an asymmetric (e.g. public-key) cryptosystem, based on modular exponentiation with big exponents and modulus. RSA can be used both for signature and encryption.

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Why is elliptic curve cryptography not widely used, compared to RSA?

I recently ran across elliptic curve crypto-systems: An Introduction to the Theory of Elliptic Curves (Brown University) Elliptic Curve Cryptography (Wikipedia) Performance analysis of identity ...
58
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5answers
26k views

How big an RSA key is considered secure today?

I think 1024 bit RSA keys were considered secure ~5 years ago, but I assume that's not true anymore. Can 2048 or 4096 keys still be relied upon, or have we gained too much computing power in the ...
48
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6answers
36k views

How can I generate large prime numbers for RSA?

What is the currently industry-standard algorithm used to generate large prime numbers to be used in RSA encryption? I'm aware that I can find any number of articles on the Internet that explain how ...
47
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4answers
16k views

How can I use asymmetric encryption, such as RSA, to encrypt an arbitrary length of plaintext?

RSA is not designed to be used on long blocks of plaintext like a block cipher, but I need to use it to send a large (encrypted) message. How can I do this?
47
votes
1answer
3k views

Why does my SSH private key still work after changing some bytes in the file?

I (for a test) just randomly altered a private RSA key by opening it up in Vim and changing a few bytes. It is the private part of an SSH key pair used for logging in on a remote system. Puzzlingly, ...
40
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5answers
9k views

How are primes generated for RSA?

As I understand it, the RSA algorithm is based on finding two large primes (p and q) and multiplying them. The security aspect is based on the fact that it's difficult to factor it back into p and q. ...
38
votes
6answers
281 views

How to check that you got the right key when brute forcing an encryption?

How do you know when you have the right key when brute-forcing? Let's say that they test the right key. They then have to check that the decrypted text makes sense. To do so, they can test whether ...
35
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11answers
8k views

Now that quantum computers have been out for a while, has RSA been cracked?

D-wave systems has released a commercially viable quantum computer. This means in theory, that all asymmetric encryption algorithms — such as RSA — are now useless due to the speed at which quantum ...
34
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5answers
6k views

In RSA, how does the CPU deal with this huge modulus (8192 bits)?

Whilst I understand how the RSA algorithm works, I don't understand how the CPU operates when it needs to use the mod function with a huge number $n$, for example. ...
32
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3answers
463 views

How does RSA compute such enormous numbers?

So I have been reading and learning a lot about cryptography lately and in particular asymmetric ciphers such as RSA. One thing that I am curious about but never seems to be mentioned is how the ...
27
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2answers
12k views

Impacts of not using RSA exponent of 65537

This RFC says the RSA Exponent should be 65537. Why is that number recommended and what are the theoretical and practical impacts & risks of making that number higher or lower? What are the ...
27
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2answers
2k views

What security authorities and standards reject $e=3$ in RSA, when, and with what rationale?

In RSA, some security authorities and/or standards allow the public exponent $e=3$, others require or recommend $e>2^{16}$ (or perhaps some other minimum). I gathered the following: PKCS#1 allows ...
25
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1answer
32k views

How does RSA signature verification work?

I understand how the RSA algorithm works for encryption and decryption purposes but I don't get how signing is done. Here's what I (think) I know and is common practice: If I have a message that I ...
22
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6answers
2k views

Is it feasible to build an index of prime factors?

Would it be possible to break an RSA key, in for example 1 week of time, if the cracker have already spent X number of years building an index of primes by performing every permutation of existing ...
21
votes
3answers
42k views

RSA encryption with private key and decryption with a public key

When using the RSA cryptosystem, does it still work if you instead encrypt with the private key and decrypt with the public key? What about in the case of using RSA for sender authentication?
21
votes
2answers
2k views

Why is PuTTYgen-created RSA public exponent(e) not in {3,17,65537}?

As far as I know, RSA public exponent(e) should be one of {3,17,65537}. However, I found PuTTYgen-created RSA public exponent(e) is 0x25(37) by default,as follows, (PuTTYgen version: 0.66) I am ...
21
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1answer
10k views

Can you explain Bleichenbacher's CCA attack on PKCS#1 v1.5?

I've studied that the Bleichenbacher's CCA attack on PKCS#1 v1.5. is a base to many versions of attacks in the area. I'm trying to understand that attack, but every explanation I saw starts with the ...
21
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2answers
11k views

Definition of Textbook RSA

What is the definition of Textbook RSA? What are some of the properties of textbook RSA? How does it differ from other RSAs?
21
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2answers
20k views

Signatures: RSA compared to ECDSA

I'm signing very small messages using RSA, and the signature and public key are added to every message, which requires a lot of space compared to the actual content. I'm considering switching to ...
20
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6answers
8k views

Is Diffie-Hellman mathematically the same as RSA?

Is the Diffie-Hellman key exchange the same as RSA? Diffie Hellman allows key exchange on a observed wire – but so can RSA. Alice and Bob want to exchange a key – Big brother is watching everything. ...
20
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4answers
7k views

Security strength of RSA in relation with the modulus size

NIST SP 800-57 §5.6.1 p.62–64 specifies a correspondence between RSA modulus size $n$ and expected security strength $s$ in bits: ...
19
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4answers
4k views

Why does RSA need p and q to be prime numbers?

This is a clarifying thread after finding: What makes RSA secure by using prime numbers?. Despite the fact that this question was answered already, I am still struggling to really grasp the ...
18
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5answers
4k views

Does RSA work for any message M?

I decided to read the original RSA paper A Method for Obtaining Digital Signatures and Public-Key Cryptosystem because of a question I had about RSA (which is not the question I'm about to ask, but ...
16
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9answers
5k views

RSA with small exponents?

Just to establish notation with respect to the RSA protocol, let $n = pq$ be the product of two large primes and let $e$ and $d$ be the public and private exponents, respectively ($e$ is the inverse ...
16
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4answers
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Is Wiener's attack on RSA extendable to larger keys with low hamming weight?

Using small private exponents with RSA improves performance. However, it has been shown (Wiener, 1990) that if $\log d \leq \frac14 \log N$, the private exponent $d$ can be reconstructed from the ...
16
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3answers
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Why hash the message before signing it with RSA?

The diagram below illustrates the process of digitally signing a message with RSA: As diagram shows, the message is first hashed, and the signature is then computed on the hash, rather than on the ...
16
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2answers
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When to use RSA and when ElGamal asymmetric encryption

If i am not wrong in cryptography there are 2 basic cryptographic schemes for public key cryptography. RSA encryption whose security is based on the infeasibility of solving the factoring of big ...
16
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2answers
23k views

What's the fundamental difference between Diffie-Hellman and RSA?

What is the difference in the purpose of DH and RSA? Aren't they both public-key encryption?
15
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1answer
22k views

Calculating RSA private exponent when given public exponent and the modulus factors using extended euclid

When given $p = 5, q = 11, N = 55$ and $e = 17$, I'm trying to compute the RSA private key $d$. I can calculate $\varphi(N) = 40$, but my lecturer then says to use the extended Euclidean algorithm to ...
14
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3answers
2k views

Is there any more information on this RSA backdoor?

Someone posted this on reddit a few days ago, and I haven't found much about it except for links back to the same page. Here's the article http://programmingpraxis.com/2016/01/19/rsa-encryption-...
14
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2answers
7k views

Are there practical upper limits of RSA key lengths?

Suppose one wanted to use RSA encryption for the sole purpose of sending key bits for use in symmetric crypto systems, a dedicated key exchange system so to speak. And say you didn't think that the ...
14
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4answers
4k views

Is it possible to validate a Public Key in RSA?

If I have a 1024-bit number, and someone is telling me that it is in fact a valid RSA public key, is there any way I can quickly validate that it is indeed so (without cracking RSA)? (I suppose I am ...
14
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1answer
4k views

Assuming a 1024qb quantum computer, how long to brute force 1024bit RSA, 256bit AES and 512bit SHA512

Assuming in the future there was a functioning 1024 qubit quantum supercomputer and it could run Shor's algorithm or Grover's algorithm to crack encryption very quickly. I'm interested in how the ...
13
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4answers
3k views

Why is RSA encryption key based on modulo $\varphi(n)$ rather than modulo $n$?

While calculating RSA encryption key we take modulo $\varphi(n)$ rather than modulo $n$. I can’t understand why it’s done this way.
13
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4answers
2k views

Why are RSA key sizes almost always a power of two?

I know that other bit sizes are possible, e.g. this HTTPS server seems to have a 9000 bit key https://www.ssllabs.com/ssltest/analyze.html?d=qqq.gg, but it's very rare that one sees a key not of size ...
12
votes
1answer
5k views

Why do all SSH-RSA Keys begin with “AAAAB3NzaC1yc”?

My Friend and I have been generating a few ssh2-rsa keys and noticed that all the public keys began with "AAAAB3NzaC1yc". The similarity extended to "AAAAB3NzaC1yc2EAAAABIwAAAQEA" between two keys I ...
12
votes
1answer
646 views

RSA with probable primes

I am a bit of a newbie to RSA encryption, so please be patient. I understand that for a 4096 bit RSA, the numbers p and q should be prime. And to have the best security, the p and q should both be ...
12
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4answers
1k views

Are any of the major asymmetric ciphers distinguishable (EG, RSA, ECC)?

Related to "Is it possible to derive the encryption method from encrypted text?". Given ciphertexts generated by any of the major asymmetric ciphers (RSA, ElGamal, ECC, etc..) can these ciphertexts ...
12
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2answers
18k views

Difference between “Signature Algorithm” and “Signature Hash Algorithm” in X.509

What's the difference between the "Signature Algorithm" and the "Signature Hash Algorithm" found in an X.509 certificate? Why does it need a "Signature Hash Algorithm"? Edit: I'm creating the X....
11
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3answers
8k views

What is the relation between RSA & Fermat's little theorem?

I came across this while refreshing my cryptography brain cells. From the RSA algorithm I understand that it somehow depends on the fact that, given a large number (A) it is computationally ...
11
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4answers
1k views

RSA & DH at risk due to math advances, will this eventually affect elliptic curves too?

I was looking into the predictions by some researchers that RSA and Diffie-Hellman may not be secure in the next few years due to advances in math and being able to calculate the discrete logarithm ...
11
votes
5answers
14k views

What is the SSL private key file format?

I was researching about how to encrypt with RSA. I understood everything but not the format of the private keys. In the phpseclib (RSA in PHP), you can import your private key (private.key format) ...
11
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2answers
3k views

Can two different pairs of RSA key have the same modulus?

Can $n=pq$ be part of two different pairs of RSA keys? If such keys exist, say $(e_1,n)$ and $(e_2,n)$, how are they related? What will be the security concerns for the two users?
11
votes
2answers
1k views

RSA encryption using multiplication

Generally in RSA we encrypt as $m^e \pmod n$. Will RSA work if we replace the power by normal multiplication? $E = (m \times e) \mod n$ and decryption as $c \times d \mod n$. What will be $d$ ...
11
votes
4answers
17k views

Why is padding used for RSA encryption given that it is not a block cipher?

In AES we use some padded bytes at end of message to fit 128/256 byte blocks. But as RSA is not a block cipher why is padding used? Can the message size be any byte length (is the encrypting agent ...
11
votes
3answers
356 views

Is it possible to create an asymmetric cryptosystem where the private keys are not easily verifiable as such?

Plaintext that consists of an RSA key is easily recognizable as such, because it satisfies certain mathematical properties, in particular (See the answer for Why can an encrypted private key be brute ...
11
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2answers
269 views

Attacking RSA signature verification that ignores padding

I am looking at a RSA signature verification that is quite obviously flawed and am wondering if there's a way to exploit that flaw in practice. Signature is generated using RSA with PKCS 1.5 padding, ...
11
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1answer
2k views

What is the history of recommended RSA key sizes?

One can find up to date recommended key sizes for RSA at http://csrc.nist.gov/publications/nistpubs/800-131A/sp800-131A.pdf for example. In short it suggests a key size of at least 2048 bits. Is it ...
11
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1answer
2k views

Bleichenbacher 1998 “Million message attack” on RSA

I have been reading Bleichenbacher's 1998 paper on a forged message attack on RSA. The paper assumes access to an Oracle that takes a ciphertext $c$ and will check the decrypted text for valid PKCS #1 ...
10
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4answers
6k views

Why is it important that phi(n) is kept a secret, in RSA?

Why is it important that $\phi(n)$ is kept a secret, in RSA?