1
vote
1answer
64 views

Does Zhang's result on primes makes RSA weaker?

I read from Finnish newspaper ( http://www.uusisuomi.fi/tiede-ja-ymparisto/72212-matemaattinen-ongelma-eli-2-300-vuotta-mies-subway-tiskin-takaa-ratkaisi#.VBwhYp09F2k.facebook ) the article of Zhang's ...
1
vote
0answers
31 views

Finding a point on an elliptic curve

I have an elliptic curve with the equation $ y^2 = x^3 + ax + b $ in modulo p, where p is prime. I also have a point G on that curve. How can I find another point that isn't a multiple of G? I ...
0
votes
0answers
11 views

Affine cipher and shift cipher

I have the following question: $$An\;affine\;cipher\;with\;key\;K(0,b)\;is\;equivalent\;to\;a\;shift\;cipher\;explain\;why$$ I don't think this is true, and assume it is a typo, $K(1,b)$ I would ...
0
votes
1answer
15 views

How to determine the key-matrix of a Hill cipher where the encrypted-message-matrix is not invertible?

I am new to this subject and I have a homework problem based on Hill cipher, where encryption is done on di-graphs (a pair of alphabets and not on individuals). The alphabet domain is $\{A\dots ...
1
vote
1answer
13 views

Exponention cipher - prove unique mapping from plain text to cipher text

At the heart of RSA, is the exponention cipher: C=M^e mod P (where C=ciphertext, M=Plaintext e=exponent and P=modulus.) How do you prove that two different plaintexts don't map to same ciphertext?
2
votes
1answer
199 views

How to show that the $x^a \equiv 1 \pmod p$ has exactly $\gcd(a,p-1)$ solutions at $Z^*_{p}$?

It is given that $p$ is prime number and $a\ge1$ solution so far: $x^{\gcd(a,p-1)} ≡ 1$ because it known that a group of units of $Z/pZ$ is cyclic and of order $n=p-1$ for $p$ prime, and also in ...
1
vote
1answer
57 views

Proof of an alternative form of Fermat-Euler's theorem.

I want to know a proof of an alternative form of Fermat-Euler's theorem $$a^{\phi (n) +1} \equiv a (mod \; n)$$ when a and n are not relatively prime. I searched some number theory books and a ...
0
votes
0answers
57 views

Factor a big number by Pollard Rho method

How to factor $2^{2^8}+1$ by Pollard Rho algorithm? I have tried this question,but I have no clue. In order to use Pollard Rho, I should know some factor of this number right? But how can I find one?
0
votes
2answers
51 views

key generation in RSA cryptosystem: why it can be performed in polynomial time?

Suppose that I want to generate the keys of the RSA cryptosystem: the public key will be the couple $(n,e)$ where $n$ is the product of two primes $p$ and $q$ and gcd$(\phi(n),e)=1$.The private key ...
3
votes
4answers
60 views

choose two prime numbers of length $k$

Maybe the following is a stupid question, if it is I apologize, and I encourage you to close my post. Suppose that I want to encrypt a message with the RSA cryptosystem; the starting rule is the ...
0
votes
2answers
42 views

To solve for the decryption exponent, why do we solve the congruence $de = 1 (mod (p-1)(q-1))$

So we choose two large primes p and q and multiply them together to get n. We also pick an encryption exponent e and so for any message m, we can compute m^e (mod n) which is our ciphertext c. So ...
1
vote
1answer
26 views

Problem on Miller's Primality test

I am reading Miller's paper entitle "RIEMANN's HYPOTHESIS and Tests for Primality". In the last page, it is defined Dirichlet's L function by $L(S,\chi)=\sum_{n=1}^{\infty} \chi(n)/n^s$ and ...
0
votes
3answers
35 views

RSA decryption problem

(e,n) = (17,323), with ciphertext 185 First compute $\phi(323) = \phi(17*19) = 16*18 = 288$ In order to find the decryption exponent, we must solve 17*d = 1 mod 288 This is equal to $d = ...
0
votes
2answers
39 views

How do I compute Euler phi function efficiently for repeated prime factors?

In RSA decryption problems, you have to compute $\phi(n)$ and then sometimes $\phi(\phi(n))$ quickly. For example, I had to compute $\phi(2^5)$ for one particular problem and it seems to me (for ...
0
votes
1answer
54 views

Determining if a number is an nth root

I am working on a proof that depends on if an adversary can determine if a number is an $nth$ power for some large prime $p$. My intuition tells me that for a sufficiently large value of $n$ this is ...
1
vote
1answer
31 views

Confused about discrete logarithm question

For purposes of explaining the notation for those unfamiliar, if we fix a prime $q$, as well as $a,b$ nonzero integers $\mod{q}$, $L_a(b) = x$ is the solution to the equation $b = a^x \mod{p}$ We are ...
0
votes
1answer
45 views

Figuring out RSA Encryption from 1 encrypted and decrypted message

Suppose that you have an encrypted message and a decrypted message (just one). M (the public key) and k (the exponent you raise each number to) are public. Does having one copy both version of a ...
0
votes
3answers
65 views

How large do my $2$ primes need to be to “guarantee” a longevity of security for my RSA-encrypted plaintext?

I am currently attempting to learn RSA. Most of the literature I am using is at least a few years old, if not older. Given the advancements in computing and improvements in attacking RSA, I am wanting ...
1
vote
3answers
79 views

Inverse Totient Function, given $n$ find all possible is for $\phi(i)=n$

I am trying to figure out easy understandable approach to given small number of $n$, list all possible is with proof, I read this paper but it is really beyond my level to fathom, attempt for ...
0
votes
1answer
38 views

Modular arithmetic to find the mod of a large number

If $x \equiv 23 \bmod 317$ and $x \equiv 25 \bmod 331$, what is $x \bmod 104927$? What techniques are typically used to solve problems of this nature? It doesn't seem clear to me that it can be solved ...
1
vote
0answers
42 views

Closest vector problem

Given is a vector $v=\begin{pmatrix}2,&-1,&0,&1\end{pmatrix}$ as the shortest vector of the lattice $\Lambda (B)$, where $B$ is determined as $B=\begin{pmatrix}4 &-3 & 2 & 0\\ ...
1
vote
4answers
109 views

Using Euler Totient to compute digits in $3^{40000005}$

I'm trying to computer the two rightmost digits in $3^{40000005}$. Can this be done using the Euler Totient function alone as: For every digit $m >1$, $$m = \prod_{i = 1}^{n}p_i^{e_i}$$ where the ...
3
votes
1answer
40 views

Why is the RSA exponentiation function a permutation (i.e. a bijection) over $\mathbb{Z}^*_N$

Why is the RSA exponentiation function a permutation (i.e. a bijection) over $\mathbb{Z}^*_N$? My doubt was specifically why, when raising to the power of the decryption key d we get a unique number ...
1
vote
2answers
69 views

Determine a generator of $\mathbb{Z}^*_{11}$ manually.

What is the best/standard way to do this manually? Could you describe a solution in a step-by-step fashion.
0
votes
1answer
84 views

ElGamal Public Key Cryptosystem and Digital Signature Scheme

I'm tryting to understand how ElGamal algorithm works, and I got the following example, and I couldn't understand one part of this: A) P=23, g=5. B) x=3, then y=10 (for 53 mod 23=10 ). C) Sign for ...
2
votes
2answers
60 views

Manually performing the Miller-Rabin probabilistic primality test

What is the standard/best way to do that manually? Could you give an example with $n=241$ and $a = 3$.
1
vote
1answer
68 views

Manipulating square roots mod p (prime) and when is $g^{ \frac{x}{2}} = p - z_1 \pmod p$ true?

tl;dr: If $z_1 = g^t \pmod p$ is one of the square roots of $g^x \pmod p$ such that $ \frac{p-1}{2} \leq t < p-1$. Then, does $p-z_1 = g^{\frac{x}{2}} \pmod p$ hold true? Say that we define a ...
2
votes
1answer
42 views

Strong primes in cryptography, their relation to complexity theory and security

According to the lecture slide by Shafi Goldwasser a prime is a strong prime if: $$p = 2q + 1$$ for some prime q. For me it, seems a bit arbitrary that is the definition of a strong prime in ...
8
votes
2answers
340 views

Why is Euler's totient function equal to $(p-1)(q-1)$ when $N=pq$ and $p$ and $q$ are prime in a clean intuitive way?

Why is does euler's totient function equal to $(p-1)(q-1)$ when $N=pq$ and $p$ and $q$ are prime? I had my own proof for it but I really don't like it (it feels not intuitive at all because it ...
2
votes
1answer
37 views

Uniqueness of points in Elliptic Curve addition

When working on a curve E, is the point yielded by P + Q (some P and Q on E) completely unique? What I mean is there are no other points on E sharing the same x or y value. Thanks!
0
votes
1answer
58 views

Finding the private key: Attack against El Gamal

El Gamal encryption involves picking $(p,g,b)$ which is our public key. We compute $b=a^x$ $mod$ $p$. Here, $x$ is the private key which we don't know. What are some efficient and strong algorithms ...
1
vote
2answers
66 views

problem about modulus root and quadratic reciprocity

How to calculate $x$ from $x^{14} \equiv 26 \pmod{91}$? What I tried: Let $y=x^2$ $$y^7 \equiv 26 \mod 91$$ then $y \equiv 26 \mod 91$. Then I have $x^2 \equiv 26 \mod 91$ How to solve this? or ...
2
votes
1answer
94 views

Solve $x^2$ $mod$ $23 = 7^2$

What is the procedure to solving $x^2$ $mod$ $23 = 7^2$? According to WolframAlpha, there is no integer solution but I am completely confused as to what steps was taken to determine that. Before ...
2
votes
1answer
35 views

Solve for a variable in mod

I want to solve for $s=\frac{(M-x^y)}{r}$ mod $(p-1)$ where I know the values for $M,x,y,p,s$ but don't know $r$. How can I solve for $r$? I tried to solve for $r$ by trying to compute ...
6
votes
1answer
86 views

What are the implications of Prime Number Theorem in Cryptography?

I know that primes and prime factorization are the basis concepts in cryptography. However, I would like to know how does the Prime Number Theorem come into picture in cryptography, since it states ...
0
votes
0answers
48 views

how to encrypt “STACK” using RSA with keys $e=133,n=2160$

$n=2257=37\times 61,\phi(n)=2160$ A B C D E F G H I J K $\space $L$\space $ M$\space $ N$\space $ O$\space $ P$\space $ Q$\space $ R $\space $S$\space $ T $\space $U $\space $V$\space $ ...
1
vote
1answer
64 views

El-Gamal: Recovering random number r

For a padded message, M, using the El Gamal encryption schema, how can we determine the random number $r$, when we are given $p$, the prime number, $g$ which is the primitive root of $p$, $b$ and $x$ ...
0
votes
2answers
81 views

Computing p and q from private key

We are given n (public modulus) where $n=pq$ and $e$ (encryption exponent) using RSA. Then I was able to crack the private key $d$, using Wieners attack. So now, I have $(n,e,d)$. My question: is ...
0
votes
0answers
45 views

$c$ primitive root, $a \in \{1,\ldots,p-1\}, w/ j \in \mathbb Z^+, a \equiv c^j \pmod p), a^{\frac{p-1}{2}} \equiv 1 \pmod p\implies j\text{ even}$.

Suppose c is a primitive root modulo $p$. Suppose you have a particular integer $a \in \{1,2,\ldots,p-1\}$ and you have found $j \in \mathbb Z^+$ such that $a \equiv c^j\pmod p$. Show that if ...
1
vote
0answers
92 views

Mathematical foundation crisis and the RSA

I am currently in my last year of high school and I am writing a report on cryptography from a idea historical and mathematical perspective. I am including a few of the subjects: Cantor's diagonal ...
1
vote
1answer
34 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 ...
2
votes
3answers
62 views

computing $2^{170}+ 3^{63}\pmod {19}, 3^{175} + 2^{73} \pmod {17}$, etc… by hand

I came across several questions like this in the problem section of a book on coding theory & cryptography and I have no idea how to tackle them. There must be a certain trick that allows for ...
6
votes
4answers
140 views

Approximation of $26!$

Peltzl's Cryptology states on page 8 that $26!$ is approximately $2^{88}$. I have tried different variations of Stirling's formula to confirm this but no luck. I know the argument is hiding in there ...
0
votes
1answer
46 views

Why, in the Rabin cryptosystem, during decryption, do we get four possibilities instead of two?

The encryption algorithm : c=m^2 modn, should mean that we have two(or one) possibilities for m. Why do we get four squareroots?
1
vote
0answers
84 views

Understanding Quadratic Sieve Algorithm

I am studying Cryptography and came upon the quadratic sieve algorithm. However, I am having hard time understanding how the algorithm works. I kind of understood how the steps are followed through ...
1
vote
0answers
64 views

Decrypting a message without the Private Key

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, ...
1
vote
0answers
76 views

Decryption of an Encrypted Message

Suppose we are given sending a message to two people: A and C. A and C have the same RSA encryption modulas: R=(some arbitrary number, say) 454564515456465465465156. But A and C have two different ...
2
votes
1answer
92 views

Modular exponentiation WITHOUT modular exponentiation

Given that 719 is prime, find the least positive residue of $11^{721} (mod\ 719)$, without using modular exponentiation. So, I know how to use modular exponentiation and have done it to get the ...
0
votes
1answer
73 views

Cryptography finding $\bar{k}$

In order to transmit messages in a secure way, some sort of scrambling or encoding of the message is necessary. Without doing so, sensitive information about intended troop movements or information ...
2
votes
1answer
116 views

Problem related to Chinese Remainder Theorem

I'm not sure if there is a typo in the question or if I am incorrect (will point out as I get to it), but I am given that $a,b,m,n$ are integers with $\gcd(m,n) = 1$ and that \begin{equation} c \equiv ...