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In cryptography, a **three-pass protocol** for sending messages is a framework which allows one party to securely send a message to a second party without the need to exchange or distribute encryption keys. Such message protocols should not be confused with various other algorithms which use 3 passes for authentication.

**Cryptography** or **cryptology** is the practice and study of techniques for secure communication in the presence of third parties called adversaries. More generally, cryptography is about constructing and analyzing protocols that prevent third parties or the public from reading private messages; various aspects in information security such as data confidentiality, data integrity, authentication, and non-repudiation are central to modern cryptography. Modern cryptography exists at the intersection of the disciplines of mathematics, computer science, electrical engineering, communication science, and physics. Applications of cryptography include electronic commerce, chip-based payment cards, digital currencies, computer passwords, and military communications.

In cryptography, a **key** is a piece of information that determines the functional output of a cryptographic algorithm. For encryption algorithms, a key specifies the transformation of plaintext into ciphertext, and vice versa for decryption algorithms. Keys also specify transformations in other cryptographic algorithms, such as digital signature schemes and message authentication codes.

**Authentication** is the act of proving an assertion, such as the identity of a computer system user. In contrast with identification, the act of indicating a person or thing's identity, authentication is the process of verifying that identity. It might involve validating personal identity documents, verifying the authenticity of a website with a digital certificate, determining the age of an artifact by carbon dating, or ensuring that a product or document is not counterfeit.

It is called a *three-pass protocol* because the sender and the receiver exchange three encrypted messages. The first three-pass protocol was developed by Adi Shamir circa 1980, and is described in more detail in a later section. The basic concept of the three-pass protocol is that each party has a private encryption key and a private decryption key. The two parties use their keys independently, first to encrypt the message, and then to decrypt the message.

**Adi Shamir** is an Israeli cryptographer. He is a co-inventor of the Rivest–Shamir–Adleman (RSA) algorithm, a co-inventor of the Feige–Fiat–Shamir identification scheme, one of the inventors of differential cryptanalysis and has made numerous contributions to the fields of cryptography and computer science.

The protocol uses an encryption function *E* and a decryption function *D*. The encryption function uses an encryption key *e* to change a plaintext message *m* into an encrypted message, or ciphertext, *E(e,m)*. Corresponding to each encryption key *e* there is a decryption key *d* which allows the message to be recovered using the decryption function, *D(d,E(e,m))=m*. Sometimes the encryption function and decryption function are the same.

In cryptography, **encryption** is the process of encoding a message or information in such a way that only authorized parties can access it and those who are not authorized cannot. Encryption does not itself prevent interference, but denies the intelligible content to a would-be interceptor. In an encryption scheme, the intended information or message, referred to as plaintext, is encrypted using an encryption algorithm – a cipher – generating ciphertext that can be read only if decrypted. For technical reasons, an encryption scheme usually uses a pseudo-random encryption key generated by an algorithm. It is in principle possible to decrypt the message without possessing the key, but, for a well-designed encryption scheme, considerable computational resources and skills are required. An authorized recipient can easily decrypt the message with the key provided by the originator to recipients but not to unauthorized users.

In cryptography, **plaintext** usually means unencrypted information pending input into cryptographic algorithms, usually encryption algorithms. *Cleartext* usually refers to data that is transmitted or stored unencrypted.

In cryptography, **ciphertext** or **cyphertext** is the result of encryption performed on plaintext using an algorithm, called a cipher. Ciphertext is also known as encrypted or encoded information because it contains a form of the original plaintext that is unreadable by a human or computer without the proper cipher to decrypt it. Decryption, the inverse of encryption, is the process of turning ciphertext into readable plaintext. Ciphertext is not to be confused with codetext because the latter is a result of a code, not a cipher.

In order for the encryption function and decryption function to be suitable for the *three-pass protocol* they must have the property that for any message *m*, any encryption key *e* with corresponding decryption key *d* and any independent encryption key *k*, *D(d,E(k,E(e,m))) = E(k,m)*. In other words, it must be possible to remove the first encryption with the key *e* even though a second encryption with the key *k* has been performed. This will always be possible with a commutative encryption. A commutative encryption is an encryption that is order-independent, i.e. it satisfies *E(a,E(b,m))=E(b,E(a,m))* for all encryption keys *a* and *b* and all messages *m*. Commutative encryptions satisfy *D(d,E(k,E(e,m))) = D(d,E(e,E(k,m))) = E(k,m)*.

The three-pass protocol works as follows:

- The sender chooses a private encryption key
*s*and a corresponding decryption key*t*. The sender encrypts the message*m*with the key*s*and sends the encrypted message*E(s,m)*to the receiver. - The receiver chooses a private encryption key
*r*and a corresponding decryption key*q*and super-encrypts the first message*E(s,m)*with the key*r*and sends the doubly encrypted message*E(r,E(s,m))*back to the sender. - The sender decrypts the second message with the key
*t*. Because of the commutativity property described above*D(t,E(r,E(s,m)))=E(r,m)*which is the message encrypted with only the receiver's private key. The sender sends this to the receiver.

The receiver can now decrypt the message using the key *q*, namely *D(q,E(r,m))=m* the original message.

Notice that all of the operations involving the sender's private keys *s* and *t* are performed by the sender, and all of the operations involving the receiver's private keys *r* and *q* are performed by the receiver, so that neither party needs to know the other party's keys.

The first three-pass protocol was the Shamir three-pass protocol developed circa in 1980. It is also called the *Shamir No-Key Protocol* because the sender and the receiver do not exchange any keys, however the protocol requires the sender and receiver to have two private keys for encrypting and decrypting messages. The Shamir algorithm uses exponentiation modulo a large prime as both the encryption and decryption functions. That is *E*(*e*,*m*) = *m*^{e} mod *p* and *D*(*d*,*m*) = *m*^{d} mod *p* where *p* is a large prime. For any encryption exponent *e* in the range 1..*p*-1 with gcd(*e*,*p*-1) = 1. The corresponding decryption exponent *d* is chosen such that *de* ≡ 1 (mod *p*-1). It follows from Fermat's Little Theorem that *D*(*d*,*E*(*e*,*m*)) = *m*^{de} mod *p* = *m*.

**Exponentiation** is a mathematical operation, written as *b*^{n}, involving two numbers, the *base*b and the *exponent* or *power*n. When n is a positive integer, exponentiation corresponds to repeated multiplication of the base: that is, *b*^{n} is the product of multiplying n bases:

A **prime number** is a natural number greater than 1 that cannot be formed by multiplying two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, 1 × 5 or 5 × 1, involve 5 itself. However, 6 is composite because it is the product of two numbers that are both smaller than 6. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order.

The Shamir protocol has the desired commutativity property since *E*(*a*,*E*(*b*,*m*)) = *m*^{ab} mod *p* = *m*^{ba} mod *p* = *E*(*b*,*E*(*a*,*m*)).

The Massey–Omura Cryptosystem was proposed by James Massey and Jim K. Omura in 1982 as a possible improvement over the Shamir protocol. The Massey–Omura method uses exponentiation in the Galois field GF(2^{n}) as both the encryption and decryption functions. That is *E(e,m)=m ^{e}* and

**James Lee Massey** was an information theorist and cryptographer, Professor Emeritus of Digital Technology at ETH Zurich. His notable work includes the application of the Berlekamp–Massey algorithm to linear codes, the design of the block ciphers IDEA and SAFER, and the Massey-Omura cryptosystem.

**Jimmy K. Omura** is an electrical engineer and information theorist.

In mathematics, a **finite field** or **Galois field** is a field that contains a finite number of elements. As with any field, a finite field is a set on which the operations of multiplication, addition, subtraction and division are defined and satisfy certain basic rules. The most common examples of finite fields are given by the integers mod *p* when *p* is a prime number.

Each element of the Galois field GF(2^{n}) is represented as a binary vector over a normal basis in which each basis vector is the square of the preceding one. That is, the basis vectors are *v*^{1}, *v*^{2}, *v*^{4}, *v*^{8}, ... where *v* is a field element of maximum order. By using this representation, exponentiations by powers of 2 can be accomplished by cyclic shifts. This means that raising *m* to an arbitrary power can be accomplished with at most *n* shifts and *n* multiplications. Moreover, several multiplications can be performed in parallel. This allows faster hardware realizations at the cost of having to implement several multipliers.

In mathematics, specifically the algebraic theory of fields, a **normal basis** is a special kind of basis for Galois extensions of finite degree, characterised as forming a single orbit for the Galois group. The **normal basis theorem** states that any finite Galois extension of fields has a normal basis. In algebraic number theory, the study of the more refined question of the existence of a normal integral basis is part of Galois module theory.

In group theory, a branch of mathematics, the term *order* is used in three different senses:

In combinatorial mathematics, a **circular shift** is the operation of rearranging the entries in a tuple, either by moving the final entry to the first position, while shifting all other entries to the next position, or by performing the inverse operation. A circular shift is a special kind of cyclic permutation, which in turn is a special kind of permutation. Formally, a circular shift is a permutation σ of the *n* entries in the tuple such that either

A necessary condition for a three-pass algorithm to be secure is that an attacker cannot determine any information about the message *m* from the three transmitted messages *E(s,m)*, *E(r,E(s,m))* and *E(r,m)*.

For the encryption functions used in the Shamir algorithm and the Massey–Omura algorithm described above, the security relies on the difficulty of computing discrete logarithms in a finite field. If an attacker could compute discrete logarithms in *GF(p)* for the Shamir method or *GF(2 ^{n})* for the Massey–Omura method then the protocol could be broken. The key

The three-pass protocol as described above does not provide any authentication. Hence, without any additional authentication the protocol is susceptible to a man-in-the-middle attack if the opponent has the ability to create false messages, or to intercept and replace the genuine transmitted messages.

**Pretty Good Privacy** (**PGP**) is an encryption program that provides cryptographic privacy and authentication for data communication. PGP is used for signing, encrypting, and decrypting texts, e-mails, files, directories, and whole disk partitions and to increase the security of e-mail communications. Phil Zimmermann developed PGP in 1991.

**Public-key cryptography**, or **asymmetric cryptography**, is a cryptographic system that uses pairs of keys: *public keys* which may be disseminated widely, and *private keys* which are known only to the owner. The generation of such keys depends on cryptographic algorithms based on mathematical problems to produce one-way functions. Effective security only requires keeping the private key private; the public key can be openly distributed without compromising security.

**RSA** (**Rivest–Shamir–Adleman**) is one of the first public-key cryptosystems and is widely used for secure data transmission. In such a cryptosystem, the encryption key is public and it is different from the decryption key which is kept secret (private). In RSA, this asymmetry is based on the practical difficulty of the factorization of the product of two large prime numbers, the "factoring problem". The acronym RSA is made of the initial letters of the surnames of Ron Rivest, Adi Shamir, and Leonard Adleman, who first publicly described the algorithm in 1977. Clifford Cocks, an English mathematician working for the British intelligence agency Government Communications Headquarters (GCHQ), had developed an equivalent system in 1973, but this was not declassified until 1997.

The **Merkle–Hellman knapsack cryptosystem** was one of the earliest public key cryptosystems invented by Ralph Merkle and Martin Hellman in 1978. The ideas behind it are simpler than those involving RSA, and it has been broken.

In cryptography, a **block cipher mode of operation** is an algorithm that uses a block cipher to provide information security such as confidentiality or authenticity. A block cipher by itself is only suitable for the secure cryptographic transformation of one fixed-length group of bits called a block. A mode of operation describes how to repeatedly apply a cipher's single-block operation to securely transform amounts of data larger than a block.

**ID-based encryption**, or **identity-based encryption** (**IBE**), is an important primitive of ID-based cryptography. As such it is a type of public-key encryption in which the public key of a user is some unique information about the identity of the user. This means that a sender who has access to the public parameters of the system can encrypt a message using e.g. the text-value of the receiver's name or email address as a key. The receiver obtains its decryption key from a central authority, which needs to be trusted as it generates secret keys for every user.

The affine is a type of monoalphabetic substitution cipher, wherein each letter in an alphabet is mapped to its numeric equivalent, encrypted using a simple mathematical function, and converted back to a letter. The formula used means that each letter encrypts to one other letter, and back again, meaning the cipher is essentially a standard substitution cipher with a rule governing which letter goes to which. As such, it has the weaknesses of all substitution ciphers. Each letter is enciphered with the function (*ax* + *b*) mod 26, where b is the magnitude of the shift.

The **Rabin cryptosystem** is an asymmetric cryptographic technique, whose security, like that of RSA, is related to the difficulty of integer factorization. However the Rabin cryptosystem has the advantage that it has been mathematically proven to be computationally secure against a chosen-plaintext attack as long as the attacker cannot efficiently factor integers, while there is no such proof known for RSA. It has the disadvantage that each output of the Rabin function can be generated by any of four possible inputs; if each output is a ciphertext, extra complexity is required on decryption to identify which of the four possible inputs was the true plaintext.

The **Paillier crypto system**, invented by and named after Pascal Paillier in 1999, is a probabilistic asymmetric algorithm for public key cryptography. The problem of computing *n*-th residue classes is believed to be computationally difficult. The decisional composite residuosity assumption is the intractability hypothesis upon which this cryptosystem is based.

In cryptography, an **oblivious transfer** (**OT**) protocol is a type of protocol in which a sender transfers one of potentially many pieces of information to a receiver, but remains oblivious as to what piece has been transferred.

**Link encryption** is an approach to communications security that encrypts and decrypts all traffic at each network routing point until arrival at its final destination. This repeated decryption and encryption is necessary to allow the routing information contained in each transmission to be read and employed further to direct the transmission toward its destination, before which it is re-encrypted. This contrasts with **end-to-end encryption** where internal information, but not the header/routing information, are encrypted by the sender at the point of origin and only decrypted by the intended receiver.

**Ciphertext indistinguishability** is a property of many encryption schemes. Intuitively, if a cryptosystem possesses the property of indistinguishability, then an adversary will be unable to distinguish pairs of ciphertexts based on the message they encrypt. The property of indistinguishability under chosen plaintext attack is considered a basic requirement for most provably secure public key cryptosystems, though some schemes also provide indistinguishability under chosen ciphertext attack and adaptive chosen ciphertext attack. Indistinguishability under chosen plaintext attack is equivalent to the property of semantic security, and many cryptographic proofs use these definitions interchangeably.

**Mental poker** is the common name for a set of cryptographic problems that concerns playing a fair game over distance without the need for a trusted third party. The term is also applied to the theories surrounding these problems and their possible solutions. The name comes from the card game poker which is one of the games to which this kind of problem applies. Similar problems described as two party games are Blum's flipping a coin over a distance, Yao's Millionaires' Problem, and Rabin's Oblivious transfer.

**Authenticated encryption** (**AE**) and **authenticated encryption with associated data** (**AEAD**) are forms of encryption which simultaneously assure the confidentiality and authenticity of data. These attributes are provided under a single, easy to use programming interface.

A **threshold cryptosystem**, the basis for the field of **threshold cryptography**, is a cryptosystem that protects information by encrypting it and distributing it among a cluster of fault-tolerant computers. The message is encrypted using a public key, and the corresponding private key is shared among the participating parties. With a threshold cryptosystem, in order to decrypt an encrypted message or to sign a message, several parties must cooperate in the decryption or signature protocol.

In cryptography, a **hybrid cryptosystem** is one which combines the convenience of a public-key cryptosystem with the efficiency of a symmetric-key cryptosystem. Public-key cryptosystems are convenient in that they do not require the sender and receiver to share a common secret in order to communicate securely. However, they often rely on complicated mathematical computations and are thus generally much more inefficient than comparable symmetric-key cryptosystems. In many applications, the high cost of encrypting long messages in a public-key cryptosystem can be prohibitive. This is addressed by hybrid systems by using a combination of both.

The **Schmidt-Samoa cryptosystem** is an asymmetric cryptographic technique, whose security, like Rabin depends on the difficulty of integer factorization. Unlike Rabin this algorithm does not produce an ambiguity in the decryption at a cost of encryption speed.

The **Sakai–Kasahara scheme**, also known as the Sakai–Kasahara key encryption algorithm (**SAKKE**), is an identity-based encryption (IBE) system proposed by Ryuichi Sakai and Masao Kasahara in 2003. Alongside the Boneh–Franklin scheme, this is one of a small number of commercially implemented identity-based encryption schemes. It is an application of pairings over elliptic curves and finite fields. A security proof for the algorithm was produced in 2005 by Chen and Cheng. SAKKE is described in Internet Engineering Task Force (IETF) RFC 6508.

**Identity-based conditional proxy re-encryption** (**IBCPRE**) is a type of proxy re-encryption (PRE) scheme in the identity-based public key cryptographic setting. An IBCPRE scheme is a natural extension of proxy re-encryption on two aspects. The first aspect is to extend the proxy re-encryption notion to the identity-based public key cryptographic setting. The second aspect is to extend the feature set of proxy re-encryption to support conditional proxy re-encryption. By conditional proxy re-encryption, a proxy can use an IBCPRE scheme to re-encrypt a ciphertext but the ciphertext would only be well-formed for decryption if a condition applied onto the ciphertext together with the re-encryption key is satisfied. This allows fine-grained proxy re-encryption and can be useful for applications such as secure sharing over encrypted cloud data storage.

- U.S. Patent 4,567,600 , U.S. patent on the Massey–Omura cryptosystem
- Konheim, Alan G. (1981).
*Cryptography: A Primer*. pp. 346–347. - Menezes, A.; VanOorschot, P.; Vanstone, S. (1996).
*Handbook of Applied Cryptography*. pp. 500, 642. - Sakurai, K.; Shizuya, H. (1998). "A Structural Comparison of the Computational Difficulty of Breaking Discrete Log Cryptosystems".
*Journal of Cryptology*.**11**: 29–43. doi:10.1007/s001459900033.

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