A prime number is an integer greater than 1 with no divisors other than itself and 1. Primes and prime products play an important role in public key cryptography.

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What exactly is inside a private key?

May sound stupid to many, but I would like to have some pointers on what exactly is contained inside a private key. I have decent understanding of public/private keys/certificates (have created them ...
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why RSA uses Semiprime numbers? [duplicate]

Why does RSA use semiprime numbers? Why not just use any big number ?? What is the advantage of the two original numbers being prime? Because factoring any big number will be difficult
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191 views

Best group if one wants the discrete log problem to be hard?

Suppose one is implementing a cryptographic scheme over a group where one needs the discrete logarithm to be hard - what is the recommended group to use? I'm looking for a group where calculations are ...
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Are there public $p$ and $q$ numbers for use in DSA?

There are many RFC documents giving large primes to use in Diffie-Hellman. However, I couldn't find standards on the $p$ and $q$ large primes used in the DSA signature scheme. This is proving to be a ...
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166 views

Would this program be useful in cryptography?

I know nothing of encrypting. I'm not even sure how to tag this. I wrote a program that can calculate this pretty quickly on my macbook pro 2.3GHz IntelCore i7. The two exponents are Mersenne primes, ...
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208 views

Factoring large numbers

I am trying to factor few integers that are each between 115 and 135 digits long. I have just, little over a month ago, began my study of Cryptography. I was wondering if anyone knew of any efficient ...
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166 views

Good Encryption Exponent

I have placed a bet that I can create a public key such that my adversary will not be able to crack (decrypt) it for at least one week. For my primes $p$ and $q$, I chose very large numbers that are ...
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396 views

Prime number theorem - RSA

I am having trouble understanding the prime number theorem. As part of some revision for an exam, I am trying to answer the following questions (but seeing as I don't understand the concept of the ...
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49 views

How can I evaluate the congruency of an AKS primality test?

Despite the fact primality test is a mathematical issue, it plays a part on the security of many cryptosystems such as RSA. I was trying to understand how it works until I came to the following ...
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128 views

Why are huge prime numbers important in cryptography?

I read an article the other day about the search for prime numbers. According to the article and several online sources the biggest prime number is over 17 million digits! This made me wonder why ...
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205 views

What is the difference between, and security of $Z_p$ and $F_p$?

While reading some papers, I found that sometimes $Z_p$ was used but sometimes $F_p$ was used ($p$ is prime). Usually, the author will only use their subgroups like this: choose a big prime factor of ...
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523 views

Decrypting a message without the private key using CRT

I am given 5 different encryption modulus, $N$, each ranging from 78 to 88 numbers long. Then for the encryption exponent, each has the same which is 5. Then I am given 5 different encrypted messages, ...
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What makes RSA secure by using prime numbers?

I am just learning about the RSA algorithm. Looking at the first two steps: Choose two distinct prime numbers $p$ and $q$. Compute $n = pq$. I have some probably stupid questions: Why do $p$ ...
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231 views

What are the possible cryptographic implications of Zhang's proof of the Twin Prime Conjecture?

Earlier this year, Yitang Zhang published a proof of a weakened form of the Twin Prime Conjecture. I'm wondering if any of the new mathematical machinery he developed has uses in cryptography or could ...
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155 views

Integer factorization based password authentication

After looking at this security issue at DjangoProject, I started to think in a password-based authentication that places the burden of PBKDF2 (or whatever is the hashing function) on the client. So I ...
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491 views

Quadratic residue problem on composite integers

Its believed that the quadratic residue modulo $n=p·q$ for large primes $p$ and $q$ is intractable, which forms the basis of some cryptosystems. However, it is solvable if the factors of $n$ are ...
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264 views

why AKS is so slow in practice? [closed]

This note (http://maths-people.anu.edu.au/~brent/pd/primality4.pdf) states that AKS is not practical. However, it is known that AKS runs in polynomial-time, and I cannot understand where the slowness ...
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301 views

Primality testing (deterministic vs. non-deterministic)

I noticed that non-deterministic primality testing algorithms are more commonly used in practice while there is a deterministic algorithm e.g., AKS which runs in polynomial time? I am right? if so, ...
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526 views

Algorithm for proving Carmichael numbers

I have an application for determining if a number is a prime or not, currently I'm getting a random number, then doing the Fermat primality testing to find out if the number is probably prime (so this ...
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1answer
186 views

Proof of work for determining whether a number is prime?

I have an idea for a system that would be outsourcing some brute force calculations to many users in hopes of finding divisors of a number. However, there is a possibility that a given number would be ...
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206 views

How did they factor RSA-704?

I don't understand the 'Wiedemann algorithm' works. Can someone explain the factoring of RSA-704 in an easy way?
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70 views

Why is it impractical to generate a semiprime dictionary? [duplicate]

This might be a very simple question. However, I am just learning the concept, so just excuse me. I am wondering why there is not any attempt to generate all semiprime numbers? (as an dict. attack to ...
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179 views

Solving hard problems in $\mathbb Z_{p}^{*}$ when $\mathbb p$ is close to $\mathbb 2^{n}$

Suppose, for some security parameter $n$ you choose a prime $p$ such that $p = 2^n+c$ for some relatively small $|c| < 2^m << 2^n$. I have seen such primes being called Pseudo-Mersenne Primes ...
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How to calculate y value from ((y*y) mod prime) efficiently

i am working ECC-224 bit. can any one tell me, how to calculate y value from ((y*y) mod prime) efficiently for large bit numbers.
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154 views

Selecting a large NUMS Safe prime

Suppose I want to use the following simple hash function. For a mesage $m$, take some public $a$ and prime $p$ and raise $a^m \bmod p$ (never mind the computational expense of this operation). This ...
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235 views

Solving congruences using PARI

I'm having trouble finding info in the docs about how to solve a system of congruences. The closest I can find is 'matsolvemod' in here: ...
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433 views

Security of Pohlig-Hellman exponentation cipher?

I am looking into implementing Pohlig-Hellman exponentation cipher and I would like to know how secure that algorithm is? I am guessing it's security relates greatly to the prime number used in it. ...
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162 views

How to solve the reverse of an equation that uses MOD?

I've been tasked with reverse engineering an unknown crypto function. The function uses the following constants: $a=380951$: I noticed that this is a prime number $b=3182$: I noted that this is a ...
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RSA primes vs. largest known primes

In the context of a new largest (mersenne) prime number being found this week - The largest known prime number is now 2^57,885,161 − 1, and it took 5 years to find ...
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631 views

Visualization of cryptography

I think CrypTool is great software. And what I find most useful in it is visualization of algorithms such as Caesar, Vigenere, AES, DES. And my question is: does anyone know other tools which are ...
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Is (2^333)-1 a prime number? [closed]

How can I see if $2^{333}-1 $ is a prime number? Does this have to do with Mersenne prime numbers ($2^n-1$) ?? Thank you!
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253 views

Choosing good parameter for Lenstra's elliptic curve factorization

In Wikipedia, there is an article explaining Lenstra's factorization algorithm. As far as I got it, we choose some $e \in \mathbb{N}$ and a point $P$ on the curve and then calculate $eP$. While ...
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phi(P*Q) = (P-1) * (Q-1)

I was trying to understand RSA when I encountered the Euler Function. I do understand this: $\phi(P)$, where $P$ is a prime is $P-1$. However it seems that for a number $N$ such at $N=P\cdot Q$ where ...
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Why does Schnorr's Digital Signature scheme necessitate two prime numbers?

One of the necessary components to the Schnorr Digital Signature scheme is a pair of prime numbers p and q such that q divides p-1. However, there is never a modular inverse taken of q so why is there ...
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Necessity for finite field arithmetic and the prime number p in Shamir's Secret Sharing Scheme

Shamir's original paper (PDF, 197kb) describing a threshold secret sharing scheme states: To make this claim more precise, we use modular arithmetic instead of real arithmetic. The set of ...
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How can I use eulers totient and the chinese remainder theorem for modular exponentiation?

I'm trying to implement modular exponentiation in Java using Lagrange and the Chinese remainder theorem. The example we've been given is: Let $N = 55 = 5 · 11$ and suppose we want to compute ...
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Is there an algorithm for factoring N, which is just as simple as this one, but faster?

I found a simple algorithm for factoring semiprime numbers, you can read about it in Factoring Semiprimes and Possible Implications for RSA. It basically works like this: You reverse the digits in ...
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684 views

In layman's terms, how does Shor's algorithm work?

I've just been reading up on Shor's algorithm, and I find it both fascinating and baffling. I don't really understand much about it, other than that it can factor semiprimes in polynomial time. Could ...
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655 views

ABC Conjecture's Impact on RSA Encryption

A recent proof of the ABC Conjecture has been released by one Shinichi Mochizuki. Now, I'm not well versed in mathematics but it would appear that this proof implies that finding prime factors could ...
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767 views

What is the correct value for “certainty” in RSA key pair generation?

I'm creating an RSA key pair in Bouncy Castle and need to specify an int value for certainty. This Stack Overflow answer says it is a relative test for how prime the values are. There is another ...
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189 views

ECDSA - point order criterion

i am creating some primitive demostration for ECDSA over small curve (p < 229). But my implementation have some weird issues. Verify process return false even if the signature is correct. Because I ...
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2k views

Generating Random Primes

Although this has been extensively discussed around here, I'm curious whether my approach makes sense, or I should just stick to "the standard version". I'm implementing some homomorphic encryption ...
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187 views

In RSA, rationale for prime $p$ with $p-1$ having prime factor $u$ with $u-1$ having large prime factor?

In the 1978 RSA paper, it is recommended, among other things, to choose primes $p$ such that $(p-1)$ has a large prime factor $u$. This was motivated by Pollard's p-1 algorithm. Further, the authors ...
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How are the primes used to generate RSA keys?

I am confused about how keys in RSA asymmetric encryption are generated and what the implications for open communications are. Textbooks say the one-way function is merely two primes (with some ...
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Is it safer to generate your own Diffie-Hellman primes or to use those defined in RFC 3526?

I was wondering if the prime numbers defined for use with Diffie-Hellman in RFC 3526 are more trustworthy than generating one's own, especially considering the recent Arjen Lenstra paper (Ron was ...
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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. ...
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How large should a Diffie-Hellman p be?

In a Diffie-Hellman exchange, the parties need to agree on a prime p and a base g in order to continue. Assuming some ...
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How to construct a zero-knowledge proof of a number of the form $n=p^a q^b$

Let $n = p^a$$q^b$ where p and q are distinct primes and a and b are positive integers. How to construct a zero knowledge proof that n is of such form? This is actually a homework problem with a ...
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How to better generate large primes: sieving and then random picking or random picking and then checking?

I'm writing an RSA algorithm, and am wondering what is the best and/or usual way to choose the initial prime numbers (p and q). I know of two methods to achieve this, one based on a prime number ...
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Why does the PKCS1 RSA private key structure contain more than just exponent and modulus?

The ASN.1 spec for the PKCS1 RSA private key format is as follows: ...