her message; next, she encrypts the hash value with her private key. Upon receipt of the digital signature, Bob recovers the hash value calculated by Alice by decrypting the digital signature with Alice's public key. Bob can then apply the hash function to Alice's original message, which he has already decrypted see previous paragraph.
If the resultant hash value is not the same as the value supplied by Alice, then Bob knows that the message has been altered; if the hash values are the same, Bob should believe that the message he received is identical to the one that Alice sent. This scheme also provides nonrepudiation since it proves that Alice sent the message; if the hash value recovered by Bob using Alice's public key proves that the message has not been altered, then only Alice could have created the digital signature.
Bob also has proof that he is the intended receiver; if he can correctly decrypt the message, then he must have correctly decrypted the session key meaning that his is the correct private key. This diagram purposely suggests a cryptosystem where the session key is used for just a single session. Even if this session key is somehow broken, only this session will be compromised; the session key for the next session is not based upon the key for this session, just as this session's key was not dependent on the key from the previous session.
This is known as Perfect Forward Secrecy ; you might lose one session key due to a compromise but you won't lose all of them. In a article in the industry literature, a writer made the claim that bit keys did not provide as adequate protection for DES at that time as they did in because computers were times faster in than in Therefore, the writer went on, we needed 56,bit keys in instead of bit keys to provide adequate protection.
The conclusion was then drawn that because 56,bit keys are infeasible true , we should accept the fact that we have to live with weak cryptography false! The major error here is that the writer did not take into account that the number of possible key values double whenever a single bit is added to the key length; thus, a bit key has twice as many values as a bit key because 2 57 is two times 2 In fact, a bit key would have times more values than a bit key.
In cryptography, size does matter. The larger the key, the harder it is to crack a block of encrypted data. The reason that large keys offer more protection is almost obvious; computers have made it easier to attack ciphertext by using brute force methods rather than by attacking the mathematics which are generally well-known anyway.
With a brute force attack, the attacker merely generates every possible key and applies it to the ciphertext. Any resulting plaintext that makes sense offers a candidate for a legitimate key. Until the mids or so, brute force attacks were beyond the capabilities of computers that were within the budget of the attacker community. By that time, however, significant compute power was typically available and accessible. General-purpose computers such as PCs were already being used for brute force attacks.
Distributed attacks, harnessing the power of up to tens of thousands of powerful CPUs, are now commonly employed to try to brute-force crypto keys. This information was not merely academic; one of the basic tenets of any security system is to have an idea of what you are protecting and from whom are you protecting it!
The table clearly shows that a bit key was essentially worthless against even the most unsophisticated attacker. On the other hand, bit keys were fairly strong unless you might be subject to some pretty serious corporate or government espionage. But note that even bit keys were clearly on the decline in their value and that the times in the table were worst cases.
So, how big is big enough? DES, invented in , was still in use at the turn of the century, nearly 25 years later. If we take that to be a design criteria i. The DES proposal suggested bit keys; by , a bit key would have been required to offer equal protection and an bit key necessary by A or bit SKC key will probably suffice for some time because that length keeps us ahead of the brute force capabilities of the attackers. Note that while a large key is good, a huge key may not always be better; for example, expanding PKC keys beyond the current or bit lengths doesn't add any necessary protection at this time.
Weaknesses in cryptosystems are largely based upon key management rather than weak keys. Blaze, W. Diffie, R. Rivest, B. Schneier, T. Shimomura, E. Thompson, and M. Wiener The most effective large-number factoring methods today use a mathematical Number Field Sieve to find a certain number of relationships and then uses a matrix operation to solve a linear equation to produce the two prime factors.
The sieve step actually involves a large number of operations that can be performed in parallel; solving the linear equation, however, requires a supercomputer. In early , Shamir of RSA fame described a new machine that could increase factorization speed by orders of magnitude. There still appear to be many engineering details that have to be worked out before such a machine could be built.
Furthermore, the hardware improves the sieve step only; the matrix operation is not optimized at all by this design and the complexity of this step grows rapidly with key length, both in terms of processing time and memory requirements. Nevertheless, this plan conceptually puts bit keys within reach of being factored.
It is also interesting to note that while cryptography is good and strong cryptography is better, long keys may disrupt the nature of the randomness of data files. Shamir and van Someren "Playing hide and seek with stored keys" have noted that a new generation of viruses can be written that will find files encrypted with long keys, making them easier to find by intruders and, therefore, more prone to attack.
Finally, U. Until the mids, export outside of North America of cryptographic products using keys greater than 40 bits in length was prohibited, which made those products essentially worthless in the marketplace, particularly for electronic commerce; today, crypto products are widely available on the Internet without restriction.
The U. Department of Commerce Bureau of Industry and Security maintains an Encryption FAQ web page with more information about the current state of encryption registration. Without meaning to editorialize too much in this tutorial, a bit of historical context might be helpful. In the mids, the U. Department of Commerce still classified cryptography as a munition and limited the export of any products that contained crypto.
For that reason, browsers in the era, such as Internet Explorer and Netscape, had a domestic version with bit encryption downloadable only in the U. Many cryptographers felt that the export limitations should be lifted because they only applied to U. Those restrictions were lifted by or , but there is still a prevailing attitude, apparently, that U.
On a related topic, public key crypto schemes can be used for several purposes, including key exchange, digital signatures, authentication, and more. The length of the secret keys exchanged via that system have to have at least the same level of attack resistance. Secure use of cryptography requires trust. While secret key cryptography can ensure message confidentiality and hash codes can ensure integrity, none of this works without trust. PKC solved the secret distribution problem, but how does Alice really know that Bob is who he says he is?
Just because Bob has a public and private key, and purports to be "Bob," how does Alice know that a malicious person Mallory is not pretending to be Bob? There are a number of trust models employed by various cryptographic schemes. This section will explore three of them:. Each of these trust models differs in complexity, general applicability, scope, and scalability. Pretty Good Privacy described more below in Section 5. A PGP user maintains a local keyring of all their known and trusted public keys.
The user makes their own determination about the trustworthiness of a key using what is called a "web of trust. If Alice needs Bob's public key, Alice can ask Bob for it in another e-mail or, in many cases, download the public key from an advertised server; this server might a well-known PGP key repository or a site that Bob maintains himself. In fact, Bob's public key might be stored or listed in many places. Alice is prepared to believe that Bob's public key, as stored at these locations, is valid.
Suppose Carol claims to hold Bob's public key and offers to give the key to Alice. How does Alice know that Carol's version of Bob's key is valid or if Carol is actually giving Alice a key that will allow Mallory access to messages? The answer is, "It depends. And trust is not necessarily transitive; if Dave has a copy of Bob's key and Carol trusts Dave, it does not necessarily follow that Alice trusts Dave even if she does trust Carol.
The point here is that who Alice trusts and how she makes that determination is strictly up to Alice. PGP makes no statement and has no protocol about how one user determines whether they trust another user or not. In any case, encryption and signatures based on public keys can only be used when the appropriate public key is on the user's keyring.
Kerberos is a commonly used authentication scheme on the Internet. Developed by MIT's Project Athena, Kerberos is named for the three-headed dog who, according to Greek mythology, guards the entrance of Hades rather than the exit, for some reason! In this model, security and authentication will be based on secret key technology where every host on the network has its own secret key. It would clearly be unmanageable if every host had to know the keys of all other hosts so a secure, trusted host somewhere on the network, known as a Key Distribution Center KDC , knows the keys for all of the hosts or at least some of the hosts within a portion of the network, called a realm.
In this way, when a new node is brought online, only the KDC and the new node need to be configured with the node's key; keys can be distributed physically or by some other secure means. While the details of their operation, functional capabilities, and message formats are different, the conceptual overview above pretty much holds for both.
One primary difference is that Kerberos V4 uses only DES to generate keys and encrypt messages, while V5 allows other schemes to be employed although DES is still the most widely algorithm used. Certificates and Certificate Authorities CA are necessary for widespread use of cryptography for e-commerce applications. While a combination of secret and public key cryptography can solve the business issues discussed above, crypto cannot alone address the trust issues that must exist between a customer and vendor in the very fluid, very dynamic e-commerce relationship.
How, for example, does one site obtain another party's public key? How does a recipient determine if a public key really belongs to the sender? How does the recipient know that the sender is using their public key for a legitimate purpose for which they are authorized?
When does a public key expire? How can a key be revoked in case of compromise or loss? The basic concept of a certificate is one that is familiar to all of us. A driver's license, credit card, or SCUBA certification, for example, identify us to others, indicate something that we are authorized to do, have an expiration date, and identify the authority that granted the certificate. As complicated as this may sound, it really isn't. Consider driver's licenses.
I have one issued by the State of Florida. The license establishes my identity, indicates the type of vehicles that I can operate and the fact that I must wear corrective lenses while doing so, identifies the issuing authority, and notes that I am an organ donor.
When I drive in other states, the other jurisdictions throughout the U. When I leave the U. When I am in Aruba, Australia, Canada, Israel, and many other countries, they will accept not the Florida license, per se, but any license issued in the U. This analogy represents the certificate trust chain, where even certificates carry certificates. For purposes of electronic transactions, certificates are digital documents.
The specific functions of the certificate include:. A sample abbreviated certificate is shown in Figure 6. While this is a certificate issued by VeriSign, many root-level certificates can be found shipped with browsers. When the browser makes a connection to a secure Web site, the Web server sends its public key certificate to the browser. The browser then checks the certificate's signature against the public key that it has stored; if there is a match, the certificate is taken as valid and the Web site verified by this certificate is considered to be "trusted.
TABLE 2. Contents of an X. Most certificates today comply with X. Certificate authorities are the repositories for public keys and can be any agency that issues certificates. When a sender needs an intended receiver's public key, the sender must get that key from the receiver's CA. That scheme is straight-forward if the sender and receiver have certificates issued by the same CA.
If not, how does the sender know to trust the foreign CA? One industry wag has noted, about trust: "You are either born with it or have it granted upon you. CAs, in turn, form trust relationships with other CAs.
Thus, if a user queries a foreign CA for information, the user may ask to see a list of CAs that establish a "chain of trust" back to the user. One major feature to look for in a CA is their identification policies and procedures. When a user generates a key pair and forwards the public key to a CA, the CA has to check the sender's identification and takes any steps necessary to assure itself that the request is really coming from the advertised sender.
Different CAs have different identification policies and will, therefore, be trusted differently by other CAs. Verification of identity is just one of many issues that are part of a CA's Certification Practice Statement CPS and policies; other issues include how the CA protects the public keys in its care, how lost or compromised keys are revoked, and how the CA protects its own private keys. As a final note, CAs are not immune to attack and certificates themselves are able to be counterfeited.
Problems have continued over the years; good write-ups on this can be found at " Another Certification Authority Breached the 12th! The paragraphs above describe three very different trust models. It is hard to say that any one is better than the others; it depends upon your application.
One of the biggest and fastest growing applications of cryptography today, though, is electronic commerce e-commerce , a term that itself begs for a formal definition. PGP's web of trust is easy to maintain and very much based on the reality of users as people. The model, however, is limited; just how many public keys can a single user reliably store and maintain? And what if you are using the "wrong" computer when you want to send a message and can't access your keyring?
How easy it is to revoke a key if it is compromised? PGP may also not scale well to an e-commerce scenario of secure communication between total strangers on short-notice. Kerberos overcomes many of the problems of PGP's web of trust, in that it is scalable and its scope can be very large. In the early days of the Internet, every host had to maintain a list of every other host; the Domain Name System DNS introduced the idea of a distributed database for this purpose and the DNS is one of the key reasons that the Internet has grown as it has.
While certificates and the benefits of a PKI are most often associated with electronic commerce, the applications for PKI are much broader and include secure electronic mail, payments and electronic checks, Electronic Data Interchange EDI , secure transfer of Domain Name System DNS and routing information, electronic forms, and digitally signed documents.
A single "global PKI" is still many years away, that is the ultimate goal of today's work as international electronic commerce changes the way in which we do business in a similar way in which the Internet has changed the way in which we communicate. The paragraphs above have provided an overview of the different types of cryptographic algorithms, as well as some examples of some available protocols and schemes.
The paragraphs below will show several real cryptographic applications that many of us employ knowingly or not everyday for password protection and private communication. Some of the schemes described below never were widely deployed but are still historically interesting, thus remain included here. But passwords are not typically kept on a host or server in plaintext, but are generally encrypted using some sort of hash scheme. Note that each password is stored as a byte string.
The first two characters are actually a salt , randomness added to each password so that if two users have the same password, they will still be encrypted differently; the salt, in fact, provides a means so that a single password might have different encryptions. The remaining 11 bytes are the password hash, calculated using DES. This fact, coupled with the weak encryption of the passwords, resulted in the development of the shadow password system where passwords are kept in a separate, non-world-readable file used in conjunction with the normal password file.
In the NT case, all passwords are hashed using the MD4 algorithm, resulting in a bit byte hash value they are then obscured using an undocumented mathematical transformation that was a secret until distributed on the Internet. The password password , for example, might be stored as the hash value in hexadecimal b22d73c34bd4aa79c8b09f Passwords are not saved in plaintext on computer systems precisely so they cannot be easily compromised. For similar reasons, we don't want passwords sent in plaintext across a network.
But for remote logon applications, how does a client system identify itself or a user to the server? One mechanism, of course, is to send the password as a hash value and that, indeed, may be done. A weakness of that approach, however, is that an intruder can grab the password off of the network and use an off-line attack such as a dictionary attack where an attacker takes every known word and encrypts it with the network's encryption algorithm, hoping eventually to find a match with a purloined password hash.
In some situations, an attacker only has to copy the hashed password value and use it later on to gain unauthorized entry without ever learning the actual password. An even stronger authentication method uses the password to modify a shared secret between the client and server, but never allows the password in any form to go across the network.
As suggested above, Windows NT passwords are stored in a security file on a server as a byte hash value. When a user logs on to a server from a remote workstation, the user is identified by the username, sent across the network in plaintext no worries here; it's not a secret anyway!
The server then generates a bit random number and sends it to the client also in plaintext. This number is the challenge. Recall that DES employs a bit key, acts on a bit block of data, and produces a bit output.
In this case, the bit data block is the random number. The client actually uses three different DES keys to encrypt the random number, producing three different bit outputs. The first key is the first seven bytes 56 bits of the password's hash value, the second key is the next seven bytes in the password's hash, and the third key is the remaining two bytes of the password's hash concatenated with five zero-filled bytes.
So, for the example above, the three DES keys would be b22d73c34 , bd4aa79c8b0 , and 9f Each key is applied to the random number resulting in three bit outputs, which comprise the response. Thus, the server's 8-byte challenge yields a byte response from the client and this is all that would be seen on the network.
The server, for its part, does the same calculation to ensure that the values match. There is, however, a significant weakness to this system. Specifically, the response is generated in such a way as to effectively reduce byte hash to three smaller hashes, of length seven, seven, and two, respectively. Thus, a password cracker has to break at most a 7-byte hash. One Windows NT vulnerability test program that I used in the past reported passwords that were "too short," defined as "less than 8 characters.
This was, in fact, not the case at all; all the software really had to do was to look at the last eight bytes of the Windows NT LanMan hash to see that the password was seven or fewer characters. Consider the following example, showing the LanMan hash of two different short passwords take a close look at the last 8 bytes :. MS-CHAP assumes that it is working with hashed values of the password as the key to encrypting the challenge. Diffie and Hellman introduced the concept of public key cryptography.
The mathematical "trick" of Diffie-Hellman key exchange is that it is relatively easy to compute exponents compared to computing discrete logarithms. Diffie-Hellman works like this. Alice and Bob start by agreeing on a large prime number, N. There is actually another constraint on G, namely that it must be primitive with respect to N. As an example, 2 is not primitive to 7 because the set of powers of 2 from 1 to 6, mod 7 i. The definition of primitive introduced a new term to some readers, namely mod.
The phrase x mod y and read as written! Read more about the modulo function in the appendix. Anyway, either Alice or Bob selects N and G; they then tell the other party what the values are. Alice and Bob then work independently:. Perhaps a small example will help here. In this example, then, Alice and Bob will both find the secret key 1 which is, indeed, 3 6 mod 7 i. A short digression on modulo arithmetic. This can be confirmed, of course, by noting that:.
Diffie-Hellman can also be used to allow key sharing amongst multiple users. Note again that the Diffie-Hellman algorithm is used to generate secret keys, not to encrypt and decrypt messages. Unlike Diffie-Hellman, RSA can be used for key exchange as well as digital signatures and the encryption of small blocks of data. Today, RSA is primarily used to encrypt the session key used for secret key encryption message integrity or the message's hash value digital signature.
RSA's mathematical hardness comes from the ease in calculating large numbers and the difficulty in finding the prime factors of those large numbers. Although employed with numbers using hundreds of digits, the math behind RSA is relatively straight-forward.
The public key is the number pair n,e. Although these values are publicly known, it is computationally infeasible to determine d from n and e if p and q are large enough. Now, this might look a bit complex and, indeed, the mathematics does take a lot of computer power given the large size of the numbers; since p and q may be digits decimal or more, d and e will be about the same size and n may be over digits.
Nevertheless, a simple example may help. In this example, the values for p, q, e, and d are purposely chosen to be very small and the reader will see exactly how badly these values perform, but hopefully the algorithm will be adequately demonstrated:. I choose this trivial example because the value of n is so small in particular, the value M cannot exceed n.
But here is a more realistic example using larger d, e, and n values, as well as a more meaningful message; thanks to Barry Steyn for permission to use values from his How RSA Works With Examples page. Now suppose that our message M is the character string "attack at dawn" which has the numeric value after converting the ASCII characters to a bit string and interpreting that bit string as a decimal number of This more realistic example gives just a clue as to how large the numbers are that are used in the real world implementations.
RSA keylengths of and bits are considered to be pretty weak. The minimum suggested RSA key is bits; and bits are even better. It employs dc , an arbitrary precision arithmetic package that ships with most UNIX systems:. Despite all of these options, ECB is the most commonly deployed mode of operation.
Although other block ciphers have replaced DES, it is still interesting to see how DES encryption is performed; not only is it sort of neat, but DES was the first crypto scheme commonly seen in non-governmental applications and was the catalyst for modern "public" cryptography and the first public Feistel cipher. DES uses a bit key. In fact, the bit key is divided into eight 7-bit blocks and an 8th odd parity bit is added to each block i. By using the 8 parity bits for rudimentary error detection, a DES key is actually 64 bits in length for computational purposes although it only has 56 bits worth of randomness, or entropy See Section A.
DES then acts on bit blocks of the plaintext, invoking 16 rounds of permutations, swaps, and substitutes, as shown in Figure 8. The standard includes tables describing all of the selection, permutation, and expansion operations mentioned below; these aspects of the algorithm are not secrets. The basic DES steps are:. At any given step in the process, then, the new L block value is merely taken from the prior R block value. K n is a bit value derived from the bit DES key.
Each round uses a different 48 bits according to the standard's Key Schedule algorithm. The cipher function, f, combines the bit R block value and the bit subkey in the following way. First, the 32 bits in the R block are expanded to 48 bits by an expansion function E ; the extra 16 bits are found by repeating the bits in 16 predefined positions. The bit expanded R-block is then ORed with the bit subkey.
The result is a bit value that is then divided into eight 6-bit blocks. These are fed as input into 8 selection S boxes, denoted S 1 , Each 6-bit input yields a 4-bit output using a table lookup based on the 64 possible inputs; this results in a bit output from the S-box. The 32 bits are then rearranged by a permutation function P , producing the results from the cipher function. Observe that we start with a byte input message.
DES acts on eight bytes at a time, so this message is padded to 24 bytes and provides three "inputs" to the cipher algorithm we don't see the padding here; it is appended by the DES code. Since we have three input blocks, we get 24 bytes of output from the three bit eight byte output blocks. An excellent step-by-step example of DES can also be found at J.
The mainstream cryptographic community has long held that DES's bit key was too short to withstand a brute-force attack from modern computers. Remember Moore's Law: computer power doubles every 18 months. Given that increase in power, a key that could withstand a brute-force guessing attack in could hardly be expected to withstand the same attack a quarter century later. DES is even more vulnerable to a brute-force attack because it is often used to encrypt words, meaning that the entropy of the bit block is, effectively, greatly reduced.
That is, if we are encrypting random bit streams, then a given byte might contain any one of 2 8 possible values and the entire bit block has 2 64 , or about If we are encrypting words, however, we are most likely to find a limited set of bit patterns; perhaps 70 or so if we account for upper and lower case letters, the numbers, space, and some punctuation.
Despite this criticism, the U. It was completed in 84 days by R. Verser in a collaborative effort using thousands of computers on the Internet. This problem was solved by distributed. The distributed. Information about the hardware design and all software can be obtained from the EFF.
This is widely considered to have been the final nail in DES's coffin. The Deep Crack algorithm is actually quite interesting. The general approach that the DES Cracker Project took was not to break the algorithm mathematically but instead to launch a brute-force attack by guessing every possible key. A bit key yields 2 56 , or about 72 quadrillion, possible values.
So the DES cracker team looked for any shortcuts they could find! First, they assumed that some recognizable plaintext would appear in the decrypted string even though they didn't have a specific known plaintext block. They then applied all 2 56 possible key values to the bit block I don't mean to make this sound simple! The system checked to see if the decrypted value of the block was "interesting," which they defined as bytes containing one of the alphanumeric characters, space, or some punctuation.
This dropped the number of possible keys that might yield positive results to about 2 40 , or about a trillion. They then made the assumption that an "interesting" 8-byte block would be followed by another "interesting" block. So, if the first block of ciphertext decrypted to something interesting, they decrypted the next block; otherwise, they abandoned this key.
Only if the second block was also "interesting" did they examine the key closer. Looking for 16 consecutive bytes that were "interesting" meant that only 2 24 , or 16 million, keys needed to be examined further. This further examination was primarily to see if the text made any sense. And even a slow laptop today can search through lists of only a few million items in a relatively short period of time.
It is well beyond the scope of this paper to discuss other forms of breaking DES and other codes. Nevertheless, it is worth mentioning a couple of forms of cryptanalysis that have been shown to be effective against DES. Differential cryptanalysis , invented in by E. Biham and A. Shamir of RSA fame , is a chosen-plaintext attack. By selecting pairs of plaintext with particular differences, the cryptanalyst examines the differences in the resultant ciphertext pairs.
Linear plaintext , invented by M. Matsui, uses a linear approximation to analyze the actions of a block cipher including DES. Both of these attacks can be more efficient than brute force. Once DES was "officially" broken, several variants appeared. But none of them came overnight; work at hardening DES had already been underway. In the early s, there was a proposal to increase the security of DES by effectively increasing the key length by using multiple keys with multiple passes. But for this scheme to work, it had to first be shown that the DES function is not a group , as defined in mathematics.
If DES were a group, it wouldn't matter how many keys and passes we applied to some plaintext; we could always find a single bit key that would provide the same result. As it happens, DES was proven to not be a group so that as we apply additional keys and passes, the effective key length increases.
One obvious choice, then, might be to use two keys and two passes, yielding an effective key length of bits. Let's call this Double-DES. The two keys, Y1 and Y2, might be applied as follows:. So far, so good. But there's an interesting attack that can be launched against this "Double-DES" scheme.
First, notice that the applications of the formula above can be thought of with the following individual steps where C' and P' are intermediate results :. That leaves us vulnerable to a simple known plaintext attack sometimes called "Meet-in-the-middle" where the attacker knows some plaintext P and its matching ciphertext C.
To obtain C', the attacker needs to try all 2 56 possible values of Y1 applied to P; to obtain P', the attacker needs to try all 2 56 possible values of Y2 applied to C. So "Double-DES" is not a good solution. Generation of the ciphertext C from a block of plaintext P is accomplished by:.
For obvious reasons, this is sometimes referred to as an encrypt-decrypt-encrypt mode operation. The use of three, independent bit keys provides 3DES with an effective key length of bits. Given the relatively low cost of key storage and the modest increase in processing due to the use of longer keys, the best recommended practices are that 3DES be employed with three keys.
Developed in , DESX is a very simple algorithm that greatly increases DES's resistance to brute-force attacks without increasing its computational complexity. As it happens, DESX is no more immune to other types of more sophisticated attacks, such as differential or linear cryptanalysis, but brute-force is the primary attack vector on DES.
Although DES has been deprecated and replaced by the Advanced Encryption Standard AES because of its vulnerability to a modestly-priced brute-force attack, many applications continue to rely on DES for security, and many software designers and implementers continue to include DES in new applications. In some cases, use of DES is wholly appropriate but, in general, DES should not continue to be promulgated in production software and hardware.
Pretty Good Privacy PGP is one of today's most widely used public key cryptography programs and was the first open cryptosystem that combined hashing, compression, SKC, and PKC into a method to protect files, devices, and e-mail. Public keys were shared via a concept known as a Web of Trust; individuals would directly exchange their public keyrings and then share their keyrings with other trusted parties. PGP secret keys, however, were bits or larger, making it a "strong" cryptography product.
Yet, in , perhaps as a harbinger of the mixed feelings that this technology engendered, the Electronic Frontier Foundation EFF awarded Zimmermann the Pioneer Award and Newsweek Magazine named him one of the 50 most influential people on the Internet. Signcryption has since been adopted by the International Standardization Organization ISO as a global standard for cyber security. Follow the linked course names to see sample syllabi which describe course aims, learning objectives, requirements, and schedules.
These are samples only. The instructor may make changes to these syllabi in future courses. Cryptocurrency Applied Cryptography. Puwen Wei and Yuliang Zheng. International Journal of Foundations of Computer Science 26, 1 , Graham Enos and Yuliang Zheng. Information Processing Letters , Alex Dent and Yuliang Zheng. Applied Signcryption. Springer Verlag. Formal Proofs for the Security of Signcryption.
Journal of Cryptology 20 No. Ron Steinfeld and Yuliang Zheng. Yuliang Zheng.
He is widely known as the father of the signcryption technology, which is now an ISO international standard for cyber security. A prolific researcher with more than refereed publications, Dr. Zheng has been at the forefront of establishing international standards for cryptographic engineering. Department of Computer Science. Software Resources. Yuliang Zheng Professor and Chair This email address is being protected from spambots.
These are samples only. The instructor may make changes to these syllabi in future courses. Cryptocurrency Applied Cryptography. Puwen Wei and Yuliang Zheng. International Journal of Foundations of Computer Science 26, 1 , Graham Enos and Yuliang Zheng. Kostianinen, S. Meiklejohn, A. Miller, E. Prasad, K. Brookings Institute Working Paper, Kelkar, F. Zhang, S. Order-Fairness for Byzantine Consensus. Mirkin, Y. Ji, J. Pang, A.
Klages-Mundt, I. Eyal, and A. Malvai, S. Goldfeder, and A. Project website: deco. Daian, S. Goldfeder, T. Kell, Y. Li, X. Zhao, I. Bentov, L. Breidenbach, and A. Flash Boys 2. Bloomberg article here. Juels, L. Breidenbach, A. Nazarov, S. Ellis, and B. Mixicles: Simple Private Decentralized Finance.
Work performed in my non-academic role as technical advisor to Chainlink. Chatterjee, M. Riazi, T. Chowdhury, E. Marasco, F. Koushanfar, and A. Maram, F. Zhang, L. Wang, A. Low, Y. Zhang, A. Juels, and D. Cecchetti, B. Fisch, I. Miers, and A. Bentov, Y. Ji, F. Zhang, Y. Zhao, L. Breidenbach, P. Daian, and A. Zhang, P. Daian, I. Bentov, and A. Bitcoin Magazine article here. Cheng, F. Zhang, J. Kos, W. He, N. Hynes, N. Johnson, A.
Juels, A. Miller, and D. Note: Used in Oasis Labs products. Matetic, M. Schneider, A. Miller, A, Juels, and S. Tramer, and A. Visit the Hydra Project website for code. Video presentation here. Cecchetti, F. Ji, A. Kosba, A. Juels, and E. Ahmed, K. Kostiainen, A. Dhar, D. Sommer, A. Gervais, A. Juels, and S. Zhang, I. Eyal, R.
Escriva, A. Juels, and R. Aikat, A. Akella, J. Chase, A. Juels, M. Reiter, T. Ristenpart, V. Sekar, M. Rethinking Security in the Era of Cloud Computing. Woodage, R. Chatterjee, Y. Dodis, A. Juels, and T. Sekar, and M. Zhang, H. Lin, J. Atlidakis, R. Geambasu, D. Hubaux, M. Humbert, A. Juels, and H. Eyal, A. Juels, and G. Zhang, E. Cecchetti, K. Croman, A. Note: Licensed by Chainlink.
Slides here. Kosba, and E. Reiter, and T. Marino and A. Rule ML, pp. Hubaux and A. Privacy is dead, long live privacy: Protecting social norms as confidentiality wanes. Chatterjee, A. Athayle, D. Akawhe, A. Croman, C. Decker, I. Gencer, A. Miller, P. Saxena, E. Shi, E. Sirer, D. Song, and R. On Scaling Decentralized Blockchains. Estrin and A. Reassembling Our Digital Selves. Everspaugh, R. Chatterjee, S. Scott, A. Huang, E. Ayday, J. Hubaux, J. Chatterjee, J. Dodis, C. Ganesh, A.
Golovnev, A. Juels and T. Yen, V. Heorhiadi, A. Oprea, M. Reiter, and A. Bowers, C. Hart, A. Juels, and N. RAID, pp. A bodyguard of lies: the use of honey objects in information security. Miller, A. Juels, E. Shi, B. Parno, and J. Juels and B. The Interplay of Neuroscience and Cryptography. Yen, A. Oprea, K. Onarlioglu, T. Leetham, A. Kirda, and W.
Juels and R. Honeywords: Making Password-Cracking Detectable. Rostami, A. Juels, and F. Oprea, and R. Rostami, W. Burleson, A.
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Koushanfar, and A. Maram, F. Zhang, L. Wang, A. Low, Y. Zhang, A. Juels, and D. Cecchetti, B. Fisch, I. Miers, and A. Bentov, Y. Ji, F. Zhang, Y. Zhao, L. Breidenbach, P. Daian, and A. Zhang, P. Daian, I. Bentov, and A. Bitcoin Magazine article here. Cheng, F. Zhang, J. Kos, W. He, N. Hynes, N. Johnson, A. Juels, A. Miller, and D. Note: Used in Oasis Labs products. Matetic, M. Schneider, A. Miller, A, Juels, and S. Tramer, and A. Visit the Hydra Project website for code.
Video presentation here. Cecchetti, F. Ji, A. Kosba, A. Juels, and E. Ahmed, K. Kostiainen, A. Dhar, D. Sommer, A. Gervais, A. Juels, and S. Zhang, I. Eyal, R. Escriva, A. Juels, and R. Aikat, A. Akella, J. Chase, A. Juels, M. Reiter, T. Ristenpart, V. Sekar, M. Rethinking Security in the Era of Cloud Computing.
Woodage, R. Chatterjee, Y. Dodis, A. Juels, and T. Sekar, and M. Zhang, H. Lin, J. Atlidakis, R. Geambasu, D. Hubaux, M. Humbert, A. Juels, and H. Eyal, A. Juels, and G. Zhang, E. Cecchetti, K. Croman, A. Note: Licensed by Chainlink. Slides here. Kosba, and E. Reiter, and T. Marino and A. Rule ML, pp. Hubaux and A. Privacy is dead, long live privacy: Protecting social norms as confidentiality wanes. Chatterjee, A. Athayle, D. Akawhe, A. Croman, C.
Decker, I. Gencer, A. Miller, P. Saxena, E. Shi, E. Sirer, D. Song, and R. On Scaling Decentralized Blockchains. Estrin and A. Reassembling Our Digital Selves. Everspaugh, R. Chatterjee, S. Scott, A. Huang, E. Ayday, J. Hubaux, J. Chatterjee, J. Dodis, C. Ganesh, A. Golovnev, A. Juels and T. Yen, V. Heorhiadi, A. Oprea, M. Reiter, and A. Bowers, C. Hart, A. Juels, and N. RAID, pp.
A bodyguard of lies: the use of honey objects in information security. Miller, A. Juels, E. Shi, B. Parno, and J. Juels and B. The Interplay of Neuroscience and Cryptography. Yen, A. Oprea, K. Onarlioglu, T. Leetham, A. Kirda, and W. Juels and R. Honeywords: Making Password-Cracking Detectable. Rostami, A.
Juels, and F. Oprea, and R. Rostami, W. Burleson, A. Balancing Security and Utility in Medical Devices. Bowers, A. Juels, R. Rivest, and E. Juels and A. Zanetti, S. Capkun, and A. Stefanov, M. Oprea, and A. Oprea, R. Rivest, E.
Stefanov, and N. Farley, V. Varajarajan, K. Bowers, M. Griffin, A. Rivest, and N. GameSec, pp. Denning, K. Fu, and C. Paar, Eds. Oprea and A. Future tense: The Primal Cue. Juels, D. Catalano, and M. Coercion-Resistant Electronic Elections.
Towards Trustworthy Elections , pp. Workshop version appeared as [JCJ05]. Abstract of keynote talk. Koscher, A. Juels, T. Kohno, and V. FAQ available here. Juels and S. Salajegheh, S. What is a Private Key? Public Key Cryptography PKC is a way to validate the authenticity of something using asymmetric encryption. PKC was first used primarily to encrypt and decrypt messages.
Cryptocurrencies now use this technology to encrypt and decrypt transactions. While it may be possible, it would likely take a supercomputer — and thousands of years — to reverse engineer these functions. A public key allows you to receive cryptocurrency transactions. The public key that can receive transactions is usually an address, which is simply a shortened form of your public key. Therefore, you can freely share your public key without worry. You may have seen donation pages for content-creators or charities with the public keys for their crypto addresses online.
Before we start, there is one crucial piece of advice to remember: NEVER share your private key with anyone. A private key gives you the ability to prove ownership or spend the funds associated with your public address. A private key can take many forms:.
QR code. Mnemonic phrase. You can have any number of public keys connected to a private key. To complete a transaction on the blockchain , it needs to be signed. The steps for someone to send you a transaction are:. A transaction is encrypted using a public key.
The transaction can only be decrypted by the accompanying private key. The digital signature is generated through combining the private key with the data being sent in the transaction. Finally, the transaction can be verified as authentic using the accompanying public key.
Nodes check and authenticate transactions automatically. Any unauthenticated transactions get rejected by the network and an authentic, mined transaction on the blockchain is irreversible. Your private keys are in a cryptocurrency wallet , which is typically mobile or desktop software or a specialized hardware device. Your private keys are not on the cryptocurrency blockchain network. If you transfer your cryptocurrency from an exchange to a non-custodial wallet , you are in control of your keys.
How public and private keys work together is fundamental to understanding how cryptocurrency transactions function. If you hold your own private keys, consider modern HD wallets, which can do a great job of managing your private keys, and remember to never share them.
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