An Encryption Key That Can Be Shared and Does Not Need to Be Kept Private Is Referred to as a:

Algorithm for public-central cryptography

This article is about a cryptosystem. For the company, run across RSA Security.

RSA

Full general
Designers	Ron Rivest, Adi Shamir, and Leonard Adleman
Offset published	1977
Certification	PKCS#i, ANSI X9.31, IEEE 1363
Zilch item
Primal sizes	two,048 to 4,096 bit typical
Rounds	one
Best public cryptanalysis
Full general number field sieve for classical computers; Shor's algorithm for quantum computers. An 829-scrap primal has been cleaved.

RSA (Rivest–Shamir–Adleman) is a public-central cryptosystem that is widely used for secure information manual. It is also i of the oldest. The acronym "RSA" comes from the surnames of Ron Rivest, Adi Shamir and Leonard Adleman, who publicly described the algorithm in 1977. An equivalent organization was developed secretly in 1973 at GCHQ (the British signals intelligence agency) past the English mathematician Clifford Cocks. That organisation was declassified in 1997.^[1]

In a public-cardinal cryptosystem, the encryption key is public and distinct from the decryption key, which is kept secret (private). An RSA user creates and publishes a public cardinal based on two big prime number numbers, forth with an auxiliary value. The prime numbers are kept secret. Messages can exist encrypted by anyone, via the public key, but tin only be decoded past someone who knows the prime numbers.^[2]

The security of RSA relies on the applied difficulty of factoring the product of two large prime numbers, the "factoring trouble". Breaking RSA encryption is known every bit the RSA problem. Whether it is as difficult as the factoring problem is an open question.^[three] In that location are no published methods to defeat the system if a large plenty key is used.

RSA is a relatively wearisome algorithm. Considering of this, it is not commonly used to straight encrypt user information. More often, RSA is used to transmit shared keys for symmetric-cardinal cryptography, which are then used for majority encryption–decryption.

History [edit]

The idea of an asymmetric public-private key cryptosystem is attributed to Whitfield Diffie and Martin Hellman, who published this concept in 1976. They also introduced digital signatures and attempted to use number theory. Their formulation used a shared-secret-key created from exponentiation of some number, modulo a prime. Notwithstanding, they left open the problem of realizing a ane-way role, possibly considering the difficulty of factoring was not well-studied at the time.^[4]

Ron Rivest, Adi Shamir, and Leonard Adleman at the Massachusetts Found of Technology fabricated several attempts over the course of a twelvemonth to create a 1-manner function that was hard to invert. Rivest and Shamir, as calculator scientists, proposed many potential functions, while Adleman, as a mathematician, was responsible for finding their weaknesses. They tried many approaches, including "knapsack-based" and "permutation polynomials". For a time, they thought what they wanted to attain was incommunicable due to contradictory requirements.^[v] In April 1977, they spent Passover at the house of a student and drank a proficient bargain of Manischewitz wine earlier returning to their homes at around midnight.^{[half dozen]} Rivest, unable to slumber, lay on the couch with a math textbook and started thinking about their one-mode function. He spent the remainder of the night formalizing his idea, and he had much of the newspaper prepare past daybreak. The algorithm is at present known as RSA – the initials of their surnames in same order as their paper.^[7]

Clifford Cocks, an English mathematician working for the British intelligence agency Government Communications Headquarters (GCHQ), described an equivalent system in an internal document in 1973.^[eight] Nevertheless, given the relatively expensive computers needed to implement information technology at the fourth dimension, information technology was considered to exist mostly a curiosity and, as far as is publicly known, was never deployed. His discovery, still, was non revealed until 1997 due to its top-secret classification.

Kid-RSA (KRSA) is a simplified public-central cipher published in 1997, designed for educational purposes. Some people experience that learning Kid-RSA gives insight into RSA and other public-primal ciphers, analogous to simplified DES.^{[9] [10] [11] [12] [13]}

Patent [edit]

A patent describing the RSA algorithm was granted to MIT on 20 September 1983: U.S. Patent 4,405,829 "Cryptographic communications organization and method". From DWPI's abstract of the patent:

The system includes a communications channel coupled to at to the lowest degree one last having an encoding device and to at least ane terminal having a decoding device. A bulletin-to-be-transferred is enciphered to ciphertext at the encoding terminal past encoding the message every bit a number Chiliad in a predetermined set. That number is then raised to a beginning predetermined power (associated with the intended receiver) and finally computed. The remainder or residue, C, is... computed when the exponentiated number is divided by the product of two predetermined prime number numbers (associated with the intended receiver).

A detailed description of the algorithm was published in August 1977, in Scientific American's Mathematical Games column.^[7] This preceded the patent's filing appointment of December 1977. Consequently, the patent had no legal continuing outside the The states. Had Cocks's work been publicly known, a patent in the United States would non have been legal either.

When the patent was issued, terms of patent were 17 years. The patent was most to elapse on 21 September 2000, only RSA Security released the algorithm to the public domain on 6 September 2000.^[14]

Operation [edit]

The RSA algorithm involves four steps: key generation, key distribution, encryption, and decryption.

A basic principle behind RSA is the observation that it is practical to find three very big positive integers e, d, and northward, such that with modular exponentiation for all integers m (with 0 ≤ m < northward ):

${\displaystyle (m^{e})^{d}\equiv k{\pmod {northward}}}$

and that knowing eastward and northward, or even m, it tin be extremely difficult to find d. The triple bar (≡) hither denotes modular congruence.

In improver, for some operations information technology is user-friendly that the society of the two exponentiations tin be changed and that this relation also implies

${\displaystyle (chiliad^{d})^{due east}\equiv m{\pmod {n}}.}$

RSA involves a public key and a individual fundamental. The public key can be known past everyone and is used for encrypting messages. The intention is that messages encrypted with the public fundamental can only be decrypted in a reasonable amount of fourth dimension by using the private key. The public key is represented by the integers n and due east, and the individual key by the integer d (although northward is also used during the decryption process, and so it might be considered to be a part of the individual key too). m represents the bulletin (previously prepared with a certain technique explained beneath).

Key generation [edit]

The keys for the RSA algorithm are generated in the following way:

1. Choose ii distinct prime number numbers p and q.
   • For security purposes, the integers p and q should be chosen at random and should be like in magnitude simply differ in length by a few digits to make factoring harder.^[two] Prime number integers can be efficiently establish using a primality examination.
   • p and q are kept secret.
2. Compute north = pq .
   • north is used as the modulus for both the public and private keys. Its length, normally expressed in bits, is the key length.
   • n is released as part of the public key.
3. Compute λ(n), where λ is Carmichael's totient function. Since n = pq, λ(north) = lcm(λ(p),λ(q)), and since p and q are prime number, λ(p) = φ(p) = p − one, and also λ(q) = q − 1. Hence λ(n) = lcm(p − 1, q − 1).
   • λ(n) is kept secret.
   • The lcm may be calculated through the Euclidean algorithm, since lcm(a,b) = |ab|/gcd(a,b).
4. Choose an integer due east such that 1 < e < λ(north) and gcd(e, λ(n)) = one; that is, e and λ(due north) are coprime.
   • east having a brusk bit-length and small Hamming weight results in more efficient encryption – the most commonly chosen value for east is 2¹⁶ + ane = 65537 . The smallest (and fastest) possible value for eastward is 3, simply such a minor value for due east has been shown to be less secure in some settings.^[xv]
   • due east is released as function of the public key.
5. Determine d as d ≡ east ⁻¹ (modern λ(north)); that is, d is the modular multiplicative changed of e modulo λ(n).
   • This ways: solve for d the equation d⋅east ≡ 1 (mod λ(n)); d can be computed efficiently by using the extended Euclidean algorithm, since, thanks to due east and λ(n) being coprime, said equation is a form of Bézout's identity, where d is 1 of the coefficients.
   • d is kept secret as the private key exponent.

The public fundamental consists of the modulus n and the public (or encryption) exponent e. The individual key consists of the private (or decryption) exponent d, which must be kept secret. p, q, and λ(due north) must likewise be kept underground because they can exist used to calculate d. In fact, they can all be discarded after d has been computed.^[sixteen]

In the original RSA newspaper,^[2] the Euler totient part φ(northward) = (p − 1)(q − 1) is used instead of λ(n) for computing the individual exponent d. Since φ(n) is ever divisible by λ(due north), the algorithm works as well. The possibility of using Euler totient office results as well from Lagrange'southward theorem practical to the multiplicative group of integers modulo pq. Thus any d satisfying d⋅east ≡ 1 (mod φ(n)) also satisfies d⋅e ≡ ane (modern λ(n)). However, computing d modulo φ(north) will sometimes yield a upshot that is larger than necessary (i.eastward. d > λ(due north)). Most of the implementations of RSA will have exponents generated using either method (if they utilize the private exponent d at all, rather than using the optimized decryption method based on the Chinese remainder theorem described below), but some standards such as FIPS 186-4 may crave that d < λ(n). Whatsoever "oversized" individual exponents not meeting this benchmark may always be reduced modulo λ(n) to obtain a smaller equivalent exponent.

Since any common factors of (p − ane) and (q − ane) are present in the factorisation of n − 1 = pq − 1 = (p − 1)(q − 1) + (p − ane) + (q − ane),^[17] it is recommended that (p − one) and (q − 1) have but very minor common factors, if any, besides the necessary ii.^{[ii] [eighteen] [19] [twenty]}

Annotation: The authors of the original RSA paper carry out the cardinal generation by choosing d and and then computing e as the modular multiplicative changed of d modulo φ(n), whereas nigh current implementations of RSA, such equally those post-obit PKCS#i, exercise the contrary (choose e and compute d). Since the chosen fundamental tin be pocket-size, whereas the computed primal normally is not, the RSA paper's algorithm optimizes decryption compared to encryption, while the modernistic algorithm optimizes encryption instead.^{[2] [21]}

Key distribution [edit]

Suppose that Bob wants to transport 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 letters, Alice transmits her public cardinal (north, eastward) to Bob via a reliable, but not necessarily secret, route. Alice's private key (d) is never distributed.

Encryption [edit]

Later on Bob obtains Alice'south public primal, he can transport a message M to Alice.

To do it, he outset turns M (strictly speaking, the un-padded plaintext) into an integer thousand (strictly speaking, the padded plaintext), such that 0 ≤ yard < n past using an agreed-upon reversible protocol known as a padding scheme. He then computes the ciphertext c, using Alice'southward public primal e, corresponding to

${\displaystyle c\equiv thou^{e}{\pmod {n}}.}$

This can be washed reasonably quickly, even for very large numbers, using modular exponentiation. Bob then transmits c to Alice. Notation that at least nine values of m volition yield a ciphertext c equal to k,^[22] but this is very unlikely to occur in practice.

Decryption [edit]

Alice can recover one thousand from c by using her private key exponent d by calculating

${\displaystyle c^{d}\equiv (one thousand^{e})^{d}\equiv k{\pmod {n}}.}$

Given thou, she can recover the original message M past reversing the padding scheme.

Example [edit]

Here is an example of RSA encryption and decryption. The parameters used hither are artificially pocket-size, simply one can besides use OpenSSL to generate and examine a real keypair.

1. Choose two distinct prime numbers, such equally

   ${\displaystyle p=61}$ and ${\displaystyle q=53.}$

2. Compute n = pq giving

   ${\displaystyle n=61\times 53=3233.}$

3. Compute the Carmichael's totient office of the product as λ(due north) = lcm(p − 1, q − 1) giving

   ${\displaystyle \lambda (3233)=\operatorname {lcm} (60,52)=780.}$

4. Choose any number 1 < e < 780 that is coprime to 780. Choosing a prime number for e leaves us simply to check that eastward is not a divisor of 780.

   Permit ${\displaystyle e=17}$ .

5. Compute d, the modular multiplicative inverse of e (mod λ(n)), yielding
   

   $${\displaystyle d=413,}$$

   

   as ${\displaystyle 1=(17\times 413){\bmod {7}}eighty.}$

The public fundamental is ( n = 3233, due east = 17). For a padded plaintext message m, the encryption function is

${\displaystyle {\begin{aligned}c(thou)&=m^{eastward}{\bmod {n}}\\&=grand^{17}{\bmod {3}}233.\end{aligned}}}$

The private key is ( n = 3233, d = 413). For an encrypted ciphertext c, the decryption role is

${\displaystyle {\begin{aligned}grand(c)&=c^{d}{\bmod {n}}\\&=c^{413}{\bmod {3}}233.\end{aligned}}}$

For case, in club to encrypt k = 65, nosotros summate

${\displaystyle c=65^{17}{\bmod {3}}233=2790.}$

To decrypt c = 2790, we summate

${\displaystyle one thousand=2790^{413}{\bmod {3}}233=65.}$

Both of these calculations can be computed efficiently using the square-and-multiply algorithm for modular exponentiation. In real-life situations the primes selected would be much larger; in our example it would exist trivial to factor n = 3233 (obtained from the freely available public central) back to the primes p and q. e, also from the public key, is then inverted to get d, thus acquiring the private key.

Practical implementations employ the Chinese residue theorem to speed up the calculation using modulus of factors (mod pq using mod p and mod q).

The values d _p , d _q and q _inv, which are part of the private central are computed as follows:

${\displaystyle {\begin{aligned}d_{p}&=d{\bmod {(}}p-ane)=413{\bmod {(}}61-1)=53,\\d_{q}&=d{\bmod {(}}q-one)=413{\bmod {(}}53-1)=49,\\q_{\text{inv}}&=q^{-1}{\bmod {p}}=53^{-1}{\bmod {6}}1=38\\&\Rightarrow (q_{\text{inv}}\times q){\bmod {p}}=38\times 53{\bmod {half dozen}}1=1.\end{aligned}}}$

Here is how d _p , d _q and q _inv are used for efficient decryption (encryption is efficient by choice of a suitable d and e pair):

${\displaystyle {\brainstorm{aligned}m_{1}&=c^{d_{p}}{\bmod {p}}=2790^{53}{\bmod {six}}ane=iv,\\m_{2}&=c^{d_{q}}{\bmod {q}}=2790^{49}{\bmod {5}}3=12,\\h&=(q_{\text{inv}}\times (m_{ane}-m_{2})){\bmod {p}}=(38\times -8){\bmod {vi}}1=one,\\m&=m_{2}+h\times q=12+1\times 53=65.\end{aligned}}}$

Signing messages [edit]

Suppose Alice uses Bob's public primal to send him an encrypted message. In the message, she tin claim to exist Alice, but Bob has no way of verifying that the message was from Alice, since anyone can use Bob'due south public key to send him encrypted messages. In society to verify the origin of a message, RSA tin can also be used to sign a message.

Suppose Alice wishes to ship a signed message to Bob. She can employ her own private key to do so. She produces a hash value of the message, raises it to the power of d (modulo n) (every bit she does when decrypting a message), and attaches it every bit a "signature" to the message. When Bob receives the signed message, he uses the aforementioned hash algorithm in conjunction with Alice's public cardinal. He raises the signature to the power of east (modulo due north) (as he does when encrypting a message), and compares the resulting hash value with the message's hash value. If the 2 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 rules:

${\displaystyle h=\operatorname {hash} (thou),}$

${\displaystyle (h^{eastward})^{d}=h^{ed}=h^{de}=(h^{d})^{due east}\equiv h{\pmod {n}}.}$

Thus the keys may be swapped without loss of generality, that is, a private cardinal of a cardinal pair may be used either to:

1. Decrypt a bulletin but intended for the recipient, which may be encrypted by anyone having the public primal (asymmetric encrypted transport).
2. Encrypt a message which may exist decrypted by anyone, only which can only be encrypted by one person; this provides a digital signature.

Proofs of correctness [edit]

Proof using Fermat's fiddling theorem [edit]

The proof of the correctness of RSA is based on Fermat's little theorem, stating that a ^{p − 1} ≡ 1 (mod p) for any integer a and prime number p, not dividing a.^{[note 1]}

We want to testify that

${\displaystyle (k^{due east})^{d}\equiv thou{\pmod {pq}}}$

for every integer 1000 when p and q are singled-out prime numbers and e and d are positive integers satisfying ed ≡ 1 (mod λ(pq)).

Since λ(pq) = lcm(p − 1, q − 1) is, by construction, divisible by both p − 1 and q − 1, we can write

${\displaystyle ed-1=h(p-1)=k(q-1)}$

for some nonnegative integers h and k.^{[note 2]}

To check whether two numbers, such as k^ed and m, are congruent modpq, it suffices (and in fact is equivalent) to cheque that they are congruent modp and modq separately. ^{[note iii]}

To show m^ed ≡ yard (mod p), we consider 2 cases:

1. If m ≡ 0 (mod p), m is a multiple of p. Thus m^ed is a multiple of p. So m^ed ≡ 0 ≡ m (modernistic p).
2. If m ${\displaystyle \non \equiv }$ 0 (mod p),

${\displaystyle one thousand^{ed}=k^{ed-1}one thousand=m^{h(p-1)}g=(m^{p-1})^{h}k\equiv 1^{h}1000\equiv m{\pmod {p}},}$

where we used Fermat'south little theorem to replace g ^p−ane mod p with 1.

The verification that m^ed ≡ k (mod q) proceeds in a completely analogous fashion:

1. If thou ≡ 0 (mod q), m^ed is a multiple of q. So m^ed ≡ 0 ≡ 1000 (mod q).
2. If m ${\displaystyle \non \equiv }$ 0 (mod q),

${\displaystyle one thousand^{ed}=thou^{ed-1}m=yard^{k(q-1)}thou=(m^{q-1})^{k}chiliad\equiv 1^{k}grand\equiv thou{\pmod {q}}.}$

This completes the proof that, for whatever integer 1000, and integers e, d such that ed ≡ i (modern λ(pq)),

${\displaystyle (1000^{due east})^{d}\equiv one thousand{\pmod {pq}}.}$

Notes:

1. ^ We cannot trivially interruption RSA past applying the theorem (mod pq) considering pq is not prime number.
2. ^ In particular, the statement to a higher place holds for any e and d that satisfy ed ≡ i (modernistic (p − ane)(q − 1)), since (p − 1)(q − ane) is divisible by λ(pq), and thus trivially besides by p − 1 and q − ane. Notwithstanding, in modern implementations of RSA, it is common to utilize a reduced private exponent d that but satisfies the weaker, but sufficient condition ed ≡ 1 (modernistic λ(pq)).
3. ^ This is office of the Chinese remainder theorem, although information technology is not the pregnant function of that theorem.

Proof using Euler'southward theorem [edit]

Although the original newspaper of Rivest, Shamir, and Adleman used Fermat'southward piffling theorem to explicate why RSA works, it is mutual to find proofs that rely instead on Euler'south theorem.

We want to testify that thou^ed ≡ m (mod n), where n = pq is a product of two different prime numbers, and e and d are positive integers satisfying ed ≡ 1 (mod φ(n)). Since eastward and d are positive, we tin can write ed = 1 + hφ(n) for some not-negative integer h. Assuming that 1000 is relatively prime to n, we have

${\displaystyle thou^{ed}=m^{1+h\varphi (north)}=yard(k^{\varphi (n)})^{h}\equiv thousand(1)^{h}\equiv m{\pmod {n}},}$

where the second-final congruence follows from Euler's theorem.

More than generally, for any east and d satisfying ed ≡ ane (modern λ(n)), the same conclusion follows from Carmichael's generalization of Euler's theorem, which states that chiliad ^λ(n) ≡ 1 (mod n) for all chiliad relatively prime number to due north.

When m is non relatively prime to n, the statement only given is invalid. This is highly improbable (only a proportion of one/p + 1/q − one/(pq) numbers take this belongings), but even in this case, the desired congruence is nonetheless truthful. Either yard ≡ 0 (mod p) or m ≡ 0 (modern q), and these cases tin exist treated using the previous proof.

Padding [edit]

Attacks against plain RSA [edit]

There are a number of attacks against plain RSA as described beneath.

• When encrypting with depression encryption exponents (e.chiliad., e = iii) and small values of the 1000 (i.e., m < n ^1/e ), the effect of thou ^e is strictly less than the modulus n. In this case, ciphertexts can exist decrypted easily by taking the eth 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 eastward, but different p, q, and therefore n, and so it is piece of cake to decrypt the original clear-text message via the Chinese remainder theorem. Johan Håstad noticed that this attack is possible even if the clear texts are non equal, but the attacker knows a linear relation between them.^[23] This attack was later improved by Don Coppersmith (come across Coppersmith'south attack).^[24]
• Because RSA encryption is a deterministic encryption algorithm (i.due east., has no random component) an assaulter can successfully launch a called plaintext attack against the cryptosystem, by encrypting likely plaintexts under the public cardinal and test whether they are equal to the ciphertext. A cryptosystem is called semantically secure if an attacker cannot distinguish two encryptions from each other, even if the assailant knows (or has chosen) the respective plaintexts. RSA without padding is non semantically secure.^[25]
• RSA has the property that the production of 2 ciphertexts is equal to the encryption of the production of the respective plaintexts. That is, yard ₁ ^east g _two ^{due east} ≡ (m _one m _two) ^east (mod n). Because of this multiplicative holding, a called-ciphertext set on is possible. East.g., an attacker who wants to know the decryption of a ciphertext c ≡ m ^e (mod n) may ask the holder of the private key d to decrypt an unsuspicious-looking ciphertext c′ ≡ cr ^e (mod n) for some value r chosen past the assaulter. Because of the multiplicative property, c′ is the encryption of mr (mod n). Hence, if the aggressor is successful with the attack, they will acquire mr (mod due north), from which they can derive the message m past multiplying mr with the modular changed of r modulo n.^{[ citation needed ]}
• Given the individual exponent d, i can efficiently cistron the modulus n = pq. And given factorization of the modulus n = pq, one tin can obtain any individual fundamental (d′,n) generated against a public cardinal (e′, northward).^[fifteen]

Padding schemes [edit]

To avoid these issues, practical RSA implementations typically embed some course of structured, randomized padding into the value k before encrypting information technology. This padding ensures that chiliad does not autumn into the range of insecure plaintexts, and that a given bulletin, once padded, will encrypt to 1 of a big number of different possible ciphertexts.

Standards such equally 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 united nations-padded message M must be somewhat smaller. RSA padding schemes must be carefully designed so as to foreclose sophisticated attacks that may be facilitated by a predictable message construction. Early on versions of the PKCS#1 standard (upward to version 1.5) used a structure that appears to brand RSA semantically secure. Still, at Crypto 1998, Bleichenbacher showed that this version is vulnerable to a practical adaptive called-ciphertext attack. Furthermore, at Eurocrypt 2000, Coron et al.^[26] showed that for some types of messages, this padding does not provide a high enough level of security. Afterwards versions of the standard include Optimal Asymmetric Encryption Padding (OAEP), which prevents these attacks. Equally such, OAEP should be used in any new application, and PKCS#1 v1.5 padding should be replaced wherever possible. The PKCS#1 standard too incorporates processing schemes designed to provide additional security for RSA signatures, due east.1000. the Probabilistic Signature Scheme for RSA (RSA-PSS).

Secure padding schemes such every bit RSA-PSS are as essential for the security of message signing as they are for bulletin encryption. 2 United states of america patents on PSS were granted (U.Southward. Patent 6,266,771 and U.Southward. Patent 7,036,014); however, these patents expired on 24 July 2009 and 25 April 2010 respectively. Employ of PSS no longer seems to be burdened by patents.^{[ original research? ]} Notation that using unlike RSA key pairs for encryption and signing is potentially more secure.^[27]

Security and practical considerations [edit]

Using the Chinese remainder algorithm [edit]

For efficiency, many popular crypto libraries (such every bit OpenSSL, Coffee and .Net) use for decryption and signing the following optimization based on the Chinese rest theorem. The post-obit values are precomputed and stored as part of the private fundamental:

${\displaystyle p}$ and ${\displaystyle q}$  – the primes from the central generation,

${\displaystyle d_{P}=d{\pmod {p-1}},}$

${\displaystyle d_{Q}=d{\pmod {q-1}},}$

${\displaystyle q_{\text{inv}}=q^{-1}{\pmod {p}}.}$

These values let the recipient to compute the exponentiation g = c ^d (mod pq) more efficiently as follows:

${\displaystyle m_{ane}=c^{d_{P}}{\pmod {p}},}$

${\displaystyle m_{2}=c^{d_{Q}}{\pmod {q}},}$

${\displaystyle h=q_{\text{inv}}(m_{1}-m_{2}){\pmod {p}}}$ ,^[28]

${\displaystyle thou=m_{2}+hq{\pmod {pq}}.}$

This is more efficient than computing exponentiation by squaring, even though two modular exponentiations have to be computed. The reason is that these ii modular exponentiations both use a smaller exponent and a smaller modulus.

Integer factorization and RSA trouble [edit]

The security of the RSA cryptosystem is based on 2 mathematical issues: the problem of factoring big numbers and the RSA problem. Full decryption of an RSA ciphertext is thought to be infeasible on the supposition that both of these problems are difficult, i.due east., no efficient algorithm exists for solving them. Providing security against partial decryption may require the add-on of a secure padding scheme.^[29]

The RSA trouble is defined as the task of taking eth roots modulo a composite northward: recovering a value m such that c ≡ m ^e (modernistic north), 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 number factors, an attacker can compute the secret exponent d from a public key (northward, east), then decrypt c using the standard procedure. To reach this, an attacker factors n into p and q, and computes lcm(p − one, q − i) that allows the determination of d from e. No polynomial-time method for factoring large integers on a classical reckoner has yet been found, merely it has not been proven that none exists; see integer factorization for a discussion of this problem.

Multiple polynomial quadratic sieve (MPQS) tin can be used to factor the public modulus northward.

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.^[thirty] By 2009, Benjamin Moody could gene an 512-bit RSA key in 73 days using only public software (GGNFS) and his desktop computer (a dual-core Athlon64 with a ane,900 MHz CPU). Just less than v gigabytes of disk storage was required and about 2.v gigabytes of RAM for the sieving process.

Rivest, Shamir, and Adleman noted^[ii] that Miller has shown that – assuming the truth of the extended Riemann hypothesis – finding d from northward and e is as hard as factoring n into p and q (up to a polynomial time departure).^[31] However, Rivest, Shamir, and Adleman noted, in section Nine/D of their paper, that they had not institute a proof that inverting RSA is as hard every bit factoring.

Equally of 2020^[update], the largest publicly known factored RSA number had 829 bits (250 decimal digits, RSA-250).^[32] Its factorization, by a state-of-the-art distributed implementation, took approximately 2700 CPU years. In exercise, RSA keys are typically 1024 to 4096 bits long. In 2003, RSA Security estimated that 1024-bit keys were likely to become crackable by 2010.^[33] Equally of 2020, information technology is not known whether such keys can be cracked, just minimum recommendations have moved to at least 2048 bits.^[34] It is mostly presumed that RSA is secure if north is sufficiently big, outside of quantum calculating.

If n is 300 bits or shorter, it can be factored in a few hours in a personal computer, using software already freely bachelor. Keys of 512 bits have been shown to exist practically brittle in 1999, when RSA-155 was factored by using several hundred computers, and these are now factored in a few weeks using common hardware. Exploits using 512-bit lawmaking-signing certificates that may have been factored were reported in 2011.^[35] A theoretical hardware device named TWIRL, described past Shamir and Tromer in 2003, chosen into question the security of 1024-scrap keys.^[33]

In 1994, Peter Shor showed that a breakthrough calculator – if ane could ever be practically created for the purpose – would be able to factor in polynomial time, breaking RSA; see Shor'south algorithm.

Faulty key generation [edit]

Finding the large primes p and q is commonly done by testing random numbers of the correct size with probabilistic primality tests that quickly eliminate near all of the nonprimes.

The numbers p and q should not be "too close", lest the Fermat factorization for n be successful. If p − q is less than 2n ^i/iv (n = p⋅q, which even for "pocket-sized" 1024-bit values of n is 3×10⁷⁷ ), solving for p and q is trivial. Furthermore, if either p − one or q − 1 has only small prime factors, due north can be factored quickly by Pollard'due south p − 1 algorithm, and hence such values of p or q should be discarded.

It is important that the private exponent d be large enough. Michael J. Wiener showed that if p is between q and 2q (which is quite typical) and d < northward ^i/4/three, then d can be computed efficiently from n ande.^[36]

At that place is no known attack against small public exponents such equally e = 3, provided that the proper padding is used. Coppersmith's set on has many applications in attacking RSA specifically if the public exponent e is small and if the encrypted message is short and not padded. 65537 is a ordinarily used value fore; this value can be regarded as a compromise between avoiding potential minor-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 non allow public exponents due east smaller than 65537, simply does not state a reason for this restriction.

In October 2017, a team of researchers from Masaryk University appear the ROCA vulnerability, which affects RSA keys generated by an algorithm embodied in a library from Infineon known equally RSALib. A large number of smart cards and trusted platform modules (TPM) were shown to be affected. Vulnerable RSA keys are easily identified using a exam programme the team released.^[37]

Importance of strong random number generation [edit]

A cryptographically strong random number generator, which has been properly seeded with adequate entropy, must be used to generate the primes p and q. An analysis comparing millions of public keys gathered from the Net was carried out in early on 2012 past Arjen K. Lenstra, James P. Hughes, Maxime Augier, Joppe W. Bos, Thorsten Kleinjung and Christophe Wachter. They were able to factor 0.two% of the keys using merely Euclid's algorithm.^{[38] [39]}

They exploited a weakness unique to cryptosystems based on integer factorization. If n = pq is ane public key, and n′ = p′q′ is another, so if by risk p = p′ (but q is non equal to q′), then a simple computation of gcd(n, n′) = p factors both north and n′, totally compromising both keys. Lenstra et al. note that this trouble can exist minimized by using a strong random seed of flake length twice the intended security level, or by employing a deterministic function to choose q given p, instead of choosing p and q independently.

Nadia Heninger was role of a group that did a similar experiment. They used an idea of Daniel J. Bernstein to compute the GCD of each RSA fundamental n against the product of all the other keys n′ they had plant (a 729-million-digit number), instead of computing each gcd(n, north′) separately, thereby achieving a very pregnant speedup, since after one large sectionalization, 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 30 manufacturers. Heninger explains that the i-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 2d primes. Using seeds of sufficiently loftier entropy obtained from key stroke timings or electronic diode dissonance or atmospheric noise from a radio receiver tuned between stations should solve the trouble.^[40]

Strong random number generation is important throughout every stage of public-key cryptography. For instance, if a weak generator is used for the symmetric keys that are being distributed past RSA, then an eavesdropper could bypass RSA and guess the symmetric keys direct.

Timing attacks [edit]

Kocher described a new assail on RSA in 1995: if the assailant Eve knows Alice'south hardware in sufficient detail and is able to measure out the decryption times for several known ciphertexts, Eve tin can deduce the decryption key d speedily. This attack tin can also be applied against the RSA signature scheme. In 2003, Boneh and Brumley demonstrated a more than applied attack capable of recovering RSA factorizations over a network connection (e.yard., from a Secure Sockets Layer (SSL)-enabled webserver).^[41] This attack takes reward of information leaked by the Chinese remainder theorem optimization used past many RSA implementations.

One fashion to thwart these attacks is to ensure that the decryption operation takes a constant amount of time for every ciphertext. Nevertheless, this approach can significantly reduce operation. Instead, well-nigh 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 due north), Alice first chooses a hole-and-corner random value r and computes (r ^e c) ^d (modern due north). The issue of this computation, after applying Euler'due south theorem, is rc ^d (modern north), and and then the effect of r tin can exist removed past 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 set on fails.

Adaptive chosen-ciphertext attacks [edit]

In 1998, Daniel Bleichenbacher described the get-go practical adaptive chosen-ciphertext assail confronting RSA-encrypted messages using the PKCS #1 v1 padding scheme (a padding scheme randomizes and adds construction to an RSA-encrypted message, then information technology is possible to determine whether a decrypted bulletin is valid). Due to flaws with the PKCS #1 scheme, Bleichenbacher was able to mount a practical assault against RSA implementations of the Secure Sockets Layer protocol and to recover session keys. As a consequence of this piece of work, cryptographers at present recommend the employ of provably secure padding schemes such as Optimal Disproportionate 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.^{[42] [43]} It impacted the Mozilla NSS Crypto Library, which was used notably past Firefox and Chrome.

Side-channel analysis attacks [edit]

A side-channel attack using co-operative-prediction assay (BPA) has been described. Many processors use a branch predictor to determine whether a provisional branch in the instruction period of a program is probable to be taken or not. Often these processors too implement simultaneous multithreading (SMT). Co-operative-prediction analysis attacks use a spy procedure to discover (statistically) the private cardinal when candy with these processors.

Uncomplicated Co-operative Prediction Analysis (SBPA) claims to meliorate BPA in a non-statistical way. In their paper, "On the Power of Simple Branch Prediction Analysis",^[44] the authors of SBPA (Onur Aciicmez and Cetin Kaya Koc) claim to accept discovered 508 out of 512 bits of an RSA key in 10 iterations.

A power-mistake attack on RSA implementations was described in 2010.^[45] The author recovered the key by varying the CPU power voltage outside limits; this acquired multiple power faults on the server.

Catchy implementation [edit]

There are many details to go along in mind in order to implement RSA deeply (strong PRNG, acceptable public exponent...) ^[46]. This makes the implementation challenging, to the point the book Applied Cryptography With Get suggest to avert RSA if possible.

Implementations [edit]

Some cryptography libraries that provide support for RSA include:

• Botan
• Bouncy Castle
• cryptlib
• Crypto++
• Libgcrypt
• Nettle
• OpenSSL
• wolfCrypt
• GnuTLS
• mbed TLS
• LibreSSL

Run into too [edit]

• Audio-visual cryptanalysis
• Computational complexity theory
• Cryptographic primal length
• Diffie–Hellman cardinal exchange
• Key exchange
• Key management
• Elliptic-curve cryptography
• Public-key cryptography
• Trapdoor part

References [edit]

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2. ^ ^{a b c d east f} Rivest, R.; Shamir, A.; Adleman, Fifty. (Feb 1978). "A Method for Obtaining Digital Signatures and Public-Key Cryptosystems" (PDF). Communications of the ACM. 21 (two): 120–126. CiteSeerX10.one.1.607.2677. doi:10.1145/359340.359342. S2CID 2873616.
3. ^ Casteivecchi, Davide, Quantum-computing pioneer warns of complacency over Internet security, Nature, Oct 30, 2022 interview of Peter Shor.
4. ^ Diffie, W.; Hellman, One thousand. Eastward. (Nov 1976). "New directions in cryptography". IEEE Transactions on Information Theory. 22 (6): 644–654. CiteSeerXx.1.1.37.9720. doi:ten.1109/TIT.1976.1055638. ISSN 0018-9448.
5. ^ Rivest, Ronald. "The Early Days of RSA – History and Lessons" (PDF).
6. ^ Calderbank, Michael (2007-08-xx). "The RSA Cryptosystem: History, Algorithm, Primes" (PDF).
7. ^ ^{a b} Robinson, Sara (June 2003). "Still Guarding Secrets after Years of Attacks, RSA Earns Accolades for its Founders" (PDF). SIAM News. 36 (five).
8. ^ Cocks, C. C. (20 Nov 1973). "A Notation on Non-Secret Encryption" (PDF). world wide web.gchq.gov.uk . Retrieved 2017-05-30 .
9. ^ Jim Sauerberg. "From Private to Public Key Ciphers in 3 Piece of cake Steps".
10. ^ Margaret Cozzens and Steven J. Miller. "The Mathematics of Encryption: An Unproblematic Introduction". p. 180.
11. ^ Alasdair McAndrew. "Introduction to Cryptography with Open-Source Software". p. 12.
12. ^ Surender R. Chiluka. "Public fundamental Cryptography".
13. ^ Neal Koblitz. "Cryptography As a Teaching Tool". Cryptologia, Vol. 21, No. 4 (1997).
14. ^ "RSA Security Releases RSA Encryption Algorithm into Public Domain". Archived from the original on June 21, 2007. Retrieved 2010-03-03 .
15. ^ ^{a b} Boneh, Dan (1999). "Twenty Years of attacks on the RSA Cryptosystem". Notices of the American Mathematical Lodge. 46 (2): 203–213.
16. ^ Applied Cryptography, John Wiley & Sons, New York, 1996. Bruce Schneier, p. 467.
17. ^ McKee, James; Pinch, Richard (1998). "Further Attacks on Server-Aided RSA Cryptosystems". CiteSeerXten.1.i.33.1333.
18. ^ A Course in Number Theory and Cryptography, Graduate Texts in Math. No. 114, Springer-Verlag, New York, 1987. Neal Koblitz, 2nd edition, 1994. p. 94.
19. ^ Dukhovni, Viktor (July 31, 2015). "mutual factors in (p − 1) and (q − 1)". openssl-dev (Mailing list).
20. ^ Dukhovni, Viktor (Baronial i, 2015). "mutual factors in (p − 1) and (q − 1)". openssl-dev (Mailing list).
21. ^ Johnson, J.; Kaliski, B. (February 2003). Public-Key Cryptography Standards (PKCS) #1: RSA Cryptography Specifications Version 2.one. Network Working Group. doi:10.17487/RFC3447. RFC 3447. Retrieved 9 March 2016.
22. ^ Namely, the values of chiliad which are equal to −ane, 0, or ane modulo p while also equal to −1, 0, or one modulo q. At that place volition exist more values of m having c = m if p − ane or q − 1 has other divisors in common with due east − 1 as well 2 because this gives more values of thousand such that ${\displaystyle m^{e-ane}{\bmod {p}}=ane}$ or ${\displaystyle m^{due east-1}{\bmod {q}}=i}$ respectively.
23. ^ Håstad, Johan (1986). "On using RSA with Low Exponent in a Public Key Network". Advances in Cryptology — CRYPTO '85 Proceedings. Lecture Notes in Computer Scientific discipline. Vol. 218. pp. 403–408. doi:10.1007/iii-540-39799-X_29. ISBN978-3-540-16463-0.
24. ^ Coppersmith, Don (1997). "Minor Solutions to Polynomial Equations, and Low Exponent RSA Vulnerabilities" (PDF). Periodical of Cryptology. 10 (4): 233–260. CiteSeerXx.1.1.298.4806. doi:10.1007/s001459900030. S2CID 15726802.
25. ^ Due south. Goldwasser and S. Micali, Probabilistic encryption & how to play mental poker keeping secret all partial information, Almanac ACM Symposium on Theory of Computing, 1982.
26. ^ Coron, Jean-Sébastien; Joye, Marc; Naccache, David; Paillier, Pascal (2000). Preneel, Bart (ed.). "New Attacks on PKCS#i v1.5 Encryption". Advances in Cryptology — EUROCRYPT 2000. Lecture Notes in Information science. Berlin, Heidelberg: Springer. 1807: 369–381. doi:10.1007/three-540-45539-6_25. ISBN978-3-540-45539-4.
27. ^ "RSA Algorithm".
28. ^ If ${\displaystyle m_{1}<m_{2}}$ , then some^{[ clarification needed ]} libraries compute h as ${\displaystyle q_{\text{inv}}\left[\left(m_{1}+\left\lceil {\frac {q}{p}}\right\rceil p\correct)-m_{2}\right]{\pmod {p}}}$ .
29. ^ Machie, Edmond K. (29 March 2013). Network security traceback attack and react in the The states Department of Defense network. p. 167. ISBN978-1466985742.
30. ^ Lenstra, Arjen; et al. (Group) (2000). "Factorization of a 512-bit RSA Modulus" (PDF). Eurocrypt.
31. ^ Miller, Gary Fifty. (1975). "Riemann's Hypothesis and Tests for Primality" (PDF). Proceedings of Seventh Almanac ACM Symposium on Theory of Computing. pp. 234–239.
32. ^ Zimmermann, Paul (2020-02-28). "Factorization of RSA-250". Cado-nfs-discuss.
33. ^ ^{a b} Kaliski, Burt (2003-05-06). "TWIRL and RSA Primal Size". RSA Laboratories. Archived from the original on 2017-04-17. Retrieved 2017-eleven-24 .
34. ^ Barker, Elaine; Dang, Quynh (2015-01-22). "NIST Special Publication 800-57 Role 3 Revision one: Recommendation for Key Management: Awarding-Specific Key Management Guidance" (PDF). National Plant of Standards and Engineering science: 12. doi:ten.6028/NIST.SP.800-57pt3r1. Retrieved 2017-eleven-24 .
35. ^ Sandee, Michael (November 21, 2011). "RSA-512 certificates driveling in-the-wild". Fox-It International blog.
36. ^ Wiener, Michael J. (May 1990). "Cryptanalysis of brusk RSA secret exponents" (PDF). IEEE Transactions on Information Theory. 36 (3): 553–558. doi:10.1109/18.54902.
37. ^ Nemec, Matus; Sys, Marek; Svenda, Petr; Klinec, Dusan; Matyas, Vashek (November 2017). "The Return of Coppersmith's Attack: Applied Factorization of Widely Used RSA Moduli" (PDF). Proceedings of the 2022 ACM SIGSAC Briefing on Computer and Communications Security. CCS '17. doi:10.1145/3133956.3133969.
38. ^ Markoff, John (February xiv, 2012). "Flaw Found in an Online Encryption Method". The New York Times.
39. ^ Lenstra, Arjen Thousand.; Hughes, James P.; Augier, Maxime; Bos, Joppe Due west.; Kleinjung, Thorsten; Wachter, Christophe (2012). "Ron was wrong, Whit is right" (PDF).
40. ^ Heninger, Nadia (February 15, 2012). "New research: There's no need to panic over factorable keys–simply mind your Ps and Qs". Freedom to Tinker.
41. ^ Brumley, David; Boneh, Dan (2003). "Remote timing attacks are practical" (PDF). Proceedings of the 12th Conference on USENIX Security Symposium. SSYM'03.
42. ^ "'BERserk' Bug Uncovered In Mozilla NSS Crypto Library Impacts Firefox, Chrome". Retrieved four January 2022.
43. ^ "Mozilla Foundation Security Advisory 2014-73 - RSA Signature Forgery in NSS".
44. ^ Acıiçmez, Onur; Koç, Çetin Kaya; Seifert, Jean-Pierre (2007). "On the power of simple co-operative prediction assay". Proceedings of the 2d ACM Symposium on Information, Reckoner and Communications Security. ASIACCS '07. pp. 312–320. CiteSeerXx.1.1.80.1438. doi:x.1145/1229285.1266999 (inactive 31 October 2021). {{cite conference}}: CS1 maint: DOI inactive equally of October 2022 (link)
45. ^ Pellegrini, Andrea; Bertacco, Valeria; Austin, Todd (2010). "Fault-Based Attack of RSA Hallmark" (PDF).
46. ^ Isom, Kyle. "Pratical Cryptography With Become". Retrieved iv Jan 2022.

Further reading [edit]

• Menezes, Alfred; van Oorschot, Paul C.; Vanstone, Scott A. (October 1996). Handbook of Practical Cryptography . CRC Printing. ISBN978-0-8493-8523-0.
• Cormen, Thomas H.; Leiserson, Charles E.; Rivest, Ronald Fifty.; Stein, Clifford (2001). Introduction to Algorithms (2nd ed.). MIT Printing and McGraw-Hill. pp. 881–887. ISBN978-0-262-03293-3.

External links [edit]

• The Original RSA Patent as filed with the U.Due south. Patent Role past Rivest; Ronald L. (Belmont, MA), Shamir; Adi (Cambridge, MA), Adleman; Leonard M. (Arlington, MA), Dec 14, 1977, U.Due south. Patent iv,405,829 .
• PKCS #one: RSA Cryptography Standard (RSA Laboratories website)
  • The PKCS #1 standard "provides recommendations for the implementation of public-key cryptography based on the RSA algorithm, covering the post-obit aspects: cryptographic primitives; encryption schemes; signature schemes with appendix; ASN.1 syntax for representing keys and for identifying the schemes".
• Explanation of RSA using colored lamps on YouTube
• Thorough walk through of RSA
• Prime Hide-And-Seek: How the RSA Cipher Works
• Onur Aciicmez, Cetin Kaya Koc, Jean-Pierre Seifert: On the Power of Elementary Branch Prediction Analysis
• Example of an RSA implementation with PKCS#1 padding (GPL source lawmaking)
• Kocher'south article about timing attacks
• An animated explanation of RSA with its mathematical background past CrypTool
• Crud, James. "RSA Encryption". Numberphile. Brady Haran.
• How RSA Key used for Encryption in real world

crunkuncerew.blogspot.com

Source: https://en.wikipedia.org/wiki/RSA_(cryptosystem)

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