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# Questions tagged [multi-prime-rsa]

A variant of RSA where the modulus is the product of more than two primes.

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### Who first published the interest of more than two prime factors in RSA?

Multi-prime RSA is now a well known technique (described here): it uses $k>2$ distinct secret prime factors in the public RSA modulus, with the advantage that, using the CRT, we can gain a speed ...
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17 votes
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### RSA with 3 primes

I was trying to understand how does RSA with 3 primes work. I have checked Wikipedia but yet I didn’t fully understand their solution. I would like to know how do you encrypt for $n=p*q*r$ How do you ...
9 votes
1 answer
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### RSA encryption and decryption with multiple prime modulus using CRT

Every information I found on internet about RSA-CRT encryption/decryption uses only two primes. I'm interested in my project in doing that using multiple (up to 8) primes. The general idea is to ...
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7 votes
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### Are there any standards of multi-prime RSA key generation?

FIPS 186-3 specifies a method to generate DSA parameters. Is there anything similar (official standard or widely-accepted recommendation) that shows how to generate the primes for multi-prime RSA?
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### Subverting the key generation step in RSA public key cryptography

I am interested in the ways an RSA key generation implementation can or cannot be subverted so that the subverter has an advantage, but the owner of the key, and the people who interact with the owner,...
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5 votes
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### RSA with composite numbers

It has been said that RSA uses a modulus product of two prime numbers for security reasons. But does RSA even work correctly if we allow composite integers instead? I think that the answer is "NO". ...
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5 votes
2 answers
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### Does any proof exist for the optimal number of primes in a RSA key?

My guess is: Known attack algorithms only work on 2 primes factorization, they don't work on 3+ RSA primes. More than 3 primes is cpu waste time, better is to increase key length. So 3 primes will be ...
4 votes
2 answers
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### What is multi-prime RSA (RSA-MP)?

I found some related question but no real explanation of what it is and when and why to use it. What are the benefits and downsides and is it recommended?
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2 votes
1 answer
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### Decrypting Multi-Prime RSA with e, N, and factors of N given

I was wondering if there was any way to compute the private key $d$ when knowing only $e$ and $N$, and being able to factor $N$ as 4 prime numbers $p, q, r$ and $s$. I've been searching for days and I ...
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2 votes
1 answer
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### what are multi-primes and how are they different from semiprimes?

I came to know of semiprimes recently. The simplest explanation of semiprimes is you take any two prime numbers and you multiply them, say 3*11 = 33 in which case 33 would be a semiprime. The numbers ...
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2 votes
1 answer
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### RSA: exploiting consecutive primes

It's given 2 plaintexts $m_1$ and $m_2$, and 5 different values of $n\quad\{n_1, n_2, n_3, n_4, n_5\}$ which are generated as follows: $n_1$ is a a product of two relatively small 128-bit $p$ and $q$ ...
2 votes
1 answer
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### Can multi-prime RSA be used to create an abuse-resistant lawful interception mechanism?

New to site so this may have been asked before: Can multi-prime RSA, i.e. where N is product of three or more distinct primes, be used for secure communication while allowing distinct authoritative ...
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### Chinese Remainder Theorem and Elgamal

I am studying an encryption scheme which is Elgamal-like where I think CRT can help optimise the encryption and decryption but I am not sure if I am applying CRT the correct way. I have a cyclic ...
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1 vote
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### Why are the primes all the same size in pqRSA?

pqRSA is MP-RSA with 4096-bit primes to build up a modulus of up to 1TB. If the objective is to make processing the modulus as expensive as possible for the quantum computer why not use just two 4096-...
1 vote
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### Lower bound for the size of prime factors?

We all know classic RSA and that we should pick moduli of at least 2048-bit length to get decent (112 bit) security. Now there's also multi-prime RSA, which can yield significant speed-ups using the ...
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### Of what use is my code for finding prime numbers of a certain size? [closed]

I've developed a bit of Mathematica code that can find primes within a range of numbers. For example, if I wanted all the primes between one million and two million, it could do that. Of what use is ...
0 votes
1 answer
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### Can we do Multi-Prime Diffie-Hellman?

What is the correct way, if any, to do Multi-Prime DH? From security point of view, is there any benefit to do it? Multi-Prime is not about multi participants. Multi prime is when we use two or more ...
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### Plain RSA with $3$ primes, find all integers $x$ so that $x$ is equal to $y$ when $y= e_K(x) = x^{b} \bmod n$

I'm using the notation from the book: Cryptography: Theory and Practice, Third Edition. I have a cryptosystem with the following information: $n = p_1 p_2 p_3$ and integer $x$ is encrypted using \$y = ...
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