RSA (Rivest–Shamir–Adleman) is a
public-key cryptosystem
Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic alg ...
that is widely used for secure data transmission. It is also one of the oldest. The
acronym
An acronym is a word or name formed from the initial components of a longer name or phrase. Acronyms are usually formed from the initial letters of words, as in ''NATO'' (''North Atlantic Treaty Organization''), but sometimes use syllables, as ...
"RSA" comes from the surnames of
Ron Rivest
Ronald Linn Rivest (; born May 6, 1947) is a cryptographer and an Institute Professor at MIT. He is a member of MIT's Department of Electrical Engineering and Computer Science (EECS) and a member of MIT's Computer Science and Artificial Intell ...
,
Adi Shamir
Adi Shamir ( he, עדי שמיר; born July 6, 1952) is an Israeli cryptographer. He is a co-inventor of the Rivest–Shamir–Adleman (RSA) algorithm (along with Ron Rivest and Len Adleman), a co-inventor of the Feige–Fiat–Shamir identificat ...
and
Leonard Adleman
Leonard Adleman (born December 31, 1945) is an American computer scientist. He is one of the creators of the RSA encryption algorithm, for which he received the 2002 Turing Award, often called the Nobel prize of Computer science. He is also kno ...
, who publicly described the algorithm in 1977. An equivalent system was developed secretly in 1973 at
Government Communications Headquarters
Government Communications Headquarters, commonly known as GCHQ, is an intelligence and security organisation responsible for providing signals intelligence (SIGINT) and information assurance (IA) to the government and armed forces of the Un ...
(GCHQ) (the British
signals intelligence
Signals intelligence (SIGINT) is intelligence-gathering by interception of ''signals'', whether communications between people (communications intelligence—abbreviated to COMINT) or from electronic signals not directly used in communication ( ...
agency) by the English mathematician
Clifford Cocks
Clifford Christopher Cocks (born 28 December 1950) is a British mathematician and cryptographer.
In 1973, while working at the United Kingdom Government Communications Headquarters (GCHQ), he invented a public-key cryptography algorithm equiv ...
. That system was
declassified
Declassification is the process of ceasing a protective classification, often under the principle of freedom of information. Procedures for declassification vary by country. Papers may be withheld without being classified as secret, and event ...
in 1997.
In a public-key
cryptosystem
In cryptography, a cryptosystem is a suite of cryptographic algorithms needed to implement a particular security service, such as confidentiality (encryption).
Typically, a cryptosystem consists of three algorithms: one for key generation, one for ...
, the
encryption key
A key in cryptography is a piece of information, usually a string of numbers or letters that are stored in a file, which, when processed through a cryptographic algorithm, can encode or decode cryptographic data. Based on the used method, the key c ...
is public and distinct from the
decryption key
In cryptography, encryption is the process of encoding information. This process converts the original representation of the information, known as plaintext, into an alternative form known as ciphertext. Ideally, only authorized parties can decip ...
, which is kept secret (private).
An RSA user creates and publishes a public key based on two large
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of 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 ...
s, along with an auxiliary value. The prime numbers are kept secret. Messages can be encrypted by anyone, via the public key, but can only be decoded by someone who knows the prime numbers.
The security of RSA relies on the practical difficulty of
factoring the product of two large
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of 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 ...
s, the "
factoring problem
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization.
When the numbers are suf ...
". Breaking RSA encryption is known as the
RSA problem
In cryptography, the RSA problem summarizes the task of performing an RSA private-key operation given only the public key. The RSA algorithm raises a ''message'' to an '' exponent'', modulo a composite number ''N'' whose factors are not known. T ...
. Whether it is as difficult as the factoring problem is an open question. There are no published methods to defeat the system if a large enough key is used.
RSA is a relatively slow algorithm. Because of this, it is not commonly used to directly encrypt user data. More often, RSA is used to transmit shared keys for
symmetric-key cryptography, which are then used for bulk encryption–decryption.
History
![Adi Shamir 2009 crop](https://upload.wikimedia.org/wikipedia/commons/0/06/Adi_Shamir_2009_crop.jpg)
The idea of an asymmetric public-private key cryptosystem is attributed to
Whitfield Diffie
Bailey Whitfield 'Whit' Diffie (born June 5, 1944), ForMemRS, is an American cryptographer and mathematician and one of the pioneers of public-key cryptography along with Martin Hellman and Ralph Merkle. Diffie and Hellman's 1976 paper ''New Dire ...
and
Martin Hellman
Martin Edward Hellman (born October 2, 1945) is an American cryptologist and mathematician, best known for his involvement with public key cryptography in cooperation with Whitfield Diffie and Ralph Merkle. Hellman is a longtime contributor to th ...
, who published this concept in 1976. They also introduced digital signatures and attempted to apply number theory. Their formulation used a shared-secret-key created from exponentiation of some number, modulo a prime number. However, they left open the problem of realizing a one-way function, possibly because the difficulty of factoring was not well-studied at the time.
Ron Rivest
Ronald Linn Rivest (; born May 6, 1947) is a cryptographer and an Institute Professor at MIT. He is a member of MIT's Department of Electrical Engineering and Computer Science (EECS) and a member of MIT's Computer Science and Artificial Intell ...
,
Adi Shamir
Adi Shamir ( he, עדי שמיר; born July 6, 1952) is an Israeli cryptographer. He is a co-inventor of the Rivest–Shamir–Adleman (RSA) algorithm (along with Ron Rivest and Len Adleman), a co-inventor of the Feige–Fiat–Shamir identificat ...
, and
Leonard Adleman
Leonard Adleman (born December 31, 1945) is an American computer scientist. He is one of the creators of the RSA encryption algorithm, for which he received the 2002 Turing Award, often called the Nobel prize of Computer science. He is also kno ...
at the
Massachusetts Institute of Technology
The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the ...
made several attempts over the course of a year to create a one-way function that was hard to invert. Rivest and Shamir, as computer scientists, proposed many potential functions, while Adleman, as a mathematician, was responsible for finding their weaknesses. They tried many approaches, including "
knapsack
A backpack—also called knapsack, schoolbag, rucksack, rucksac, pack, sackpack, booksack, bookbag or backsack—is, in its simplest frameless form, a fabric sack carried on one's back and secured with two straps that go over the shoulders ...
-based" and "permutation polynomials". For a time, they thought what they wanted to achieve was impossible due to contradictory requirements. In April 1977, they spent
Passover
Passover, also called Pesach (; ), is a major Jewish holidays, Jewish holiday that celebrates the The Exodus, Biblical story of the Israelites escape from slavery in Ancient Egypt, Egypt, which occurs on the 15th day of the Hebrew calendar, He ...
at the house of a student and drank a good deal of
Manischewitz wine before returning to their homes at around midnight. Rivest, unable to sleep, lay on the couch with a math textbook and started thinking about their one-way function. He spent the rest of the night formalizing his idea, and he had much of the paper ready by daybreak. The algorithm is now known as RSA the initials of their surnames in same order as their paper.
Clifford Cocks
Clifford Christopher Cocks (born 28 December 1950) is a British mathematician and cryptographer.
In 1973, while working at the United Kingdom Government Communications Headquarters (GCHQ), he invented a public-key cryptography algorithm equiv ...
, an English
mathematician
A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems.
Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change.
History
On ...
working for the
British
British may refer to:
Peoples, culture, and language
* British people, nationals or natives of the United Kingdom, British Overseas Territories, and Crown Dependencies.
** Britishness, the British identity and common culture
* British English, ...
intelligence agency
Government Communications Headquarters
Government Communications Headquarters, commonly known as GCHQ, is an intelligence and security organisation responsible for providing signals intelligence (SIGINT) and information assurance (IA) to the government and armed forces of the Un ...
(GCHQ), described an equivalent system in an internal document in 1973. However, given the relatively expensive computers needed to implement it at the time, it was considered to be mostly a curiosity and, as far as is publicly known, was never deployed. His discovery, however, was not revealed until 1997 due to its top-secret classification.
Kid-RSA (KRSA) is a simplified, insecure public-key cipher published in 1997, designed for educational purposes. Some people feel that learning Kid-RSA gives insight into RSA and other public-key ciphers, analogous to
simplified DES.
Patent
A
patent
A patent is a type of intellectual property that gives its owner the legal right to exclude others from making, using, or selling an invention for a limited period of time in exchange for publishing an enabling disclosure of the invention."A p ...
describing the RSA algorithm was granted to
MIT
The Massachusetts Institute of Technology (MIT) is a private land-grant research university in Cambridge, Massachusetts. Established in 1861, MIT has played a key role in the development of modern technology and science, and is one of the m ...
on 20 September 1983: "Cryptographic communications system and method". From
DWPI's abstract of the patent:
A detailed description of the algorithm was published in August 1977, in
Scientific American
''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many famous scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it. In print since 1845, it i ...
's
Mathematical Games
A mathematical game is a game whose rules, strategies, and outcomes are defined by clear mathematical parameters. Often, such games have simple rules and match procedures, such as Tic-tac-toe and Dots and Boxes. Generally, mathematical games ne ...
column.
This preceded the patent's filing date of December 1977. Consequently, the patent had no legal standing outside the
United States
The United States of America (U.S.A. or USA), commonly known as the United States (U.S. or US) or America, is a country primarily located in North America. It consists of 50 states, a federal district, five major unincorporated territorie ...
. Had Cocks's work been publicly known, a patent in the United States would not have been legal either.
When the patent was issued,
terms of patent were 17 years. The patent was about to expire on 21 September 2000, but
RSA Security
RSA Security LLC, formerly RSA Security, Inc. and doing business as RSA, is an American computer and network security company with a focus on encryption and encryption standards. RSA was named after the initials of its co-founders, Ron Rivest, ...
released the algorithm to the public domain on 6 September 2000.
Operation
The RSA algorithm involves four steps:
key
Key or The Key may refer to:
Common meanings
* Key (cryptography), a piece of information that controls the operation of a cryptography algorithm
* Key (lock), device used to control access to places or facilities restricted by a lock
* Key (map ...
generation, key distribution, encryption, and decryption.
A basic principle behind RSA is the observation that it is practical to find three very large positive integers , , and , such that with
modular exponentiation
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie-Hellman Key Exchange and RSA public/private keys.
Modul ...
for all integers (with ):
and that knowing and , or even , it can be extremely difficult to find . The
triple bar
The triple bar, or tribar ≡, is a symbol with multiple, context-dependent meanings. It has the appearance of an equals sign sign with a third line. The triple bar character in Unicode is code point .. The closely related code point ...
(≡) here denotes
modular congruence (which is to say that when you divide by and by , they both have the same remainder).
In addition, for some operations it is convenient that the order of the two exponentiations can be changed and that this relation also implies
RSA involves a ''public key'' and a ''
private key
Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic alg ...
''. The public key can be known by everyone and is used for encrypting messages. The intention is that messages encrypted with the public key can only be decrypted in a reasonable amount of time by using the private key. The public key is represented by the integers and , and the private key by the integer (although is also used during the decryption process, so it might be considered to be a part of the private key too). represents the message (previously prepared with a certain technique explained below).
Key generation
The keys for the RSA algorithm are generated in the following way:
# Choose two large
prime number
A prime number (or a prime) is a natural number greater than 1 that is not a product of 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 ...
s and .
#* To make factoring harder, and should be chosen at random, be similar in magnitude, but differ in length.
Prime integers can be efficiently found using a
primality test
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whet ...
.
#* and should be kept secret.
# Compute .
#* is used as the
modulus for both the public and private keys. Its length, usually expressed in bits, is the
key length
In cryptography, key size, key length, or key space refer to the number of bits in a key used by a cryptographic algorithm (such as a cipher).
Key length defines the upper-bound on an algorithm's security (i.e. a logarithmic measure of the fastest ...
.
#* is released as part of the public key.
# Compute , where is
Carmichael's totient function. Since , and since and are prime, , and likewise . Hence .
#* The may be calculated through the
Euclidean algorithm
In mathematics, the Euclidean algorithm,Some widely used textbooks, such as I. N. Herstein's ''Topics in Algebra'' and Serge Lang's ''Algebra'', use the term "Euclidean algorithm" to refer to Euclidean division or Euclid's algorithm, is an effi ...
, since .
#* is kept secret.
# Choose an integer such that and ; that is, and are
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
.
#* having a short
bit-length
Bit-length or bit width is the number of binary digits, called bits, necessary to represent an unsigned integer as a binary number. Formally, the bit-length of a natural number n \geq 0 is
:\ell(n) = \lceil \log_2(n+1) \rceil
where \log_2 is th ...
and small
Hamming weight
The Hamming weight of a string is the number of symbols that are different from the zero-symbol of the alphabet used. It is thus equivalent to the Hamming distance from the all-zero string of the same length. For the most typical case, a string o ...
results in more efficient encryption the most commonly chosen value for is . The smallest (and fastest) possible value for is 3, but such a small value for has been shown to be less secure in some settings.
[
]
#* is released as part of the public key.
# Determine as ; that is, is the
modular multiplicative inverse In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer is an integer such that the product is congruent to 1 with respect to the modulus .. In the standard notation of modular arithmetic this congr ...
of modulo .
#* This means: solve for the equation ; can be computed efficiently by using the
extended Euclidean algorithm
In arithmetic and computer programming, the extended Euclidean algorithm is an extension to the Euclidean algorithm, and computes, in addition to the greatest common divisor (gcd) of integers ''a'' and ''b'', also the coefficients of Bézout's ide ...
, since, thanks to and being coprime, said equation is a form of
Bézout's identity
In mathematics, Bézout's identity (also called Bézout's lemma), named after Étienne Bézout, is the following theorem:
Here the greatest common divisor of and is taken to be . The integers and are called Bézout coefficients for ; they ...
, where is one of the coefficients.
#* is kept secret as the ''private key exponent''.
The ''public key'' consists of the modulus and the public (or encryption) exponent . The ''private key'' consists of the private (or decryption) exponent , which must be kept secret. , , and must also be kept secret because they can be used to calculate . In fact, they can all be discarded after has been computed.
In the original RSA paper,
the
Euler totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
is used instead of for calculating the private exponent . Since is always divisible by , the algorithm works as well. The possibility of using
Euler totient function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In ot ...
results also from
Lagrange's theorem applied to the
multiplicative group of integers modulo ''pq''. Thus any satisfying also satisfies . However, computing modulo will sometimes yield a result that is larger than necessary (i.e. ). Most of the implementations of RSA will accept exponents generated using either method (if they use the private exponent at all, rather than using the optimized decryption method
based on the Chinese remainder theorem described below), but some standards such a
FIPS 186-4may require that . Any "oversized" private exponents not meeting this criterion may always be reduced modulo to obtain a smaller equivalent exponent.
Since any common factors of and are present in the factorisation of = = , it is recommended that and have only very small common factors, if any, besides the necessary 2.
Note: The authors of the original RSA paper carry out the key generation by choosing and then computing as the
modular multiplicative inverse In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer is an integer such that the product is congruent to 1 with respect to the modulus .. In the standard notation of modular arithmetic this congr ...
of modulo , whereas most current implementations of RSA, such as those following
PKCS#1, do the reverse (choose and compute ). Since the chosen key can be small, whereas the computed key normally is not, the RSA paper's algorithm optimizes decryption compared to encryption, while the modern algorithm optimizes encryption instead.
Key distribution
Suppose that
Bob
Bob, BOB, or B.O.B. may refer to:
Places
* Mount Bob, New York, United States
*Bob Island, Palmer Archipelago, Antarctica
People, fictional characters, and named animals
*Bob (given name), a list of people and fictional characters
*Bob (surname ...
wants to send information to
Alice. If they decide to use RSA, Bob must know Alice's public key to encrypt the message, and Alice must use her private key to decrypt the message.
To enable Bob to send his encrypted messages, Alice transmits her public key to Bob via a reliable, but not necessarily secret, route. Alice's private key is never distributed.
Encryption
After Bob obtains Alice's public key, he can send a message to Alice.
To do it, he first turns (strictly speaking, the un-padded plaintext) into an integer (strictly speaking, the
padded plaintext), such that by using an agreed-upon reversible protocol known as a
padding scheme. He then computes the ciphertext , using Alice's public key , corresponding to
This can be done reasonably quickly, even for very large numbers, using
modular exponentiation
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie-Hellman Key Exchange and RSA public/private keys.
Modul ...
. Bob then transmits to Alice. Note that at least nine values of will yield a ciphertext equal to
,
but this is very unlikely to occur in practice.
Decryption
Alice can recover from by using her private key exponent by computing
Given , she can recover the original message by reversing the padding scheme.
Example
Here is an example of RSA encryption and decryption. The parameters used here are artificially small, but one can also
use OpenSSL to generate and examine a real keypair.
# Choose two distinct prime numbers, such as
#:
and
.
# Compute giving
#:
# Compute the
Carmichael's totient function of the product as giving
#:
# Choose any number that is
coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivale ...
to 780. Choosing a prime number for leaves us only to check that is not a divisor of 780.
#: Let
.
# Compute , the
modular multiplicative inverse In mathematics, particularly in the area of arithmetic, a modular multiplicative inverse of an integer is an integer such that the product is congruent to 1 with respect to the modulus .. In the standard notation of modular arithmetic this congr ...
of , yielding
as
The public key is . For a padded
plaintext
In cryptography, plaintext usually means unencrypted information pending input into cryptographic algorithms, usually encryption algorithms. This usually refers to data that is transmitted or stored unencrypted.
Overview
With the advent of comp ...
message , the encryption function is
The private key is . For an encrypted
ciphertext
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 ...
, the decryption function is
For instance, in order to encrypt , one calculates
To decrypt , one calculates
Both of these calculations can be computed efficiently using the
square-and-multiply algorithm for
modular exponentiation
Modular exponentiation is exponentiation performed over a modulus. It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie-Hellman Key Exchange and RSA public/private keys.
Modul ...
. In real-life situations the primes selected would be much larger; in our example it would be trivial to factor (obtained from the freely available public key) back to the primes and . , also from the public key, is then inverted to get , thus acquiring the private key.
Practical implementations use the
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
to speed up the calculation using modulus of factors (mod ''pq'' using mod ''p'' and mod ''q'').
The values , and , which are part of the private key are computed as follows:
Here is how , and are used for efficient decryption (encryption is efficient by choice of a suitable and pair):
Signing messages
Suppose
Alice uses
Bob
Bob, BOB, or B.O.B. may refer to:
Places
* Mount Bob, New York, United States
*Bob Island, Palmer Archipelago, Antarctica
People, fictional characters, and named animals
*Bob (given name), a list of people and fictional characters
*Bob (surname ...
's public key to send him an encrypted message. In the message, she can claim to be Alice, but Bob has no way of verifying that the message was from Alice, since anyone can use Bob's public key to send him encrypted messages. In order to verify the origin of a message, RSA can also be used to
sign
A sign is an object, quality, event, or entity whose presence or occurrence indicates the probable presence or occurrence of something else. A natural sign bears a causal relation to its object—for instance, thunder is a sign of storm, or me ...
a message.
Suppose Alice wishes to send a signed message to Bob. She can use her own private key to do so. She produces a
hash value
A hash function is any function that can be used to map data of arbitrary size to fixed-size values. The values returned by a hash function are called ''hash values'', ''hash codes'', ''digests'', or simply ''hashes''. The values are usually u ...
of the message, raises it to the power of (modulo ) (as she does when decrypting a message), and attaches it as a "signature" to the message. When Bob receives the signed message, he uses the same hash algorithm in conjunction with Alice's public key. He raises the signature to the power of (modulo ) (as he does when encrypting a message), and compares the resulting hash value with the message's hash value. If the two agree, he knows that the author of the message was in possession of Alice's private key and that the message has not been tampered with since being sent.
This works because of
exponentiation
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
rules:
Thus the keys may be swapped without loss of generality, that is, a private key of a key pair may be used either to:
# Decrypt a message only intended for the recipient, which may be encrypted by anyone having the public key (asymmetric encrypted transport).
# Encrypt a message which may be decrypted by anyone, but which can only be encrypted by one person; this provides a digital signature.
Proofs of correctness
Proof using Fermat's little theorem
The proof of the correctness of RSA is based on
Fermat's little theorem
Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as
: a^p \equiv a \pmod p.
For example, if = ...
, stating that for any integer and prime , not dividing .
We want to show that
for every integer when and are distinct prime numbers and and are positive integers satisfying .
Since is, by construction, divisible by both and , we can write
for some nonnegative integers and .
To check whether two numbers, such as and , are congruent , it suffices (and in fact is equivalent) to check that they are congruent and separately.
To show , we consider two cases:
# If , is a multiple of . Thus ''m
ed'' is a multiple of . So .
# If ,
#:
#: where we used
Fermat's little theorem
Fermat's little theorem states that if ''p'' is a prime number, then for any integer ''a'', the number a^p - a is an integer multiple of ''p''. In the notation of modular arithmetic, this is expressed as
: a^p \equiv a \pmod p.
For example, if = ...
to replace with 1.
The verification that proceeds in a completely analogous way:
# If , ''m
ed'' is a multiple of . So .
# If ,
#:
This completes the proof that, for any integer , and integers , such that ,
Notes
Proof using Euler's theorem
Although the original paper of Rivest, Shamir, and Adleman used Fermat's little theorem to explain why RSA works, it is common to find proofs that rely instead on
Euler's theorem
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, and \varphi(n) is Euler's totient function, then raised to the power \varphi(n) is congru ...
.
We want to show that , where is a product of two different prime numbers, and and are positive integers satisfying . Since and are positive, we can write for some non-negative integer . ''Assuming'' that is relatively prime to , we have
where the second-last congruence follows from
Euler's theorem
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, and \varphi(n) is Euler's totient function, then raised to the power \varphi(n) is congru ...
.
More generally, for any and satisfying , the same conclusion follows from
Carmichael's generalization of Euler's theorem, which states that for all relatively prime to .
When is not relatively prime to , the argument just given is invalid. This is highly improbable (only a proportion of numbers have this property), but even in this case, the desired congruence is still true. Either or , and these cases can be treated using the previous proof.
Padding
Attacks against plain RSA
There are a number of attacks against plain RSA as described below.
* When encrypting with low encryption exponents (e.g., ) and small values of the (i.e., ), the result of is strictly less than the modulus . In this case, ciphertexts can be decrypted easily by taking the th root of the ciphertext over the integers.
* If the same clear-text message is sent to or more recipients in an encrypted way, and the receivers share the same exponent , but different , , and therefore , then it is easy to decrypt the original clear-text message via the
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
.
Johan Håstad
Johan Torkel Håstad (; born 19 November 1960) is a Swedish theoretical computer scientist most known for his work on computational complexity theory. He was the recipient of the Gödel Prize in 1994 and 2011 and the ACM Doctoral Dissertation Aw ...
noticed that this attack is possible even if the clear texts are not equal, but the attacker knows a linear relation between them. This attack was later improved by
Don Coppersmith
Don Coppersmith (born 1950) is a cryptographer and mathematician. He was involved in the design of the Data Encryption Standard block cipher at IBM, particularly the design of the S-boxes, strengthening them against differential cryptanalysis ...
(see
Coppersmith's attack).
* Because RSA encryption is a
deterministic encryption algorithm (i.e., has no random component) an attacker can successfully launch a
chosen plaintext attack
A chosen-plaintext attack (CPA) is an attack model for cryptanalysis which presumes that the attacker can obtain the ciphertexts for arbitrary plaintexts.Ross Anderson, ''Security Engineering: A Guide to Building Dependable Distributed Systems'' ...
against the cryptosystem, by encrypting likely plaintexts under the public key and test whether they are equal to the ciphertext. A cryptosystem is called
semantically secure In cryptography, a semantically secure cryptosystem is one where only negligible information about the plaintext can be feasibly extracted from the ciphertext. Specifically, any probabilistic, polynomial-time algorithm (PPTA) that is given the cip ...
if an attacker cannot distinguish two encryptions from each other, even if the attacker knows (or has chosen) the corresponding plaintexts. RSA without padding is not semantically secure.
* RSA has the property that the product of two ciphertexts is equal to the encryption of the product of the respective plaintexts. That is, . Because of this multiplicative property, a
chosen-ciphertext attack
A chosen-ciphertext attack (CCA) is an attack model for cryptanalysis where the cryptanalyst can gather information by obtaining the decryptions of chosen ciphertexts. From these pieces of information the adversary can attempt to recover the hidden ...
is possible. E.g., an attacker who wants to know the decryption of a ciphertext may ask the holder of the private key to decrypt an unsuspicious-looking ciphertext for some value chosen by the attacker. Because of the multiplicative property, ' is the encryption of . Hence, if the attacker is successful with the attack, they will learn from which they can derive the message by multiplying with the modular inverse of modulo .
* Given the private exponent , one can efficiently factor the modulus . And given factorization of the modulus , one can obtain any private key (', ) generated against a public key (', ).
Padding schemes
To avoid these problems, practical RSA implementations typically embed some form of structured, randomized
padding
Padding is thin cushioned material sometimes added to clothes. Padding may also be referred to as batting when used as a layer in lining quilts or as a packaging or stuffing material. When padding is used in clothes, it is often done in an attempt ...
into the value before encrypting it. This padding ensures that does not fall into the range of insecure plaintexts, and that a given message, once padded, will encrypt to one of a large number of different possible ciphertexts.
Standards such as
PKCS#1 have been carefully designed to securely pad messages prior to RSA encryption. Because these schemes pad the plaintext with some number of additional bits, the size of the un-padded message must be somewhat smaller. RSA padding schemes must be carefully designed so as to prevent sophisticated attacks that may be facilitated by a predictable message structure. Early versions of the PKCS#1 standard (up to version 1.5) used a construction that appears to make RSA semantically secure. However, at
Crypto
Crypto commonly refers to:
* Cryptocurrency, a type of digital currency secured by cryptography and decentralization
* Cryptography, the practice and study of hiding information
Crypto or Krypto may also refer to:
Cryptography
* Cryptanalysis, ...
1998, Bleichenbacher showed that this version is vulnerable to a practical
adaptive chosen-ciphertext attack
An adaptive chosen-ciphertext attack (abbreviated as CCA2) is an interactive form of chosen-ciphertext attack in which an attacker first sends a number of ciphertexts to be decrypted chosen adaptively, and then uses the results to distinguish a tar ...
. Furthermore, at
Eurocrypt 2000, Coron et al. showed that for some types of messages, this padding does not provide a high enough level of security. Later versions of the standard include
Optimal Asymmetric Encryption Padding (OAEP), which prevents these attacks. As such, OAEP should be used in any new application, and PKCS#1 v1.5 padding should be replaced wherever possible. The PKCS#1 standard also incorporates processing schemes designed to provide additional security for RSA signatures, e.g. the Probabilistic Signature Scheme for RSA (
RSA-PSS).
Secure padding schemes such as RSA-PSS are as essential for the security of message signing as they are for message encryption. Two USA patents on PSS were granted ( and ); however, these patents expired on 24 July 2009 and 25 April 2010 respectively. Use of PSS no longer seems to be encumbered by patents. Note that using different RSA key pairs for encryption and signing is potentially more secure.
Security and practical considerations
Using the Chinese remainder algorithm
For efficiency, many popular crypto libraries (such as
OpenSSL
OpenSSL is a software library for applications that provide secure communications over computer networks against eavesdropping or need to identify the party at the other end. It is widely used by Internet servers, including the majority of HTT ...
,
Java
Java (; id, Jawa, ; jv, ꦗꦮ; su, ) is one of the Greater Sunda Islands in Indonesia. It is bordered by the Indian Ocean to the south and the Java Sea to the north. With a population of 151.6 million people, Java is the world's List ...
and
.NET) use for decryption and signing the following optimization based on the
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
. The following values are precomputed and stored as part of the private key:
*
and
the primes from the key generation,
*
*
*
These values allow the recipient to compute the exponentiation more efficiently as follows:
,
,
,
.
This is more efficient than computing
exponentiation by squaring
Exponentiation is a mathematical operation, written as , involving two numbers, the '' base'' and the ''exponent'' or ''power'' , and pronounced as " (raised) to the (power of) ". When is a positive integer, exponentiation corresponds to re ...
, even though two modular exponentiations have to be computed. The reason is that these two modular exponentiations both use a smaller exponent and a smaller modulus.
Integer factorization and RSA problem
The security of the RSA cryptosystem is based on two mathematical problems: the problem of
factoring large numbers and the
RSA problem
In cryptography, the RSA problem summarizes the task of performing an RSA private-key operation given only the public key. The RSA algorithm raises a ''message'' to an '' exponent'', modulo a composite number ''N'' whose factors are not known. T ...
. Full decryption of an RSA ciphertext is thought to be infeasible on the assumption that both of these problems are
hard
Hard may refer to:
* Hardness, resistance of physical materials to deformation or fracture
* Hard water, water with high mineral content
Arts and entertainment
* ''Hard'' (TV series), a French TV series
* Hard (band), a Hungarian hard rock supe ...
, i.e., no efficient algorithm exists for solving them. Providing security against ''partial'' decryption may require the addition of a secure
padding scheme.
The
RSA problem
In cryptography, the RSA problem summarizes the task of performing an RSA private-key operation given only the public key. The RSA algorithm raises a ''message'' to an '' exponent'', modulo a composite number ''N'' whose factors are not known. T ...
is defined as the task of taking th roots modulo a composite : recovering a value such that , where is an RSA public key, and is an RSA ciphertext. Currently the most promising approach to solving the RSA problem is to factor the modulus . With the ability to recover prime factors, an attacker can compute the secret exponent from a public key , then decrypt using the standard procedure. To accomplish this, an attacker factors into and , and computes that allows the determination of from . No polynomial-time method for factoring large integers on a classical computer has yet been found, but it has not been proven that none exists; see
integer factorization
In number theory, integer factorization is the decomposition of a composite number into a product of smaller integers. If these factors are further restricted to prime numbers, the process is called prime factorization.
When the numbers are suf ...
for a discussion of this problem.
Multiple polynomial quadratic sieve (MPQS) can be used to factor the public modulus .
The first RSA-512 factorization in 1999 used hundreds of computers and required the equivalent of 8,400 MIPS years, over an elapsed time of approximately seven months. By 2009, Benjamin Moody could factor an 512-bit RSA key in 73 days using only public software (GGNFS) and his desktop computer (a dual-core
Athlon64 with a 1,900 MHz CPU). Just less than 5 gigabytes of disk storage was required and about 2.5 gigabytes of RAM for the sieving process.
Rivest, Shamir, and Adleman noted
that Miller has shown that – assuming the truth of the
extended Riemann hypothesis
The Riemann hypothesis is one of the most important conjectures in mathematics. It is a statement about the zeros of the Riemann zeta function. Various geometrical and arithmetical objects can be described by so-called global ''L''-functions, whic ...
– finding from and is as hard as factoring into and (up to a polynomial time difference). However, Rivest, Shamir, and Adleman noted, in section IX/D of their paper, that they had not found a proof that inverting RSA is as hard as factoring.
, the largest publicly known factored
RSA number
In mathematics, the RSA numbers are a set of large semiprimes (numbers with exactly two prime factors) that were part of the RSA Factoring Challenge. The challenge was to find the prime factors of each number. It was created by RSA Laboratories in ...
had 829 bits (250 decimal digits,
RSA-250
In mathematics, the RSA numbers are a set of large semiprimes (numbers with exactly two prime factors) that were part of the RSA Factoring Challenge. The challenge was to find the prime factors of each number. It was created by RSA Laboratories in ...
). Its factorization, by a state-of-the-art distributed implementation, took approximately 2700 CPU years. In practice, RSA keys are typically 1024 to 4096 bits long. In 2003,
RSA Security
RSA Security LLC, formerly RSA Security, Inc. and doing business as RSA, is an American computer and network security company with a focus on encryption and encryption standards. RSA was named after the initials of its co-founders, Ron Rivest, ...
estimated that 1024-bit keys were likely to become crackable by 2010.
As of 2020, it is not known whether such keys can be cracked, but minimum recommendations have moved to at least 2048 bits.
It is generally presumed that RSA is secure if is sufficiently large, outside of quantum computing.
If is 300
bit
The bit is the most basic unit of information in computing and digital communications. The name is a portmanteau of binary digit. The bit represents a logical state with one of two possible values. These values are most commonly represente ...
s or shorter, it can be factored in a few hours in a
personal computer
A personal computer (PC) is a multi-purpose microcomputer whose size, capabilities, and price make it feasible for individual use. Personal computers are intended to be operated directly by an end user, rather than by a computer expert or tec ...
, using software already freely available. Keys of 512 bits have been shown to be practically breakable in 1999, when
RSA-155
In mathematics, the RSA numbers are a set of large semiprimes (numbers with exactly two prime factors) that were part of the RSA Factoring Challenge. The challenge was to find the prime factors of each number. It was created by RSA Laboratories i ...
was factored by using several hundred computers, and these are now factored in a few weeks using common hardware. Exploits using 512-bit code-signing certificates that may have been factored were reported in 2011. A theoretical hardware device named
TWIRL, described by Shamir and Tromer in 2003, called into question the security of 1024-bit keys.
In 1994,
Peter Shor
Peter Williston Shor (born August 14, 1959) is an American professor of applied mathematics at MIT. He is known for his work on quantum computation, in particular for devising Shor's algorithm, a quantum algorithm for factoring exponentially fa ...
showed that a
quantum computer
Quantum computing is a type of computation whose operations can harness the phenomena of quantum mechanics, such as superposition, interference, and entanglement. Devices that perform quantum computations are known as quantum computers. Though ...
– if one could ever be practically created for the purpose – would be able to factor in
polynomial time
In computer science, the time complexity is the computational complexity that describes the amount of computer time it takes to run an algorithm. Time complexity is commonly estimated by counting the number of elementary operations performed by ...
, breaking RSA; see
Shor's algorithm
Shor's algorithm is a quantum algorithm, quantum computer algorithm for finding the prime factors of an integer. It was developed in 1994 by the American mathematician Peter Shor.
On a quantum computer, to factor an integer N , Shor's algorithm ...
.
Faulty key generation
Finding the large primes and is usually done by testing random numbers of the correct size with probabilistic
primality test
A primality test is an algorithm for determining whether an input number is prime. Among other fields of mathematics, it is used for cryptography. Unlike integer factorization, primality tests do not generally give prime factors, only stating whet ...
s that quickly eliminate virtually all of the nonprimes.
The numbers and should not be "too close", lest the
Fermat factorization for be successful. If is less than (, which even for "small" 1024-bit values of is ), solving for and is trivial. Furthermore, if either or has only small prime factors, can be factored quickly by
Pollard's ''p'' − 1 algorithm, and hence such values of or should be discarded.
It is important that the private exponent be large enough. Michael J. Wiener showed that if is between and (which is quite typical) and , then can be computed efficiently from and .
There is no known attack against small public exponents such as , provided that the proper padding is used.
Coppersmith's attack has many applications in attacking RSA specifically if the public exponent is small and if the encrypted message is short and not padded.
65537
65537 is the integer after 65536 and before 65538.
In mathematics
65537 is the largest known prime number of the form 2^ +1 (n = 4). Therefore, a regular polygon with 65537 sides is constructible with compass and unmarked straightedge. Johann ...
is a commonly used value for ; this value can be regarded as a compromise between avoiding potential small-exponent attacks and still allowing efficient encryptions (or signature verification). The NIST Special Publication on Computer Security (SP 800-78 Rev. 1 of August 2007) does not allow public exponents smaller than 65537, but does not state a reason for this restriction.
In October 2017, a team of researchers from
Masaryk University
Masaryk University (MU) ( cs, Masarykova univerzita; la, Universitas Masarykiana Brunensis) is the second largest university in the Czech Republic, a member of the Compostela Group and the Utrecht Network. Founded in 1919 in Brno as the seco ...
announced the
ROCA vulnerability
The ROCA vulnerability is a cryptographic weakness that allows the
private key of a key pair to be recovered from the public key in keys generated by devices with the vulnerability. "ROCA" is an acronym for "Return of Coppersmith's attack". Th ...
, which affects RSA keys generated by an algorithm embodied in a library from
Infineon
Infineon Technologies AG is a German semiconductor manufacturer founded in 1999, when the semiconductor operations of the former parent company Siemens AG were spun off. Infineon has about 50,280 employees and is one of the ten largest semicond ...
known as RSALib. A large number of
smart card
A smart card, chip card, or integrated circuit card (ICC or IC card) is a physical electronic authentication device, used to control access to a resource. It is typically a plastic credit card-sized card with an embedded integrated circuit (IC) c ...
s and
trusted platform module
Trusted Platform Module (TPM, also known as ISO/IEC 11889) is an international standard for a secure cryptoprocessor, a dedicated microcontroller designed to secure hardware through integrated cryptographic keys. The term can also refer to a ch ...
s (TPM) were shown to be affected. Vulnerable RSA keys are easily identified using a test program the team released.
Importance of strong random number generation
A cryptographically strong
random number generator
Random number generation is a process by which, often by means of a random number generator (RNG), a sequence of numbers or symbols that cannot be reasonably predicted better than by random chance is generated. This means that the particular out ...
, which has been properly seeded with adequate entropy, must be used to generate the primes and . An analysis comparing millions of public keys gathered from the Internet was carried out in early 2012 by
Arjen K. Lenstra, James P. Hughes, Maxime Augier, Joppe W. Bos, Thorsten Kleinjung and Christophe Wachter. They were able to factor 0.2% of the keys using only Euclid's algorithm.
They exploited a weakness unique to cryptosystems based on integer factorization. If is one public key, and is another, then if by chance (but is not equal to '), then a simple computation of factors both and ', totally compromising both keys. Lenstra et al. note that this problem can be minimized by using a strong random seed of bit length twice the intended security level, or by employing a deterministic function to choose given , instead of choosing and independently.
Nadia Heninger
Nadia Heninger (born 1982) is an American cryptographer, computer security expert, and computational number theorist at the University of California, San Diego.
Contributions
Heninger is known for her work on freezing powered-down security devic ...
was part of a group that did a similar experiment. They used an idea of
Daniel J. Bernstein to compute the GCD of each RSA key against the product of all the other keys ' they had found (a 729-million-digit number), instead of computing each separately, thereby achieving a very significant speedup, since after one large division, the GCD problem is of normal size.
Heninger says in her blog that the bad keys occurred almost entirely in embedded applications, including "firewalls, routers, VPN devices, remote server administration devices, printers, projectors, and VOIP phones" from more than 30 manufacturers. Heninger explains that the one-shared-prime problem uncovered by the two groups results from situations where the pseudorandom number generator is poorly seeded initially, and then is reseeded between the generation of the first and second primes. Using seeds of sufficiently high entropy obtained from key stroke timings or electronic diode noise or
atmospheric noise
Atmospheric noise is radio noise caused by natural atmospheric processes, primarily lightning discharges in thunderstorms. On a worldwide scale, there are about 40 lightning flashes per second – ≈3.5 million lightning discharges ...
from a radio receiver tuned between stations should solve the problem.
Strong random number generation is important throughout every phase of public-key cryptography. For instance, if a weak generator is used for the symmetric keys that are being distributed by RSA, then an eavesdropper could bypass RSA and guess the symmetric keys directly.
Timing attacks
Kocher
The Kocher () is a -longincluding its source river Schwarzer Kocher right tributary of the Neckar in the north-eastern part of Baden-Württemberg, Germany. The name "Kocher" originates from its Celtic name "cochan" and probably means winding, mea ...
described a new attack on RSA in 1995: if the attacker Eve knows Alice's hardware in sufficient detail and is able to measure the decryption times for several known ciphertexts, Eve can deduce the decryption key quickly. This attack can also be applied against the RSA signature scheme. In 2003,
Boneh and
Brumley demonstrated a more practical attack capable of recovering RSA factorizations over a network connection (e.g., from a
Secure Sockets Layer
Transport Layer Security (TLS) is a cryptographic protocol designed to provide communications security over a computer network. The protocol is widely used in applications such as email, instant messaging, and voice over IP, but its use in securi ...
(SSL)-enabled webserver).
This attack takes advantage of information leaked by the
Chinese remainder theorem
In mathematics, the Chinese remainder theorem states that if one knows the remainders of the Euclidean division of an integer ''n'' by several integers, then one can determine uniquely the remainder of the division of ''n'' by the product of thes ...
optimization used by many RSA implementations.
One way to thwart these attacks is to ensure that the decryption operation takes a constant amount of time for every ciphertext. However, this approach can significantly reduce performance. Instead, most RSA implementations use an alternate technique known as
cryptographic blinding. RSA blinding makes use of the multiplicative property of RSA. Instead of computing , Alice first chooses a secret random value and computes . The result of this computation, after applying
Euler's theorem
In number theory, Euler's theorem (also known as the Fermat–Euler theorem or Euler's totient theorem) states that, if and are coprime positive integers, and \varphi(n) is Euler's totient function, then raised to the power \varphi(n) is congru ...
, is , and so the effect of can be removed by multiplying by its inverse. A new value of is chosen for each ciphertext. With blinding applied, the decryption time is no longer correlated to the value of the input ciphertext, and so the timing attack fails.
Adaptive chosen-ciphertext attacks
In 1998,
Daniel Bleichenbacher
Daniel Bleichenbacher (born 1964) is a Swiss cryptographer, previously a researcher at Bell Labs, and currently employed at Google. He received his Ph.D. from ETH Zurich in 1996 for contributions to computational number theory, particularly conce ...
described the first practical
adaptive chosen-ciphertext attack
An adaptive chosen-ciphertext attack (abbreviated as CCA2) is an interactive form of chosen-ciphertext attack in which an attacker first sends a number of ciphertexts to be decrypted chosen adaptively, and then uses the results to distinguish a tar ...
against RSA-encrypted messages using the PKCS #1 v1
padding scheme (a padding scheme randomizes and adds structure to an RSA-encrypted message, so it is possible to determine whether a decrypted message is valid). Due to flaws with the PKCS #1 scheme, Bleichenbacher was able to mount a practical attack against RSA implementations of the
Secure Sockets Layer
Transport Layer Security (TLS) is a cryptographic protocol designed to provide communications security over a computer network. The protocol is widely used in applications such as email, instant messaging, and voice over IP, but its use in securi ...
protocol and to recover session keys. As a result of this work, cryptographers now recommend the use of provably secure padding schemes such as
Optimal Asymmetric Encryption Padding, and RSA Laboratories has released new versions of PKCS #1 that are not vulnerable to these attacks.
A variant of this attack, dubbed "BERserk", came back in 2014. It impacted the Mozilla NSS Crypto Library, which was used notably by Firefox and Chrome.
Side-channel analysis attacks
A side-channel attack using branch-prediction analysis (BPA) has been described. Many processors use a
branch predictor
In computer architecture, a branch predictor is a digital circuit that tries to guess which way a branch (e.g., an if–then–else structure) will go before this is known definitively. The purpose of the branch predictor is to improve the flow ...
to determine whether a conditional branch in the instruction flow of a program is likely to be taken or not. Often these processors also implement
simultaneous multithreading
Simultaneous multithreading (SMT) is a technique for improving the overall efficiency of superscalar CPUs with hardware multithreading. SMT permits multiple independent threads of execution to better use the resources provided by modern process ...
(SMT). Branch-prediction analysis attacks use a spy process to discover (statistically) the private key when processed with these processors.
Simple Branch Prediction Analysis (SBPA) claims to improve BPA in a non-statistical way. In their paper, "On the Power of Simple Branch Prediction Analysis", the authors of SBPA (Onur Aciicmez and Cetin Kaya Koc) claim to have discovered 508 out of 512 bits of an RSA key in 10 iterations.
A power-fault attack on RSA implementations was described in 2010. The author recovered the key by varying the CPU power voltage outside limits; this caused multiple power faults on the server.
Tricky implementation
There are many details to keep in mind in order to implement RSA securely (strong
PRNG, acceptable public exponent...)
. This makes the implementation challenging, to the point the book Practical Cryptography With Go suggests avoiding RSA if possible.
Implementations
Some cryptography libraries that provide support for RSA include:
*
Botan
*
Bouncy Castle
Bounce or The Bounce may refer to:
* Deflection (physics), the event where an object collides with and bounces against a plane surface
Books
* Mr. Bounce, a character from the Mr. Men series of children's books
Broadcasting, film and TV
* ''B ...
*
cryptlib
cryptlib is an open-source cross-platform software security toolkit library. It is distributed under the Sleepycat License, a free software license compatible with the GNU General Public License. Alternatively, cryptlib is available under a pr ...
*
Crypto++
Crypto++ (also known as CryptoPP, libcrypto++, and libcryptopp) is a free and open-source C++ class library of cryptographic algorithms and schemes written by Wei Dai. Crypto++ has been widely used in academia, student projects, open-source, and no ...
*
Libgcrypt
Libgcrypt is a cryptography library developed as a separated module of GnuPG.
It can also be used independently of GnuPG, but depends on its error-reporting library Libgpg-error.
It provides functions for all fundamental cryptographic building blo ...
*
Nettle
{{redirect, Nettle
Nettle refers to plants with stinging hairs, particularly those of the genus '' Urtica''. It can also refer to plants which resemble ''Urtica'' species in appearance but do not have stinging hairs. Plants called "nettle" includ ...
*
OpenSSL
OpenSSL is a software library for applications that provide secure communications over computer networks against eavesdropping or need to identify the party at the other end. It is widely used by Internet servers, including the majority of HTT ...
*
wolfCrypt
wolfSSL is a small, portable, embedded SSL/TLS library targeted for use by embedded systems developers. It is an open source implementation of TLS (SSL 3.0, TLS 1.0, 1.1, 1.2, 1.3, and DTLS 1.0, 1.2, and 1.3) written in the C programming langu ...
*
GnuTLS
GnuTLS (, the GNU Transport Layer Security Library) is a free software implementation of the TLS, SSL and DTLS protocols. It offers an application programming interface (API) for applications to enable secure communication over the network trans ...
*
mbed TLS
Mbed TLS (previously PolarSSL) is an implementation of the TLS and SSL protocols and the respective cryptographic algorithms and support code required. It is distributed under the Apache License version 2.0. Stated on the website is that Mbed ...
*
LibreSSL
LibreSSL is an open-source implementation of the Transport Layer Security (TLS) protocol. The implementation is named after Secure Sockets Layer (SSL), the deprecated predecessor of TLS, for which support was removed in release 2.3.0. The OpenB ...
See also
*
Acoustic cryptanalysis Acoustic cryptanalysis is a type of side channel attack that exploits sounds emitted by computers or other devices.
Most of the modern acoustic cryptanalysis focuses on the sounds produced by computer keyboards and internal computer components, bu ...
*
Computational complexity theory
In theoretical computer science and mathematics, computational complexity theory focuses on classifying computational problems according to their resource usage, and relating these classes to each other. A computational problem is a task solved by ...
*
Key size
In cryptography, key size, key length, or key space refer to the number of bits in a key used by a cryptographic algorithm (such as a cipher).
Key length defines the upper-bound on an algorithm's security (i.e. a logarithmic measure of the fastest ...
*
Diffie–Hellman key exchange
Diffie–Hellman key exchangeSynonyms of Diffie–Hellman key exchange include:
* Diffie–Hellman–Merkle key exchange
* Diffie–Hellman key agreement
* Diffie–Hellman key establishment
* Diffie–Hellman key negotiation
* Exponential key exc ...
*
Key exchange
Key exchange (also key establishment) is a method in cryptography by which cryptographic keys are exchanged between two parties, allowing use of a cryptographic algorithm.
If the sender and receiver wish to exchange encrypted messages, each m ...
*
Key management
Key management refers to management of cryptographic keys in a cryptosystem. This includes dealing with the generation, exchange, storage, use, crypto-shredding (destruction) and replacement of keys. It includes cryptographic protocol design, ...
*
Elliptic-curve cryptography
Elliptic-curve cryptography (ECC) is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys compared to non-EC cryptography (based on plain Galois fields) to provide e ...
*
Public-key cryptography
Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic alg ...
*
Trapdoor function
In theoretical computer science and cryptography, a trapdoor function is a function that is easy to compute in one direction, yet difficult to compute in the opposite direction (finding its inverse) without special information, called the "trap ...
References
Further reading
*
*
External links
* The Original RSA Patent as filed with the U.S. Patent Office by Rivest; Ronald L. (Belmont, MA), Shamir; Adi (Cambridge, MA), Adleman; Leonard M. (Arlington, MA), December 14, 1977, .
PKCS #1: RSA Cryptography Standard(
RSA Laboratories
RSA Security LLC, formerly RSA Security, Inc. and doing business as RSA, is an American computer and network security company with a focus on encryption and encryption standards. RSA was named after the initials of its co-founders, Ron Rivest, ...
website)
** The ''
PKCS #1''
standard Standard may refer to:
Symbols
* Colours, standards and guidons, kinds of military signs
* Standard (emblem), a type of a large symbol or emblem used for identification
Norms, conventions or requirements
* Standard (metrology), an object th ...
''"provides recommendations for the implementation of
public-key cryptography
Public-key cryptography, or asymmetric cryptography, is the field of cryptographic systems that use pairs of related keys. Each key pair consists of a public key and a corresponding private key. Key pairs are generated with cryptographic alg ...
based on the RSA algorithm, covering the following aspects: cryptographic
primitives;
encryption
In cryptography, encryption is the process of encoding information. This process converts the original representation of the information, known as plaintext, into an alternative form known as ciphertext. Ideally, only authorized parties can decip ...
schemes;
signature
A signature (; from la, signare, "to sign") is a handwritten (and often stylized) depiction of someone's name, nickname, or even a simple "X" or other mark that a person writes on documents as a proof of identity and intent. The writer of a ...
schemes with appendix;
ASN.1
Abstract Syntax Notation One (ASN.1) is a standard interface description language for defining data structures that can be serialized and deserialized in a cross-platform way. It is broadly used in telecommunications and computer networking, and ...
syntax for representing keys and for identifying the schemes"''.
*
Thorough walk through of RSAOnur Aciicmez, Cetin Kaya Koc, Jean-Pierre Seifert: ''On the Power of Simple Branch Prediction Analysis''Example of an RSA implementation with PKCS#1 padding (GPL source code)Kocher's article about timing attacksAn animated explanation of RSA with its mathematical background by CrypTool*
{{DEFAULTSORT:Rsa
Public-key encryption schemes
Digital signature schemes