- RSA
In

cryptography ,**RSA**is analgorithm forpublic-key cryptography . It is the first algorithm known to be suitable for signing as well as encryption, and one of the first great advances in public key cryptography. RSA is widely used inelectronic commerce protocols, and is believed to be secure given sufficiently long keys and the use of up-to-date implementations.**History**The algorithm was publicly described in 1977 by

Ron Rivest ,Adi Shamir , andLeonard Adleman at MIT; the letters**RSA**are the initials of their surnames, listed in the same order as on the paper. [*http://www.msri.org/people/members/sara/articles/rsa.pdf SIAM News, Volume 36, Number 5, June 2003*] , "Still Guarding Secrets after Years of Attacks, RSA Earns Accolades for its Founders", by Sara Robinson]Clifford Cocks , a Britishmathematician working for the UK intelligence agency GCHQ, described an equivalent system in an internal document in 1973, but given the relatively expensive computers needed to implement it at the time, it was mostly considered 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, and Rivest, Shamir, and Adleman devised RSA independently of Cocks's work.MIT was granted Cite patent|US|4405829 for a "Cryptographic communications system and method" that used the algorithm in 1983. The patent would have expired in

2003 , but was released to the public domain by RSA on21 September 2000 . Since a paper describing the algorithm had been published in August 1977, prior to the December 1977filing date of thepatent application , regulations in much of the rest of the world precludedpatent s elsewhere and only the US patent was granted. Had Cocks' work been publicly known, a patent in the US might not have been possible either.**Operation**RSA involves a public key and a private key. The public key can be known to everyone and is used for encrypting messages. Messages encrypted with the public key can only be decrypted using the private key. The keys for the RSA algorithm are generated the following way:

#Choose two distinct large randomprime number s $p$ and $q$

#Compute $n\; =\; pq,$

#*$n,$ is used as the modulus for both the public and private keys

#Compute thetotient : $varphi(n)\; =\; (p-1)(q-1)\; ,$.

#Choose an integer $e$ such that $1\; <\; e\; <\; varphi(n)$, and $e$ and $varphi\; (n)$ share no factors other than $1$ (i.e. $e$ and $varphi\; (n)$ arecoprime )

#*$e$ is released as the public key exponent

#Compute $d$ to satisfy the congruence relation $d\; e\; equiv\; 1pmod\{varphi(n)\}$; i.e. $de\; =\; 1\; +\; kvarphi(n)$ for some integer $k$.

#*$d$ is kept as the private key exponentNotes on the above steps:

* Step 1: Numbers can be probabilistically tested for primality.

* Step 3: changed in PKCS#1 v2.0 to $lambda(n)\; =\; \{\; m\; lcm\}(p-1,\; q-1)\; ,$, where lcm is theleast common multiple , instead of $varphi(n)\; =\; (p-1)(q-1)\; ,$.

* Step 4: A popular choice for the public exponents is $e,$ = 2^{16}+ 1 = 65537. Some applications choose smaller values such as $e,$ = 3, 5, 17 or 257 instead. This is done to make encryption and signature verification faster on small devices like smart cards but small public exponents can lead to greater security risks.cite journal

url = http://crypto.stanford.edu/~dabo/abstracts/RSAattack-survey.html

last = Boneh | first = Dan

title = Twenty Years of attacks on the RSA Cryptosystem

journal = Notices of the American Mathematical Society (AMS)

volume = 46 | issue = 2

pages = pp. 203-213 | year = 1999]

* Steps 4 and 5 can be performed with theextended Euclidean algorithm ; seemodular arithmetic .The

**public key**consists of the modulus $n,$ and the public (or encryption) exponent $e,$. The**private key**consists of the modulus $n,$ and the private (or decryption) exponent $d,$ which must be kept secret.*For efficiency a different form of the

**private key**can be stored:

** $p,$ and $q,$: the primes from the key generation,

** $dmod\; (p\; -\; 1),$ and $dmod(q\; -\; 1),$,

** $q^\{-1\}\; mod(p),$.

*All parts of the private key must be kept secret in this form. $p,$ and $q,$ are sensitive since they are the factors of $n,$, and allow computation of $d,$ given $e,$. If $p,$ and $q,$ are not stored in this form of the private key then they are securely deleted along with other intermediate values from key generation.

*Although this form allows faster decryption and signing by using theChinese Remainder Theorem , it is considerably less secure since it enablesside channel attack s. This is a particular problem if implemented onsmart cards , which benefit most from the improved efficiency. (Start with $y\; =\; x^e\; mod\; n$ and let the card decrypt that. So it computes $y^d\; pmod\{p\}$ or $y^d\; pmod\{q\}$ whose results give some value $z$. Now, induce an error in one of the computations. Then $gcd(z-x,n)$ will reveal $p$ or $q$.)**Messages****Encryption:**Alice transmits her public key $(n,e),$ to Bob and keeps the private key secret. Bob then wishes to send message

**M**to Alice.He first turns

**M**into a number $m,$ < $n,$ by using an agreed-upon reversible protocol known as a padding scheme. He then computes the ciphertext $c,$ corresponding to:: $c\; =\; m^e\; mod\{n\}$

This can be done quickly using the method of

exponentiation by squaring . Bob then transmits $c,$ to Alice.**Decryption:**Alice can recover $m,$ from $c,$ by using her private key exponent $d,$ by the following computation:

: $m\; =\; c^d\; mod\{n\}.$

Given $m,$, she can recover the original message

**M**.The above decryption procedure works because first

: $c^d\; equiv\; (m^e)^d\; equiv\; m^\{ed\}pmod\{n\}$.

Now, $e\; d\; equiv\; 1pmod\{(p\; -\; 1)(q\; -\; 1)\}$, and hence

: $e\; d\; equiv\; 1pmod\{p\; -\; 1\},$ and : $e\; d\; equiv\; 1pmod\{q\; -\; 1\},$

which can also be written as

: $e\; d\; =\; k\; (p\; -\; 1)\; +\; 1,$ and: $e\; d\; =\; h\; (q\; -\; 1)\; +\; 1,$

for proper values of $k,$ and $h,$. If $m,$ is not a multiple of $p,$ then $m,$ and $p,$ are coprime because $p,$ is prime; so by

Fermat's little theorem : $m^\{(p-1)\}\; equiv\; 1\; pmod\{p\}$

and therefore, using the first expression for $e\; d,$,

: $m^\{ed\}\; =\; m^\{k\; (p-1)\; +\; 1\}\; =\; (m^\{p-1\})^k\; m\; equiv\; \{1\}^k\; m\; =\; m\; pmod\{p\},$.

If instead $m,$ is a multiple of $p,$, then

: $m^\{ed\}\; equiv\; 0^\{ed\}\; =\; 0\; equiv\; m\; pmod\{p\}$.

Using the second expression for $e\; d,$, we similarly conclude that

: $m^\{ed\}\; equiv\; m\; pmod\{q\},$.

Since $p,$ and $q,$ are distinct prime numbers, they are relatively prime to each other, so the fact that both primes divide $m^\{ed\}\; -\; m$ implies their product $pq,$ divides $m^\{ed\}\; -\; m$, which means

: $m^\{ed\}\; equiv\; m\; pmod\{pq\}$.

Thus,

: $c^d\; equiv\; m\; pmod\{n\}$.

**A worked example**Here is an example of RSA encryption and decryption. The parameters used here are artificially small, but one can also .

#Choose two prime numbers

#:$p\; =\; 61$ and $q=53$

#Compute $n\; =\; p\; q\; ,$

#:$n=61*53=3233$

#Compute thetotient $varphi(n)\; =\; (p-1)(q-1)\; ,$

#:$varphi(n)\; =\; (61\; -\; 1)(53\; -\; 1)\; =\; 3120,$

#Choose $e>1$ coprime to 3120

#:$e=17$

#Compute $d,$ such that $d\; e\; equiv\; 1pmod\{varphi(n)\},$ e.g., by computing themodular multiplicative inverse of "e" modulo $varphi(n),$:

#:$d=2753$

#:17 * 2753 = 46801 = 1 + 15 * 3120.

The**public key**is ($n=3233$, $e=17$). For a padded message $m,$ the encryption function is::$c\; =\; m^emod\; \{n\}\; =\; m^\{17\}\; mod\; \{3233\}.$

The

**private key**is ($n=3233$, $d=2753$). The decryption function is::$m\; =\; c^dmod\; \{n\}\; =\; c^\{2753\}\; mod\; \{3233\}.$

For example, to encrypt $m=123$, we calculate:$c\; =\; 123^\{17\}mod\; \{3233\}\; =\; 855.$

To decrypt $c\; =\; 855$, we calculate

:$m\; =\; 855^\{2753\}mod\; \{3233\}\; =\; 123$.

Both of these calculations can be computed efficiently using the

square-and-multiply algorithm formodular exponentiation .**Padding schemes**When used in practice, RSA is generally combined with some padding scheme. The goal of the padding scheme is to prevent a number of attacks that potentially work against RSA without padding:

*When encrypting with low encryption exponents (e.g., "e" = 3) and small values of the "m", (i.e. "m"<"n"

^{1/e}) the result of $m^e$ is strictly less than the modulus "n". In this case, ciphertexts can be easily decrypted by taking the "e"th root of the ciphertext over the integers.

*If the same clear text message is sent to "e" or more recipients in an encrypted way, and the receivers share the same exponent "e", but different "p", "q", and "n", then it is easy to decrypt the original clear text message via theChinese remainder theorem .Johan Håstad noticed that this attack is possible even if the cleartexts are not equal, but the attacker knows a linear relation between them [*Johan Håstad, "On using RSA with Low Exponent in a Public Key Network", Crypto 85*] . This attack was later improved byDon Coppersmith [*Don Coppersmith, "Small Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities", Journal of Cryptology, v. 10, n. 4, Dec. 1997*] .

*Because RSA encryption is a deterministic encryption algorithm – i.e., has no random component – an attacker can successfully launch achosen plaintext attack against the cryptosystem, by encrypting likely plaintexts under the public key and test if they are equal to the ciphertext. A cryptosystem is calledsemantically secure if an attacker cannot distinguish two encryptions from each other even if the attacker knows (or has chosen) the corresponding plaintexts. As described above, 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 $m\_1^em\_2^eequiv\; (m\_1m\_2)^epmod\{n\}.$ Because of this multiplicative property achosen-ciphertext attack is possible. E.g. an attacker, who wants to know the decryption of a ciphertext "c"="m"^{e}mod "n" may ask the holder of the secret key to decrypt an unsuspicious-looking ciphertext $c\text{'}\; =\; c\; r^emod\; n$ for some value "r" chosen by the attacker. Because of the multiplicative property $c\text{'}$ is the encryption of $mr\; mod\; n$. Hence, if the attacker is successful with the attack, he will learn $mr\; mod\; n$ from which he can derive the message "m" by multiplying "mr" with the modular inverse of "r" modulo "n".To avoid these problems, practical RSA implementations typically embed some form of structured, randomized padding into the value "m" before encrypting it. This padding ensures that "m" 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 "m" with some number of additional bits, the size of the un-padded message "M" must be somewhat smaller. RSA padding schemes must be carefully designed so as to prevent sophisticated attacks which may be facilitated by a predictable message structure. Early versions of the PKCS#1 standard (up to version 1.5) used a construction that turned RSA into a semantically secure encryption scheme. This version was later found vulnerable to a practical

adaptive chosen ciphertext attack . Later versions of the standard includeOptimal Asymmetric Encryption Padding (OAEP), which prevents these attacks. 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 ).**igning messages**Suppose Alice uses Bob'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 actually from Alice since anyone can use Bob's public key to send him encrypted messages. So, in order to verify the origin of a message, RSA can also be used to sign 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 of the message, raises it to the power of "d" mod "n" (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 "e" mod "n" (as he does when encrypting a message), and compares the resulting hash value with the message's actual hash value. If the two agree, he knows that the author of the message was in possession of Alice's secret key, and that the message has not been tampered with since.

Note that secure padding schemes such as

RSA-PSS are as essential for the security of message signing as they are for message encryption, and that the same key should never be used for both encryption and signing purposes.**ecurity**The security of the RSA cryptosystem is based on two mathematical problems: the problem of factoring large numbers and the

RSA problem . Full decryption of an RSA ciphertext is thought to be infeasible on the assumption that both of these problems are hard, 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 is defined as the task of taking "e"th roots modulo a composite "n": recovering a value "m" such that "c=m^{e}" mod n, where ("n", "e") is an RSA public key and "c" is an RSA ciphertext. Currently the most promising approach to solving the RSA problem is to factor the modulus "n". With the ability to recover prime factors, an attacker can compute the secret exponent "d" from a public key ("n", "e"), then decrypt "c" using the standard procedure. To accomplish this, an attacker factors "n" into "p" and "q", and computes ("p"-1)("q"-1) which allows the determination of "d" from "e". 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. Seeinteger factorization for a discussion of this problem.As of 2005 , the largest number factored by a general-purpose factoring algorithm was 663 bits long (seeRSA-200 ), using a state-of-the-art distributed implementation. RSA keys are typically 1024–2048 bits long. Some experts believe that 1024-bit keys may become breakable in the near term (though this is disputed); few see any way that 4096-bit keys could be broken in the foreseeable future. Therefore, it is generally presumed that RSA is secure if "n" is sufficiently large. If "n" is 256bit s or shorter, it can be factored in a few hours on apersonal computer , using software already freely available. Keys of 512 bits (or less) have been shown to be practically breakable in1999 whenRSA-155 was factored by using several hundred computers. A theoretical hardware device namedTWIRL and described by Shamir and Tromer in 2003 called into question the security of 1024 bit keys. It is currently recommended that "n" be at least 2048 bits long.In 1994,

Peter Shor publishedShor's algorithm , showing that aquantum computer could in principle perform the factorization inpolynomial time . However, quantum computation is still in the early stages of development and may never prove to be practical.**Practical considerations****Key generation**Finding the large primes "p" and "q" is usually done by testing random numbers of the right size with probabilistic

primality test s which quickly eliminate virtually all non-primes."p" and "q" should not be 'too close', lest the

Fermat factorization for "n" be successful, if p-q, for instance is less than 2n^{1/4}(which for even small 1024-bit values of n is 3x10^{77}) solving for p and q is trivial. Furthermore, if either "p"-1 or "q"-1 has only small prime factors, "n" can be factored quickly by Pollard's p − 1 algorithm, and these values of "p" or "q" should therefore be discarded as well.**peed****RSA**is much slower than DES and other symmetric cryptosystems. In practice, Bob typically encrypts a secret message with a symmetric algorithm, encrypts the (comparatively short) symmetric key with RSA, and transmits both the RSA-encrypted symmetric key and the symmetrically-encrypted message to Alice.This procedure raises additional security issues. For instance, it is of utmost importance to use a strong

random number generator for the symmetric key, because otherwise Eve (an eavesdropper wanting to see what was sent) could bypass RSA by guessing the symmetric key.**Key distribution**As with all ciphers, how RSA public keys are distributed is important to security. Key distribution must be secured against a

man-in-the-middle attack . Suppose Eve has some way to give Bob arbitrary keys and make him believe they belong to Alice. Suppose further that Eve can "intercept" transmissions between Alice and Bob. Eve sends Bob her own public key, which Bob believes to be Alice's. Eve can then intercept any ciphertext sent by Bob, decrypt it with her own secret key, keep a copy of the message, encrypt the message with Alice's public key, and send the new ciphertext to Alice. In principle, neither Alice nor Bob would be able to detect Eve's presence. Defenses against such attacks are often based ondigital certificate s or other components of apublic key infrastructure .**Timing attacks**Kocher 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, she can deduce the decryption key "d" quickly. This attack can also be applied against the**RSA**signature scheme. In2003 , Boneh and Brumley demonstrated a more practical attack capable of recovering**RSA**factorizations over a network connection (e.g., from aSecure Socket Layer (SSL)-enabled webserver). This attack takes advantage of information leaked by theChinese remainder theorem 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 "c^{d}mod n", Alice first chooses a secret random value "r" and computes "(r^{e}c)^{d}mod n". The result of this computation is "r m mod n" and so the effect of "r" can be removed by multiplying by its inverse. A new value of "r" 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 described the first practicaladaptive chosen ciphertext attack , 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 theSecure Socket Layer protocol, and to recover session keys. As a result of this work, cryptographers now recommend the use of provably secure padding schemes such asOptimal Asymmetric Encryption Padding , and RSA Laboratories has released new versions of PKCS #1 that are not vulnerable to these attacks.**Branch prediction analysis attacks**Branch prediction analysis is also called BPA. Many processors use a

branch predictor to determine whether a conditional branch in the instruction flow of a program is likely to be taken or not. Usually these processors also implementsimultaneous multithreading (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 andCetin Kaya Koc ) claim to have discovered 508 out of 512 bits of an RSA key in 10 iterations.**ee also***

Clifford Cocks

*Quantum cryptography

*Cryptographic key length

*Computational complexity theory

*Diffie-Hellman key exchange

*RSA Factoring Challenge

*List of software patents **Notes****References*** cite book

last = Menezes | first = Alfred

coauthors = Paul C. van Oorschot; Scott A. Vanstone

title = Handbook of Applied Cryptography

publisher = CRC Press | month = October | year = 1996

isbn = 0-8493-8523-7

* cite journal

first = R. | last = Rivest

coauthors = A. Shamir; L. Adleman

url = http://theory.lcs.mit.edu/~rivest/rsapaper.pdf

title = A Method for Obtaining Digital Signatures and Public-Key Cryptosystems

journal = Communications of the ACM

volume = 21 | issue = 2 | pages = pp.120-126 | year = 1978

* cite book

first = Thomas H. | last = Cormen | authorlink = Thomas H. Cormen

coauthors =Charles E. Leiserson ;Ronald L. Rivest ;Clifford Stein

title =Introduction to Algorithms

edition = 2e

publisher = MIT Press and McGraw-Hill | year = 2001

isbn = 0-262-03293-7 | pages = pp.881-887**External links*** [

*http://patft.uspto.gov/netacgi/nph-Parser?Sect1=PTO2&Sect2=HITOFF&p=1&u=%2Fnetahtml%2FPTO%2Fsearch-bool.html&r=1&f=G&l=50&co1=AND&d=PTXT&s1=4,405,829.PN.&OS=PN/4,405,829&RS=PN/4,405,829 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, Patent Number 4405829.

* [*http://www.rsasecurity.com/rsalabs/node.asp?id=2125 PKCS #1: RSA Cryptography Standard*] (RSA Laboratories website)

** The "PKCS #1" standard "provides recommendations for the implementation ofpublic-key cryptography based on the**RSA**algorithm, covering the following aspects: cryptographic primitives;encryption schemes; signature schemes with appendix;ASN.1 syntax for representing keys and for identifying the schemes".

* [*http://www.di-mgt.com.au/rsa_alg.html Thorough walk through of RSA*]

* [*http://www.securecottage.com/demo/rsa2.html RSA demo with Java and Javascript*]

* [*http://cisnet.baruch.cuny.edu/holowczak/classes/9444/rsademo/ RSA demo Applet*]

* [*http://www.muppetlabs.com/~breadbox/txt/rsa.html How the RSA Cipher Works*]

* [*http://www.cacr.math.uwaterloo.ca/hac/ Menezes, Oorschot, Vanstone, Scott: "Handbook of Applied Cryptography" (free PDF downloads), see Chapter 8*]

* [*http://eprint.iacr.org/2006/351 Onur Aciicmez, Cetin Kaya Koc, Jean-Pierre Seifert: "On the Power of Simple Branch Prediction Analysis"*]

* [*http://blog.cacert.org/2006/11/193.html A New Vulnerability In RSA Cryptography, CAcert NEWS Blog*]

* [*http://xyssl.org/code/source/rsa/ Example of an RSA implementation with PKCS#1 padding (LGPL source code)*]

* [*http://www.cryptography.com/resources/whitepapers/TimingAttacks.pdf Kocher's article about timing attacks*]

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