# Tagged Questions

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 ...

2answers
220 views

### Background for modular arithmetic function

I'm investigating this function: $a := ((b\cdot c) \bmod k) - (b \cdot c)/k$ where $/$ indicates integer division. Two things I've noticed: It's equivalent to multiplying a·b, and then ...
1answer
51 views

### Get $a$ such that quadratic residue has a solution (Rabin)

My task is to implement Rabin signature. I have trouble with choosing padding a such that $$x^2 \equiv a \pmod n$$ has a solution. In that context, $n=p\cdot q$ is composite, where $p$ and $q$ are ...
1answer
88 views

### New Improved Probabilistic version of RSA

On the 2nd page of "New probabilistic public-key encryption based on the RSA cryptosystem" by Roman'kov (PDF), at last it says Alice can find "f" of order "l" with least probability of (1-1/l). I ...
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### Using quadratic residue to learn the sign of a field element

Given $x' \in \{-x, x\} \bmod q$ (where $q$ could be any prime of my choice), $s$ is a random element in the field, $y = x'\cdot s$ and $y' = \pm\sqrt{(x\cdot s)^2}\bmod q$ (i.e., both solutions to ...
1answer
39 views

### Three-Pass Protocol number theory

I've got a homework problem that I'm having a hard time understanding. It's for the Three-Pass Protocol, and we are given p, the three messages, and are told that the original plain text is one of two ...
2answers
154 views

### Is there a security problem with this prime generation algorithm?

I am facing the following algorithm to generate an RSA public key: ...
2answers
66 views

### Obtaining Diffie-Hellman generator

In the Wikipedia article on Diffie-Hellman, the algorithm calls for a large prime modulus, $p$, and a generator, $g$, which is a primitive root of $p$. As far as my knowledge of number theory goes, ...
1answer
1k 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 ...
1answer
51 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 ...
0answers
76 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 ...
1answer
116 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 ...
2answers
77 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 ...
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.
1answer
43 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 ...
1answer
58 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 ...
1answer
42 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 ...
0answers
41 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 ...
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 ...
1answer
157 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 ...
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 $q$. ...
1answer
20k 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 ...
2answers
140 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 ...
2answers
69 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 ...
1answer
223 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 ...
1answer
520 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 ...
1answer
252 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 ...
2answers
93 views