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Introduction to Cryptography
Cryptography
Cryptography or cryptology (from
https://en.wikipedia.org/wiki/Ancient_Greek_language
Ancient Greek: “hidden, secret”; and “to write”, or “study”, respectively [1] ) is the practice and study of techniques for
https://en.wikipedia.org/wiki/Cryptography#cite_note-1
https://en.wikipedia.org/wiki/Secure_communication
secure communicationin the presence of third parties called
https://en.wikipedia.org/wiki/Adversary_(cryptography)
adversaries. [2] More generally, cryptography is about constructing and analyzing
https://en.wikipedia.org/wiki/Cryptography#cite_note-rivest90-2
https://en.wikipedia.org/wiki/Communications_protocol
protocolsthat prevent third parties or the public from reading private messages; [3] various aspects in
https://en.wikipedia.org/wiki/Cryptography#cite_note-modern-crypto-3
https://en.wikipedia.org/wiki/Information_security
information securitysuch as data
https://en.wikipedia.org/wiki/Confidentiality
confidentiality,
https://en.wikipedia.org/wiki/Data_integrity
data integrity,
https://en.wikipedia.org/wiki/Authentication
authentication, and
https://en.wikipedia.org/wiki/Non-repudiation
non-repudiation [4] are central to modern cryptography.
https://en.wikipedia.org/wiki/Cryptography#cite_note-hac-4
Modern cryptography exists at the intersection of the disciplines of
https://en.wikipedia.org/wiki/Mathematics
mathematics,
https://en.wikipedia.org/wiki/Computer_science
computer science,
https://en.wikipedia.org/wiki/Electrical_engineering
electrical engineering,
https://en.wikipedia.org/wiki/Communication_science
communication science, and
https://en.wikipedia.org/wiki/Physics
physics.
Applications of cryptography include
https://en.wikipedia.org/wiki/Electronic_commerce
electronic commerce,
https://en.wikipedia.org/wiki/Credit_card_chip
chip-based payment cards,
https://en.wikipedia.org/wiki/Digital_currencies
digital currencies,
https://en.wikipedia.org/wiki/Password
computer passwords, and
https://en.wikipedia.org/wiki/Military_communications
military communications.
Cryptography prior to the modern age was effectively synonymous with
https://en.wikipedia.org/wiki/Encryption
encryption , the conversion of information from a readable state to apparent
https://en.wikipedia.org/wiki/Nonsense
nonsense.
The originator of an encrypted message shares the decoding technique only with intended recipients to preclude access from adversaries.
The cryptography literature
https://en.wikipedia.org/wiki/Alice_and_Bob
often uses the namesAlice (“A”) for the sender, Bob (“B”) for the intended recipient, and Eve (“eavesdropper“) for the adversary. [5] Since the development of
https://en.wikipedia.org/wiki/Eavesdropper
https://en.wikipedia.org/wiki/Cryptography#cite_note-codesintro-5
https://en.wikipedia.org/wiki/Rotor_machine
rotor cipher machinesin
https://en.wikipedia.org/wiki/World_War_I
World WarIand the advent of
https://en.wikipedia.org/wiki/Computer
computersin
https://en.wikipedia.org/wiki/World_War_II
World WarII, the methods used to carry out cryptology have become increasingly complex and its application more widespread.
Modern cryptography is heavily based on mathematical theory and computer science practice; cryptographic
https://en.wikipedia.org/wiki/Algorithm
algorithmsare designed around
https://en.wikipedia.org/wiki/Computational_hardness_assumption
computational hardness assumptions, making such algorithms hard to break in practice by any adversary.
It is theoretically possible to break such a system, but it is infeasible to do so by any known practical means.
These schemes are therefore termed computationally secure; theoretical advances, e.g., improvements in
https://en.wikipedia.org/wiki/Integer_factorization
integer factorizationalgorithms, and faster computing technology require these solutions to be continually adapted.
There exist
https://en.wikipedia.org/wiki/Information_theoretic_security
information-theoretically secureschemes that provably cannot be broken even with unlimited computing power—an example is the
https://en.wikipedia.org/wiki/One-time_pad
one-time pad—but these schemes are more difficult to use in practice than the best theoretically breakable but computationally secure mechanisms.
The growth of cryptographic technology has raised a number of legal issues in the information age.
Cryptography’s potential for use as a tool for
https://en.wikipedia.org/wiki/Espionage
espionageand
https://en.wikipedia.org/wiki/Sedition
seditionhas led many governments to classify it as a weapon and to limit or even prohibit its use and export. [6] In some jurisdictions where the use of cryptography is legal, laws permit investigators to
https://en.wikipedia.org/wiki/Cryptography#cite_note-cryptolaw-6
https://en.wikipedia.org/wiki/Key_disclosure_law
compel the disclosureof encryption keys for documents relevant to an investigation. [7] [8] Cryptography also plays a major role in
https://en.wikipedia.org/wiki/Cryptography#cite_note-UK_law-7
https://en.wikipedia.org/wiki/Cryptography#cite_note-RangerSteve1-8
https://en.wikipedia.org/wiki/Digital_rights_management
digital rights managementand
https://en.wikipedia.org/wiki/Copyright_infringement
copyright infringementof digital media. [9]
https://en.wikipedia.org/wiki/Cryptography#cite_note-AACS-9
Terminology
The first use of the termcryptograph(as opposed tocryptogram) dates back to the 19th century—originating from The Gold-Bug , a novel byEdgar Allan Poe.
https://en.wikipedia.org/wiki/The_Gold-Bug
https://en.wikipedia.org/wiki/Edgar_Allan_Poe
https://en.wikipedia.org/wiki/Cryptography#cite_note-10
10
Until modern times, cryptography referred almost exclusively to
encryption, which is the process of converting ordinary information (called
https://en.wikipedia.org/wiki/Plaintext
plaintext) into unintelligible form (called
https://en.wikipedia.org/wiki/Ciphertext
ciphertext).
https://en.wikipedia.org/wiki/Cryptography#cite_note-kahnbook-11
11
Decryption is the reverse, in other words, moving from the unintelligible ciphertext back to plaintext.
A
https://en.wikipedia.org/wiki/Cipher
cipher (or
cypher) is a pair of algorithms that create the encryption and the reversing decryption.
The detailed operation of a cipher is controlled both by the algorithm and in each instance by a “key“.
https://en.wikipedia.org/wiki/Key_(cryptography)
The key is a secret (ideally known only to the communicants), usually a short string of characters, which is needed to decrypt the ciphertext.
Formally, a “cryptosystem” is the ordered list of elements of finite possible plaintexts, finite possible cyphertexts, finite possible keys, and the encryption and decryption algorithms which correspond to each key.
https://en.wikipedia.org/wiki/Cryptosystem
Keys are important both formally and in actual practice, as ciphers without variable keys can be trivially broken with only the knowledge of the cipher used and are therefore useless (or even counter-productive) for most purposes.
Historically, ciphers were often used directly for encryption or decryption without additional procedures such as
https://en.wikipedia.org/wiki/Authentication
authenticationor integrity checks.
There are two kinds of cryptosystems:
https://en.wikipedia.org/wiki/Symmetric-key_algorithm
symmetricand
https://en.wikipedia.org/wiki/Public-key_cryptography
asymmetric.
In symmetric systems the same key (the secret key) is used to encrypt and decrypt a message.
Data manipulation in symmetric systems is faster than asymmetric systems as they generally use shorter key lengths.
Asymmetric systems use a public key to encrypt a message and a private key to decrypt it.
Use of asymmetric systems enhances the security of communication.
https://en.wikipedia.org/wiki/Cryptography#cite_note-12
12
Examples of asymmetric systems include RSA (Rivest-Shamir-Adleman), and ECC (Elliptic Curve Cryptography).
https://en.wikipedia.org/wiki/Rivest-Shamir-Adleman
https://en.wikipedia.org/wiki/Elliptic_Curve_Cryptography
Symmetric models include the commonly used AES (Advanced Encryption Standard) which replaced the older DES (Data Encryption Standard).
https://en.wikipedia.org/wiki/Advanced_Encryption_Standard
https://en.wikipedia.org/wiki/Data_Encryption_Standard
https://en.wikipedia.org/wiki/Cryptography#cite_note-13
13
In
https://en.wikipedia.org/wiki/Colloquial
colloquialuse, the term “code” is often used to mean any method of encryption or concealment of meaning.
https://en.wikipedia.org/wiki/Code_(cryptography)
However, in cryptography,
codehas a more specific meaning.
It means the replacement of a unit of plaintext (i.e., a meaningful word or phrase) with a
https://en.wikipedia.org/wiki/Code_word
code word(for example, “wallaby” replaces “attack at dawn”).
https://en.wikipedia.org/wiki/Cryptanalysis
Cryptanalysisis the term used for the study of methods for obtaining the meaning of encrypted information without access to the key normally required to do so; i.e., it is the study of how to crack encryption algorithms or their implementations.
Some use the terms
cryptographyand
cryptologyinterchangeably in English, while others (including US military practice generally) use
cryptographyto refer specifically to the use and practice of cryptographic techniques and
cryptologyto refer to the combined study of cryptography and cryptanalysis.
https://en.wikipedia.org/wiki/Cryptography#cite_note-goldreichbook-14
14
https://en.wikipedia.org/wiki/Cryptography#cite_note-websters-15
15
English is more flexible than several other languages in which
cryptology(done by cryptologists) is always used in the second sense above.
https://en.wikipedia.org/wiki/Request_for_Comments_(identifier)
RFC2828advises thatsteganographyis sometimes included in cryptology.
https://tools.ietf.org/html/rfc2828
https://en.wikipedia.org/wiki/Steganography
https://en.wikipedia.org/wiki/Cryptography#cite_note-16
16
The study of characteristics of languages that have some application in cryptography or cryptology (e.g. frequency data, letter combinations, universal patterns, etc.) is called cryptolinguistics.
History of Cryptography
Before the modern era, cryptography focused on message confidentiality (i.e., encryption)—conversion of
https://en.wikipedia.org/wiki/Information
messagesfrom a comprehensible form into an incomprehensible one and back again at the other end, rendering it unreadable by interceptors or eavesdroppers without secret knowledge (namely the key needed for decryption of that message).
Encryption attempted to ensure
https://en.wikipedia.org/wiki/Secrecy
secrecyin
https://en.wikipedia.org/wiki/Communications
communications, such as those of
https://en.wikipedia.org/wiki/Spy
spies, military leaders, and
https://en.wikipedia.org/wiki/Diplomat
diplomats.
In recent decades, the field has expanded beyond confidentiality concerns to include techniques for message integrity checking, sender/receiver identity
https://en.wikipedia.org/wiki/Authentication
authentication,
https://en.wikipedia.org/wiki/Digital_signature
digital signatures,
https://en.wikipedia.org/wiki/Interactive_proof_system
interactive proofsand
https://en.wikipedia.org/wiki/Secure_multiparty_computation
secure computation, among others.
Computer era
Prior to the early 20th century, cryptography was mainly concerned with
https://en.wikipedia.org/wiki/Language
linguisticand
https://en.wikipedia.org/wiki/Lexicographic_code
lexicographicpatterns.
Since then the emphasis has shifted, and cryptography now makes extensive use of mathematics, including aspects of
https://en.wikipedia.org/wiki/Information_theory
information theory,
https://en.wikipedia.org/wiki/Computational_complexity_theory
computational complexity,
https://en.wikipedia.org/wiki/Statistics
statistics,
https://en.wikipedia.org/wiki/Combinatorics
combinatorics,
https://en.wikipedia.org/wiki/Abstract_algebra
abstract algebra,
https://en.wikipedia.org/wiki/Number_theory
number theory, and finite mathematics generally.
Cryptography is also a branch of
https://en.wikipedia.org/wiki/Engineering
engineering, but an unusual one since it deals with active, intelligent, and malevolent opposition (see cryptographic engineering and
https://en.wikipedia.org/wiki/Security_engineering
security engineering); other kinds of engineering (e.g., civil or chemical engineering) need deal only with neutral natural forces.
There is also active research examining the relationship between cryptographic problems and
https://en.wikipedia.org/wiki/Quantum_physics
quantum physics(see
https://en.wikipedia.org/wiki/Quantum_cryptography
quantum cryptographyand
https://en.wikipedia.org/wiki/Quantum_computer
quantum computer).
Just as the development of digital computers and electronics helped in cryptanalysis, it made possible much more complex ciphers.
Furthermore, computers allowed for the encryption of any kind of data representable in any binary format, unlike classical ciphers which only encrypted written language texts; this was new and significant.
Computer use has thus supplanted linguistic cryptography, both for cipher design and cryptanalysis.
Many computer ciphers can be characterized by their operation on
https://en.wikipedia.org/wiki/Binary_numeral_system
binary
https://en.wikipedia.org/wiki/Bit
bitsequences (sometimes in groups or blocks), unlike classical and mechanical schemes, which generally manipulate traditional characters (i.e., letters and digits) directly.
However, computers have also assisted cryptanalysis, which has compensated to some extent for increased cipher complexity.
Nonetheless, good modern ciphers have stayed ahead of cryptanalysis; it is typically the case that use of a quality cipher is very efficient (i.e., fast and requiring few resources, such as memory or CPU capability), while breaking it requires an effort many orders of magnitude larger, and vastly larger than that required for any classical cipher, making cryptanalysis so inefficient and impractical as to be effectively impossible.
Advent of modern cryptography
https://en.wikipedia.org/wiki/Cryptanalysis
Cryptanalysisof the new mechanical devices proved to be both difficult and laborious.
In the United Kingdom, cryptanalytic efforts at
https://en.wikipedia.org/wiki/Bletchley_Park
Bletchley Parkduring WWII spurred the development of more efficient means for carrying out repetitious tasks.
This culminated in the development of the
https://en.wikipedia.org/wiki/Colossus_computer
Colossus, the world’s first fully electronic, digital,
https://en.wikipedia.org/wiki/Computer_programming
programmablecomputer, which assisted in the decryption of ciphers generated by the German Army’s
https://en.wikipedia.org/wiki/Lorenz_SZ40/42
Lorenz SZ40/42machine.
Extensive open academic research into cryptography is relatively recent; it began only in the mid-1970’s.
In recent times, IBM personnel designed the algorithm that became the Federal (i.e., US)
https://en.wikipedia.org/wiki/Data_Encryption_Standard
Data Encryption Standard;
https://en.wikipedia.org/wiki/Whitfield_Diffie
Whitfield Diffieand
https://en.wikipedia.org/wiki/Martin_Hellman
Martin Hellmanpublished
https://en.wikipedia.org/wiki/Diffie-Hellman
their key agreement algorithm;
https://en.wikipedia.org/wiki/Cryptography#cite_note-dh2-30
30
and the
https://en.wikipedia.org/wiki/RSA_(algorithm)
RSAalgorithm was published in
https://en.wikipedia.org/wiki/Martin_Gardner
Martin Gardner‘s
https://en.wikipedia.org/wiki/Scientific_American
Scientific American column.
Following their work in 1976, it became popular to consider cryptography systems based on mathematical problems that are easy to state but have been found difficult to solve.
https://en.wikipedia.org/wiki/Cryptography#cite_note-31
31
Since then, cryptography has become a widely used tool in communications,
https://en.wikipedia.org/wiki/Computer_network
computer networks, and
https://en.wikipedia.org/wiki/Computer_security
computer securitygenerally.
Some modern cryptographic techniques can only keep their keys secret if certain mathematical problems are
https://en.wikipedia.org/wiki/Computational_complexity_theory#Intractability
intractable, such as the
https://en.wikipedia.org/wiki/Integer_factorization
integer factorizationor the
https://en.wikipedia.org/wiki/Discrete_logarithm
discrete logarithmproblems, so there are deep connections with
https://en.wikipedia.org/wiki/Abstract_mathematics
abstract mathematics.
There are very few cryptosystems that are proven to be unconditionally secure.
The
https://en.wikipedia.org/wiki/One-time_pad
one-time padis one, and was proven to be so by Claude Shannon.
There are a few important algorithms that have been proven secure under certain assumptions.
For example, the infeasibility of factoring extremely large integers is the basis for believing that
https://en.wikipedia.org/wiki/RSA_(cryptosystem)
RSAis secure, and some other systems, but even so proof of unbreakability is unavailable since the underlying mathematical problem remains open.
In practice, these are widely used, and are believed unbreakable in practice by most competent observers.
There are systems similar to RSA, such as one by
https://en.wikipedia.org/wiki/Michael_O._Rabin
Michael O. Rabinthat are provably secure provided factoring
n = pqis impossible; it is quite unusable in practice.
The
https://en.wikipedia.org/wiki/Discrete_logarithm_problem
discrete logarithm problemis the basis for believing some other cryptosystems are secure, and again, there are related, less practical systems that are provably secure relative to the solvability or insolvability discrete log problem.
https://en.wikipedia.org/wiki/Cryptography#cite_note-32
32
As well as being aware of cryptographic history, cryptographic algorithm and system designers must also sensibly consider probable future developments while working on their designs.
For instance, continuous improvements in computer processing power have increased the scope of
https://en.wikipedia.org/wiki/Brute-force_attack
brute-force attacks, so when specifying
https://en.wikipedia.org/wiki/Key_length
key lengths, the required key lengths are similarly advancing.
https://en.wikipedia.org/wiki/Cryptography#cite_note-fortify-33
33
The potential effects of
https://en.wikipedia.org/wiki/Quantum_computing
quantum computingare already being considered by some cryptographic system designers developing
https://en.wikipedia.org/wiki/Post-quantum_cryptography
post-quantum cryptography; the announced imminence of small implementations of these machines may be making the need for preemptive caution rather more than merely speculative. [4]
https://en.wikipedia.org/wiki/Cryptography#cite_note-hac-4
Symmetric-key cryptography
Main article:Symmetric-key algorithm
https://en.wikipedia.org/wiki/Symmetric-key_algorithm
https://en.wikipedia.org/wiki/File:Symmetric_key_encryption.svg
https://upload.wikimedia.org/wikipedia/commons/thumb/2/27/Symmetric_key_encryption.svg/250px-Symmetric_key_encryption.svg.png
diagram showing encrypt with a key and decrypt process
Symmetric-key cryptography, where a single key is used for encryption and decryption
Symmetric-key cryptography refers to encryption methods in which both the sender and receiver share the same key (or, less commonly, in which their keys are different, but related in an easily computable way).
This was the only kind of encryption publicly known until June 1976.
https://en.wikipedia.org/wiki/Cryptography#cite_note-dh2-30
30
https://en.wikipedia.org/wiki/File:International_Data_Encryption_Algorithm_InfoBox_Diagram.svg
https://upload.wikimedia.org/wikipedia/commons/thumb/a/af/International_Data_Encryption_Algorithm_InfoBox_Diagram.svg/220px-International_Data_Encryption_Algorithm_InfoBox_Diagram.svg.png
logic diagram showing International Data Encryption Algorithm cypher process
One round (out of 8.5) of theIDEAcipher, used in most versions ofPGPand OpenPGP compatible software for time-efficient encryption of messages
https://en.wikipedia.org/wiki/International_Data_Encryption_Algorithm
https://en.wikipedia.org/wiki/Pretty_Good_Privacy
Symmetric key ciphers are implemented as either
https://en.wikipedia.org/wiki/Block_ciphers
block ciphersor
https://en.wikipedia.org/wiki/Stream_ciphers
stream ciphers.
A block cipher enciphers input in blocks of plaintext as opposed to individual characters, the input form used by a stream cipher.
The
https://en.wikipedia.org/wiki/Data_Encryption_Standard
Data Encryption Standard(DES) and the
https://en.wikipedia.org/wiki/Advanced_Encryption_Standard
Advanced Encryption Standard(AES) are block cipher designs that have been designated
https://en.wikipedia.org/wiki/Cryptography_standards
cryptography standardsby the US government (though DES’s designation was finally withdrawn after the AES was adopted).
https://en.wikipedia.org/wiki/Cryptography#cite_note-aes-34
34
Despite its deprecation as an official standard, DES (especially its still-approved and much more secure
https://en.wikipedia.org/wiki/Triple-DES
triple-DESvariant) remains quite popular; it is used across a wide range of applications, from ATM encryption
https://en.wikipedia.org/wiki/Cryptography#cite_note-atm-35
35
to
https://en.wikipedia.org/wiki/E-mail_privacy
e-mail privacy
https://en.wikipedia.org/wiki/Cryptography#cite_note-opgp-36
36
and
https://en.wikipedia.org/wiki/Secure_Shell
secure remote access.
https://en.wikipedia.org/wiki/Cryptography#cite_note-ssh-37
37
Many other block ciphers have been designed and released, with considerable variation in quality.
Many, even some designed by capable practitioners, have been thoroughly broken, such as
https://en.wikipedia.org/wiki/FEAL
FEAL. [4]
https://en.wikipedia.org/wiki/Cryptography#cite_note-hac-4
https://en.wikipedia.org/wiki/Cryptography#cite_note-schneierbook-38
38
Stream ciphers, in contrast to the ‘block’ type, create an arbitrarily long stream of key material, which is combined with the plaintext bit-by-bit or character-by-character, somewhat like the
https://en.wikipedia.org/wiki/One-time_pad
one-time pad.
In a stream cipher, the output stream is created based on a hidden internal state that changes as the cipher operates.
That internal state is initially set up using the secret key material.
https://en.wikipedia.org/wiki/RC4
RC4is a widely used stream cipher; see
https://en.wikipedia.org/wiki/Category:Stream_ciphers
Category:Stream ciphers. [4] Block ciphers can be used as stream ciphers; see
https://en.wikipedia.org/wiki/Cryptography#cite_note-hac-4
https://en.wikipedia.org/wiki/Block_cipher_modes_of_operation
Block cipher modes of operation.
https://en.wikipedia.org/wiki/Cryptographic_hash_functions
Cryptographic hash functionsare a third type of cryptographic algorithm.
They take a message of any length as input, and output a short, fixed length
https://en.wikipedia.org/wiki/Hash_function
hash, which can be used in (for example) a digital signature.
For good hash functions, an attacker cannot find two messages that produce the same hash.
https://en.wikipedia.org/wiki/MD4
MD4is a long-used hash function that is now broken;
https://en.wikipedia.org/wiki/MD5
MD5, a strengthened variant of MD4, is also widely used but broken in practice.
The US
https://en.wikipedia.org/wiki/National_Security_Agency
National Security Agencydeveloped the Secure Hash Algorithm series of MD5-like hash functions: SHA-0 was a flawed algorithm that the agency withdrew;
https://en.wikipedia.org/wiki/SHA-1
SHA-1is widely deployed and more secure than MD5, but cryptanalysts have identified attacks against it; the
https://en.wikipedia.org/wiki/SHA-2
SHA-2family improves on SHA-1, but is vulnerable to clashes as of 2011; and the US standards authority thought it “prudent” from a security perspective to develop a new standard to “significantly improve the robustness of
https://en.wikipedia.org/wiki/National_Institute_of_Standards_and_Technology
NIST‘s overall hash algorithm toolkit.”
https://en.wikipedia.org/wiki/Cryptography#cite_note-39
39
Thus, a
https://en.wikipedia.org/wiki/NIST_hash_function_competition
hash function design competitionwas meant to select a new U.S. national standard, to be called
https://en.wikipedia.org/wiki/SHA-3
SHA-3, by 2012.
The competition ended on October 2, 2012 when the NIST announced that
https://en.wikipedia.org/wiki/Keccak
Keccakwould be the new SHA-3 hash algorithm.
https://en.wikipedia.org/wiki/Cryptography#cite_note-40
40
Unlike block and stream ciphers that are invertible, cryptographic hash functions produce a hashed output that cannot be used to retrieve the original input data.
Cryptographic hash functions are used to verify the authenticity of data retrieved from an untrusted source or to add a layer of security.
https://en.wikipedia.org/wiki/Message_authentication_code
Message authentication codes(MACs) are much like cryptographic hash functions, except that a secret key can be used to authenticate the hash value upon receipt; [4] this additional complication blocks an attack scheme against baredigest algorithms, and so has been thought worth the effort.
https://en.wikipedia.org/wiki/Cryptography#cite_note-hac-4
https://en.wikipedia.org/wiki/Md5
Modern cryptography
The modern field of cryptography can be divided into several areas of study.
The chief ones are discussed here; see
https://en.wikipedia.org/wiki/Topics_in_Cryptography
Topics in Cryptographyfor more.
Public-key cryptography
https://en.wikipedia.org/wiki/File:Public_key_encryption.svg
https://upload.wikimedia.org/wikipedia/commons/thumb/f/f9/Public_key_encryption.svg/250px-Public_key_encryption.svg.png
diagram of Public-key cryptography showing public key and private key
Public-key cryptography, where different keys are used for encryption and decryption.
Symmetric-key cryptosystems use the same key for encryption and decryption of a message, although a message or group of messages can have a different key than others.
A significant disadvantage of symmetric ciphers is the
https://en.wikipedia.org/wiki/Key_management
key managementnecessary to use them securely.
Each distinct pair of communicating parties must, ideally, share a different key, and perhaps for each ciphertext exchanged as well.
The number of keys required increases as the
https://en.wikipedia.org/wiki/Square_(algebra)
squareof the number of network members, which very quickly requires complex key management schemes to keep them all consistent and secret.
https://en.wikipedia.org/wiki/File:Diffie_and_Hellman.jpg
https://upload.wikimedia.org/wikipedia/commons/thumb/8/88/Diffie_and_Hellman.jpg/220px-Diffie_and_Hellman.jpg
headshots of Whitfield Diffie and Martin Hellman