What are the increased possibilities (if any) of being able to crack a private key given the following:   The associated bitcoin (ECDSA Secp256k1-based) public key is known. The private key has been used to generate a public key from a different ECDSA curve and that public key is also known. It is known that the two public keys are derived from the same private key. It is known which ECDSA curves were used to generate the public keys. It is known which ECDSA curve was used to generate which public key.
123052
What are the increased possibilities (if any) of being able to crack a private key given the following:   The associated bitcoin (ECDSA Secp256k1-based) public key is known. The private key has been used to generate a public key from a different ECDSA curve and that public key is also known. It is known that the two public keys are derived from the same private key. It is known which ECDSA curves were used to generate the public keys. It is known which ECDSA curve was used to generate which public key.   Does it make any more difference if there is data that is known to have been signed by the private key and the signature(s) are known and the raw data is known?  EDIT: I should have pointed out that the conditions are ANDed, I.e. All conditions are known. Sorry, I should have been clearer the first time.
49880
Diffie–Hellman offers secure key exchange only if sides are authenticated. For authentication, sides are using public/private key. So if side A knows the public key of the side B, then A can simply use that key to encrypt the message (or the new session key).  Why bother to use separate key exchange algorithm?
38879
I am struck with the following problem:  Let Alice, Bob, Chris and Eve communicate over a public network. They encrypt all messages they send using RSA system. Bob and Chris have the RSA modulus $n_B$ and $n_C$ respectively with $n_B$ = $n_C$ But different public encryption exponents: $e_B \neq e_C$. Suppose $\gcd(e_B, e_C)=1$, and that Alice sends the same secret message to Bob and Chris. Show how Eve can decipher the message.
15506
It's been over 30 years since Rivest, Shamir and Adleman first  described their algorithm for public-key cryptography; and the intelligence community is thought to have known about it for around 40 years&mdash;possibly longer.  It's fair to assume that, during those 40 years, certain three-letter organisations have employed their vast resources toward "breaking" RSA.  One brute-force approach may have been to enumerate every possible key-pair such that, upon encountering a message known to be  encrypted with a particular public-key, they need merely lookup the associated private-key in order to decrypt that message.  Signatures could be forged similarly.  How reasonable is this hypothesis?  How much computing resource would have been required over those 40 years to enumerate every possible {1024,2048,4096}-bit key-pair?  I think it best to avoid discussion and leave the question of whether the spooks could have harnessed such resource as an exercise to the reader.
82581
Is there proof that the DSA construction, also used by ECDSA, is secure assuming that discrete logarithms in the relevant group representation are difficult?
33990
There exist certain attacks that can be used against RSA keys whose prime factors are of specific forms, such as one by Coppersmith.  How common are these RSA keys?  If you generate primes randomly, is there a realistic risk of choosing a weak prime number?  Does this risk, if realistic, vanish if you always choose $p$ and $q$ to be safe primes?
77109
I recently ran across elliptic curve crypto-systems:    (Brown University)  (Wikipedia)  (IEEE)  (RSA.com)   It seemed to me to be great alternative to RSA as the de-facto cryptosystems to be used in banking and financial systems and in the public key infrastructure for certificates, but is not used! If someone can explain why this is not done, it would be very helpful. A comparison between traditional RSA and an elliptic curve cryptology would be helpful.  To begin with:  Advantage of RSA:   Well established.   Advantages of elliptic curve:   Shorter keys are as strong as long key for RSA (see the IEEE paper) Low on CPU consumption. Low on memory usage.
14644
I've recently heard some people claiming that if the fact that P = NP is proven, most (all?) the current cryptographic algorithm considered secure like RSA will be unusable in secure systems.  My question is :  1. If P = NP is proven TODAY, will a RSA key (let say 3072 bits) be still enough to ensure a certain amount of security.  2. If not, what could we do to have at least the same amount of security (take a larger key length? a stronger algorithm?)
111629
While studying the RSA algorithm I referred to some books and some sites such as  and all of them chose {d,n} as the secret (private) key and release {e,n} as the public key but as d and e are multiplicative inverse of each other, can't we keep {e,n} as private key and use the other as public key? Is it a standard rule to choose the {d,n} as the private key?
71065
I see a lot of Q/A where persons are trying to generate a specific key pair using static data such as a password. Now say we use a known PRNG (dangerous assumption), seeded with a the data as a static seed, would this always generate the same key pair?  I presume it would generate the same key pair if you would use the exact same generation method - the application would be deterministic. I would however expect differences between implementations. The random data could be used differently, the sizes of p and q could differ and there may be subtle differences when choosing the prime numbers themselves.
113197
Public-key cryptographic systems are often implemented on top of an existing problem. For example, RSA is built on top of integers and the difficulty of factoring. ECDSA/ECIES are built on top of elliptic curves and the difficulty of the discrete logarithm.  My question is, what are, mathematically speaking, the exact primitives needed to implement public-key based signature and encryption cryptography in a generic way?
114121
Suppose you randomly generate large primes p and q as in RSA, and then tell me N=pq but not p or q.  Then, you would like to actually let me factor N, except you should tell me as few bits of information as possible. Say p and q are 4096 bits, then how many bits do you need to tell me in order for me to factor N in feasible amount of time?  Note that you can arbitrarily design how these bits are derived, as long as there is a PPT algorithm to compute these bits and there is a PPT algorithm to use these bits to factor N.
7698
Is it possible to encrypt a message in such a way that multiple keys can independently decrypt it?  In other words:  Having knowledge of $key_1, key_2 ...key_N$, which are different symmetrical keys, is it possible to generate a ciphertext that can be independently decrypted using either $key_1$ or $key_2$ or $key_N$, in such a way that the length of the ciphertext is less than directly proportional to $N$?
42310
It's pretty easy to generate a Curve25519 private key: generate 32 random bytes of data and then do:  e[0] &amp;= 248 e[31] &amp;= 127 e[31] |= 64   You can then generate a public key by doing  on the private key. For example, here's a random keypair I just generated:  public: e40ed268dff20db8497d0691421d00b086aad1efedcd5bd223d9eb6c0b85665f secret: e3c0e8b20ba7894882ddb37522dbe90867befcf9f31a5ce29c0d7997e0bd856d   The whole safety of public key encryption is that it is hard to take a public key and reverse the operation. Elliptic curve cryptography, as I understand it, relies on the fact that a  is hard to find. In particular, for a given large b and g, it is hard to find an integer k such that  $b^k = g$  How does that equation relate to Curve25519? Specifically, if I had a public key like the one above, how would I express, in math terms, the problem I need to solve to find the private key? How does it relate to the equation I see from time to time - $y^2 = x^3+486662x^2+x$, and the number $2^{255} - 19$?  I am guessing it is not as simple as "the private key is k and the public key is b and/or g" in the equation above. I'm also a little confused because the keys would appear to have roughly the same number of digits, which would seem to imply that there aren't many feasible values for k?
23844
When using the RSA cryptosystem, does it still work if you instead encrypt with the private key and decrypt with the public key? What about in the case of using RSA for sender authentication?
28872
It is often said that RSA encryption of a cryptographic hash with a private key is the same as signing (signature generation). And that verification consists of decryption using a public key.  Is RSA encryption with a private key over a cryptographic hash equivalent to  signature generation?
8311
Is it possible to prove key-equivalence across elliptic curves of different order? Specifically:   Suppose I have a key $x$ valid for both curves listed below On curve $g$ (for example, Curve25519) it maps to $X_1 = x*G$, where $G$ is the generator point for $g$ On curve $h$ (for example, Curve1174) it maps to $X_2 = x*H$, where $H$ is the generator point for $h$   Is it possible for me to submit some public proof that $X_1$ and $X_2$ are generated using the same key $x$, but without revealing the value of $x$?