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### Why is it not possible to increase the size of RSA keys indefinitely?

I've never heard that RSA becomes less secure when the modulus grows. Obviously the strength doesn't grow as fast as the number of bits, but that only means that it grows sub-exponentially. If it ...
• 93.3k

### Why is it not possible to increase the size of RSA keys indefinitely?

I don't understand at all what this claim is on the website. The claim that RSA becomes very expensive for large $N$ is true, but to say that the gap between encryption/decryption cost and factoring ...

• 24.9k
You don't need to explicitly calculate $a^{m-1}$. Observe that $a^{2k} =(a^k)^2$ and that $a^{2k+1} = a \cdot (a^k)^2$. This suggests a simple recursive function $\phi(k)$ to determine $a^k$ modulo $... • 171 7 votes Accepted ### Why does AES use a Binary Field? Well, there would be two possible ways to use modular arithmetic: You could do the arithmetic modulo$2^n$. However, that has some nasty properties (not all elements have multiplicative inverses, ... • 148k 7 votes Accepted ### Paillier paper: Number Theoretic Lemma doesn't seem to work Short answer: This appears to be an error in the paper, but it's not a problem in practice. The proof of Lemma 3 uses the following implication: Since$\gcd(\lambda,n)=1$,$x_2-x_1$is necessarily ... • 12.1k 7 votes Accepted ### Discrete logarithm weak group Is there any better algorithm ? Actually, your second algorithm (select a small set of primes$\{ 2, q_1, q_2, ..., q_n \}$and check if$\ 2q_1 q_2 ... q_n + 1$is prime) is quite efficient. You ... • 148k 7 votes ### Is encrypting every number separately using RSA secure? With appropriate padding (such as OAEP), using RSA to encrypt individual bytes or characters or even bits is indeed secure*. Of course it's also incredibly wasteful, as you're turning every 8 bits of ... • 46.2k 7 votes Accepted ### What is the order of the generator point G=9 in curve25519? According to this source, the points of this curve are a group of cardinality$8\cdot p'$with$p':=2^{252}+27742317777372353535851937790883648493$. This number can be computed by using the Schoof ... • 2,615 6 votes ### If A and B are co-primes, does Ax mod B (where x, any positive int) gives {0,1,2,....,B-1}? Claim: If$A$and$Bare coprime, the map \begin{align*} \{0,\dots,B-1\}\ &\to\ \{0,\dots,B-1\} ,\\ x \ &\mapsto\ A\cdot x\bmod B \end{align*} is a well-defined bijection. It is clear ... • 12.1k 6 votes Accepted ### Why does choosing the first coprime e greater than half of φ(n) result in the same d (private exponent) When choosing the public exponente$, if the value chosen is the first coprime after$\phi(n)/2\$ then the resulting public and private exponents are equal. Well, yeah, that'll always be true. ...
W is 87 in ASCII, so $$87a+b\equiv064066\pmod{256256}.$$ I is 73 in ASCII, so $$73a+b\equiv158368\pmod{256256}.$$ ...