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7

There is a method known as "Complex Multiplication". However, it is not simple at all, and tends to be overly expensive for most target orders. See this article for some details. There is also the (theoretical) concern that a curve constructed that way may have a special structure though could possibly be leveraged into an attack one day; generally speaking, ...


6

Yes, the basic idea of hardcoding a public key is secure. It is sometimes recommended as an alternative to the complexity TLS and PKI bring – otherwise it can be easy to skip a crucial step and end up with little or no security. However, the "encrypt a secret for server" scheme has some weaknesses compared to TLS. The clearest is lack of forward secrecy ...


3

For private key operations you need at least $n$ (the modulus) and $d$ (the private exponent). The primes $p$ and $q$ let you calculate those – or use some shortcuts for quicker computation – so they also suffice. In practice RSA keys often include all of those values, to avoid having to compute them as needed and to allow for optimized and unoptimized ...


3

I will give you the simplified answer. The public key encryption does not prevent adversery from computing private key from public key. It just makes it very very very hard. The algorithms use math that allows simple private->public calculation, but public->private has no good mathematical "shortcuts". It would take adversery more time to calculate this, ...


2

Basically RSA signatures work just like encryption but with the keys exchanged. If somebody tells you $m^{sk}$ you can easily test if $$ (m^{sk})^{pk} \equiv m\ (mod\ N) $$ but you cannot calculate $m^{sk}$ yourself. The problem/trick is the usual, exponentiation is easy but logarithm is hard. (I like using $sk$/$pk$ for secret-/public-key rather than ...


1

Oh, I have found an answer. PEM here is PKCS#1 (RSA) key. Not sure why ssh-keygen used this terminology. And PKCS#8 could be used for Public keys as well since RFC-5958 which obsoletes RFC-5208. A very good article is https://tls.mbed.org/kb/cryptography/asn1-key-structures-in-der-and-pem and this question is also good: ...


1

There can be much simpler methods than CM for group orders of a specific form. For exemple if $p \equiv 2 \pmod 3$ and $b \not\equiv 0 \pmod p$, the curve $Y^2 = X^3 + b$ over $\mathbf{F}_p$ has $p+1$ points. (The proof of this is easy and left as an exercise.) Such methods are also used to easily construct "good enough" pairing-friendly curves. As Thomas ...


1

Regarding the 3rd part of your question, "Can and should ordinary, run of the mill elections be conducted online ..." - If one thinks of an election as a problem around the issue of preserving the privacy of many inputs (a people or population's votes) while correctly producing the right result (i.e. correctly tabulating the outcome of the election) and ...


1

As far as I know, unknow key-share (UKS) attacks are mostly related to key exchange protocols. Presentation An UKS attack on an authenticated key exchange (AKE) protocol is an attack whereby an entity $A$ ends up believing he shares a key with entity $B$,and although this is in fact the case, $B$ mistakenly believes the key is instead shared with an ...



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