Number theory is the study of the properties and construction of numbers, particularly integers. Prime numbers are of particular interest to number theorists and consequently cryptographers as they are considered the "building blocks" of numbers and produce many interesting results which are useful ...

learn more… | top users | synonyms

14
votes
1answer
965 views

How does a non-prime modulus for Diffie-Hellman allow for a backdoor?

Recently someone found that a Diffie-Hellman modulus used in a unix tool (socat) was not prime. This led some people to shout "backdoor". What I don't understand ...
5
votes
1answer
149 views

Why $e>n^{3/2}$ prevents Wiener's attack

Here's the Wiener's attack as I understand it. Suppose $n=pq$ with $q < p < 2q$, and $d < n^{1/4}/3$ where $ed=k\phi(n)+1$ and $e < \phi(n)$. $$\begin{align*} n-\phi(n) ...
0
votes
1answer
44 views

Computing the cardinality of the co-domain of specific modular exponentiations

Consider the following function: $f: \mathbb{Z}_n \rightarrow Y,~x \mapsto x^e \bmod n$, where $n = p \cdot q$ is an RSA modulus and $gcd(\varphi(n),e) \neq 1$ (different as required for a public ...
2
votes
0answers
68 views

Can we break ECDLP with this machine?

Let $P$ and $Q$ are two points of NIST elliptic curve $E$ (defined over $F_{2^m}$ with prime $m$) and $k$ is a private key such that $k.P=Q$. Also we have a machine that is able to leak some ...
1
vote
1answer
101 views

How “hard” it is to take an e'th root mod p?

I know it's hard to find the $e$th root of a number mod $n=p_1*p_2$, and if it would be possible we could break RSA. But how hard it is to take an $e$th root mod $p$ where $p$ is a prime and ...
2
votes
2answers
61 views

Find plaintext of RSA by solving extended euclidean algorith for two encrptions with two different exponents for same plaintext

This is my homework question (but I am not asking the answer to it): Suppose two users Alice and Bob have the same RSA modulus n and suppose that their encryption exponents eA and eB are ...
12
votes
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.
3
votes
1answer
38 views

How many bits of an exponent are leaked when doing a powmod?

How many bits of the exponent $x$ are leaked when you calculate and reveal $g^x \pmod p$ for some generator $g$ of $\mathbb Z^∗_p$? The low bit of $x$ is obviously leaked: the low bit equals ...
2
votes
1answer
57 views

Why can we ignore $y$ when using the extended Euclidean algorithm to calculate an RSA decryption exponent?

I have a follow-up question about the extended Euclidean algorithm, as applied to RSA key generation, described in this answer. Let us say we have $p=5$, $q=11$ and $e=17$, so that $N=55$ and ...
0
votes
1answer
38 views

Confusion regarding computing Multiplicative Inverse Modulo P?

May be a silly doubt, please rectify my confusion regarding below problem: For concreteness assume $g=2, p=11, a=6$ and $x=9$ $$A = g^a \bmod p = 2^6 \bmod 11 = 9$$ $$X = g^x \bmod p = 2^9 \bmod 11 ...
2
votes
0answers
39 views

Is it possible to generate backdoored DH parameters?

I know it has been already asked and answered whether it's possible to generate weak DH parameters. But "recentely" we experienced the Logjam attack, which makes use of the pre-computation ...
14
votes
4answers
3k 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 ...
6
votes
1answer
150 views

What if the p and q used in keys generation of Pailler cryptosystem are composite?

I've seen a few implementations of Paillier cryptosystem that uses probable primes to choose $p$ and $q$. Assuming that a keypair is generated with $p$ and $q$ that are coprime and that $pq$ is ...
0
votes
1answer
46 views

Algorithm for factoring a number $n$ of a specific form given $n$ and $\varphi(n)$

Given the natural number $n$, which is in the form $p^2 \cdot q^2$ with $p$,$q$ prime numbers. Also $\varphi(n)$ is given. Describe a fast algorithm (polynomial time) that calculates $p$ and $q$. ...
13
votes
1answer
15k 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 ...
3
votes
2answers
117 views

What is the difference between the standard representants of $\mathbb Z/q\mathbb Z$?

The symbol $\mathbb Z/q\mathbb Z$ (given that $q$ is prime) represents the prime field $\mathbb Z_q$. Basically, the elements of this field are represented by $\{0, 1, \dots, q-1\}$, let's call this ...
1
vote
2answers
64 views

How to compute two EC point multiplication?

I would like to know how to compute multiplication of two valid EC points over a curve E with generator G. i.e. Given only P and Q points then how to compute R = P * Q where $P = p G$, $Q = q G$ and ...
4
votes
1answer
204 views

Is the strength of RSA over quadratic or other cyclotomic fields as strong as over the integers?

If we assume the strength of RSA is based on the difficulty of factoring (which I know we can't guarantee) and we compose the modulus of some other quadratic ring that is a unique factorization domain ...
1
vote
1answer
471 views

Montgomery Exponentiation - selecting input value R for a given BigInteger

I have Montgomery exponentiation working, but it's working quite slow. I suspect there are two reasons for this - I implemented it bit size instead of word size (I didn't realize at the time that ...
3
votes
1answer
249 views

How is information disclosed by modular multiplication?

Consider the case that $c = a \cdot b \mod p$ where $p$ is a known prime and $0 < a < p$ and $0 < b < p$ are unknown integers numbers. Furthermore, some bits on the value of $c$ are ...
2
votes
2answers
84 views

Is it hard to recover $p$ from $k \phi(p)$?

Given $k\phi(p)$, is it hard to recover $p$? Here, $p$ is a large prime, $\phi(\cdot)$ is Euler's totient function and $k$ is an unknown integer. Or what's the complexity to recover $p$ from $k ...
5
votes
1answer
172 views

Non adjacent form of an integer is unique

I have tried to look up the proof for NAF (Non-adjacent form) being unique for every integer, but as far as I have seen, textbooks only mention it as a property of NAF, but no proof is given. Also I ...
1
vote
2answers
136 views

Is computing roots moduli a composite $N$ a hard problem without knowing the factorization of $N$?

Suppose that we are given $\mathbb{Z}_{N}$ and an element $x^u \in \mathbb{Z}_{N}$ with $u \in (0,l]$ where $l$ is the bit-size of $N$. Is it difficult to recover $x$ by knowing $u$ without knowing ...
2
votes
1answer
57 views

In a additive group is it hard to calculate $bg$ given $ag, g, abg$

The ECDH problem defined that given $g,ag,bg$ it is difficult to calculate $abg$. But it is also difficult to calculate $bg$ given $ag,g,abg$. where $g$ is generator and a,b are elements of group.
21
votes
3answers
4k views

For Diffie-Hellman, must g be a generator?

Due to a number of recently asked questions about Diffie-Hellman, I was thinking this morning: must $g$ in Diffie-Hellman be a generator? Recall the mathematics of Diffie-Hellman: Given public ...
0
votes
1answer
63 views

Applications of GF(p) polynomials

A Galois field of the type $GF(p)$, where $p$ is prime, is normally expressed as the ring of integers modulo $p$. If my understanding is correct, it is also possible to represent its elements as a ...
-1
votes
1answer
176 views

subgroup of quadratic residue

if a number is said to be a subgroup of a quadratic residue of Z∗p, can I affirm that it is a generator of a cyclic group ? ie, say i am looking for a number which is an element of order q in Z*p, ...
0
votes
1answer
101 views

Prime Numbers in Discrete Log

I am implementing a security protocol based on discrete log. I came across the equation $p = kq + 1$. Understand that based on number theories that both $p$ and $q$ should be large enough to be ...
-1
votes
1answer
85 views

What does $\mathbb Z_p^*$ contain?

I have a prime $p = 7$ and was tasked to select a random value in $\mathbb Z_p^*$ in my signature scheme. What does the full range of $\mathbb Z_p^*$ contain in this case? Is it $\{0...7\}$ or ...
2
votes
2answers
285 views

Need 32-bit mixing function that has perfect avalanche between octets

for my hobby tinkering project, I need a mixing function that takes 32-bit input and has 32-bit output (and will, most likely, run in a 32-bit C environment) and the following property (independent of ...
1
vote
1answer
197 views

Equal length of primes in paillier cryptosystem

In the key generation step of paillier cryptosystem , In order to satisfy $\gcd(pq,(p-1)(q-1))=1$ , we can take equal length primes. Instead of taking(length as parameter to generate $p,q$) equal ...
1
vote
1answer
285 views

How costly is to find millions of large prime numbers for RSA?

Consider I need to assign a large distinct prime number to each element in a large set. This must be deterministic so the function always gives me the same prime to the same value. What is the most ...
0
votes
1answer
197 views

RSA, finding p,q [duplicate]

If the public key $(e,n)$ and the private key $(d,n)$ are known, what is the easiest way to find the primes $p$ and $q$? When $n$ and $\phi(n)$ are given this is easy to solve. But I can't manage it ...
2
votes
1answer
722 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 ...
5
votes
1answer
1k views

AES mixcolumn stage

I'm studying AES, and am having problems with the "mixcolumn" stage. I read about finite fields, but I still don't know. How do I construct $GF(2^8)$? ...
2
votes
2answers
214 views

Given $g$, $b$, $g^{ab}$, is finding $g^a$ a hard problem?

As in the title, given $g$, $g^{ab}$ are big elements in a prime group $Z_p$ and $b$ in prime group $Z_r$ ($p > r$, $g$ is one generator of $Z_p$). $a$ is unknown and also in $Z_r$, is finding ...
1
vote
1answer
141 views

Non-commutitive and nonassociative algebraic structures in cryptography

Are there any cryptographic algorithms or primitives that have been developed and studied that make use of non-commutative or non-associative algebraic structures such as quaternion integers or ...
6
votes
1answer
481 views

Prove that textbook RSA is susceptible to a chosen ciphertext attack

Given a ciphertext $y$, describe how to choose a ciphertext $\hat{y} \neq y$, such that knowledge of the plaintext $\hat{x}=d_K(\hat{y})$ allows $x=d_k(y)$ to be computed. So I use the fact that the ...
1
vote
0answers
41 views

What constitutes a “description of B” for probabilistic encryption as defined in Cryptology 6.3.4?

On page 21 of the Rivest's Cryptology chapter, he defines a trapdoor predicate as a boolean function for which it is easy to choose an x such that ...
11
votes
3answers
2k views

Which algorithms are used to factorize large integers?

Even if RSA decided to cancel the Factoring Challenge, it seems that some teams keep working on it. According to Wikipedia, RSA-768 has been factored in late 2009. What are the current large integer ...
0
votes
1answer
98 views

Existence of a map $\phi:\mathbb{Z}_{N^2}^* \mapsto \mathbb{F} $

Is there a map between the group of $\mathbb{Z}_{N^2}^*$ where $N$ is a composite number , a product of two equal size secure prime numbers $p$ and $q$ and a finite field $\mathbb{F}$, such that for ...
0
votes
1answer
142 views

What are alternatives to number theory based crypto? [closed]

Quantum crypto,lattice based crypto, Neurocryptography and cellular automata based cryptography are alternatives to number theory based crypto. I need to know what are the other hard problems like ...
4
votes
3answers
394 views

Discrete log problem, when we have many examples

Suppose I have many instances of the discrete log problem, all using the same unknown exponent. Is this problem easier than the standard discrete log problem? Oh, heck, I should be more precise. ...
1
vote
1answer
146 views

Modular exponentiation with Chinese Remainder Theorem

I'm learning modular exponentiation with Chinese remainder theorem. I found a great answer from below How can I use eulers totient and the chinese remainder theorem for modular exponentiation? But I ...
5
votes
5answers
520 views

Is there a technique to confirm that a given large integer value is a product of two primes?

Given a list of 2048 bit integer values in which one or few 2048 bit integer values may be product of two prime numbers and other values may be just 2048 bit odd integers numbers. My question is - ...
3
votes
2answers
326 views

Breaking a PRNG Scheme

Assume the following scheme for a PRNG generating decimals in [0,100) Given a message and a key, compute tag := HMAC512(message, key). Take the first 5 hexadecimal characters of the tag. If the ...
2
votes
1answer
154 views

Efficiently computing the neutral element in a ring isomorphic to Z/NZ?

Edited to clarify question So my question is whether anyone knows of an efficient way to compute the neutral element (I'm gonna call it 1, but the operation doesn't have to be multiplication) in an ...
4
votes
4answers
440 views

Where does the $\varphi(n)$ part of RSA come from?

$e d \equiv 1 \pmod{\varphi(n)}$ Where does the $\varphi(n)$ part come from? How did the inventors of RSA arrive at $\varphi(n)$?
3
votes
2answers
92 views

Possible to check if $a \in \mathrm{QR}_n$?

It is possible to check $a \in \mathrm{QR}_p \text{ iff } a^{(p-1)/2} \equiv 1\ (\bmod\ p)$ if $p$ is a prime. $n$ is a large RSA modulus. Is it also possible to check if $a \in \mathrm{QR}_n$ if the ...
13
votes
1answer
535 views

Do recent announcements about solving the DLP in $GF(2^{6120})$ apply to schemes proposed for cryptographic use?

A recent paper by Göloğlu, Granger, McGuire, and Zumbrägel: Solving a 6120-bit DLP on a Desktop Computer seems to "demonstrate a practical DLP break in the finite field of $2^{6120}$ elements, using ...