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11

I restrict to hash functions $H$ with an output of some fixed size $n\ge1$ bit(s), accepting as input some strings, including all $n$-bit strings; MD5 (resp. SHA-1, SHA-256) is an example of such function for $n=128$ (resp. $n=160$, $n=256$). Whether there exists a solution to $H(x)=x$ depends on the particular hash function. If $H$ is a random function (as ...


8

The idea of "safe curve" is somewhat overrated. What you really want is a safe implementation which won't leak secret information when employed in some practical context. Leakage may occur in a variety of ways; some examples include timing attacks and implementation behaviour when encountering anomalous input. This is not an exhaustive list, and, depending ...


8

No, because even SHA-512 was considered overkill from a security perspective. It has 256-bit collision resistance, which is unbreakable. (The link is about keys but a similar argument applies.) If you think large quantum computers will be efficient, a 512-bit hash makes some sense, but even then a 1024-bit one wouldn't. A quantum computer requires ...


7

Yes, they are (deterministically) equivalent. The original RSA paper (Section IX.C), working off Miller's results (Theorem 3), showed how knowing the secret exponent $d$ was probabilistically equivalent to factoring $n$. Later, using more advanced techniques, Coron and May showed how to deterministically reduce finding $d$ to factoring $n$.


7

I would like to ask if that is true for every AES CTR mode implementation?, Doesn't have to be. You can store the nonce anywhere. You could even send it to the recipient via a different channel (e.g., email the ciphertext and use SMS to transmit the nonce). Storing it at the beginning has its advantages. For example, if streaming the data, you can ...


6

There is no uniform permutation; there is a permutation uniformly chosen from the set of all possible permutations over $Z_2^{128}$. It is evident that AES is not a uniformly chosen permutation, since its permutation is fixed for any key. One can consider a family $\{AES_K\}$ of AES permutations under all possible keys $K$. Even if the key is chosen ...


6

Using a MAC on the plaintext may potentially leak information about the plaintext (MAC algorithms do not necessarily ensure confidentiality of the data they are applied to, although some MAC algorithms like HMAC seem pretty safe). If you want to avoid this (theoretical) problem, then you should encrypt the MAC on the plaintext (i.e. MAC-then-encrypt, not ...


5

I assume $R(x)$ is the original generator, returning $r$ uniformly distributed with $0\le r<x$ for $x<2^{n-1}$, as does Java's int nextInt(int) for $n=32$; and we want to extend that to $R'(y)$, returning $r$ uniformly distributed with $0\le r<y$ for $y<2^n$. $R$ and $R'$ should treat an argument less than $2$ in the same way (perhaps accept it ...


5

The problem doesn't lie with curves in Weierstrass form necessarily, but with naive implementations of elliptic curve arithmetic on such curves. Basically, if you implement an ECC scheme (ECDH, ECDSA or whatever) on a smart card using a curve in Weierstrass form in the most straightforward way possible (by writing a simple double-and-add loop for ...


5

No, since finding $a$ allows offline checking of passwords. $\:$ No, although I can't back this part up.


5

Informally, a signature scheme with message recovery is one where some or all of the message is embedded in the signature, allowing to conserve bandwidth when transmitting a signed message, compared to a signature scheme with appendix. Total message recovery A signature scheme with total message recovery [some sources make total implicit, e.g. the HAC ...


5

Under the assumption that $(K,\text{Msg})\to H_K(\text{Msg})$ is a secure MAC (be it HMAC or any other MAC), and $\text{Nonce}$ does not repeat and is of fixed size, both $H_K(\text{Msg}||\text{Nonce})$ and $H_K(\text{Nonce}||\text{Msg})$ are demonstrably secure, in the sense that an adversary not knowing $K$ can't distinguish either from random, even for ...


5

The Secure Hash Standard and corresponding FIPS-180/202 do not specify any hash to meet a security requirement above 256-bits (using a 512-bit hash). This is unlikely to change. SHA-2 was built with state and word sizes to meet the security requirements on commodity computers (x86 and Alpha), which use 32 and 64-bit maximum CPU word sizes for general ...


5

You basically want a full disk encryption mode for a block cipher; XTS mode seems to be the current standard. In your case each "disk block" is actually a file offset. Note that using a stream cipher or counter mode is NOT secure if the data is ever modified in the file, as it would violate the cardinal sin of using the same key and initialization vector to ...


5

If you want strict indistinguishability, then yes, you need to store the IV (initial counter) somewhere. However, there are some relaxed modes that are used in practice for things like disk encryption, where it is often very useful to decrypt things "in the middle" like you say. For instance, XEX uses a counter which is derived from the sector and offset ...


4

A key derivation function lets you derive keys from others. In this case I would use HKDF, which means using HMAC in a predefined way. Your key material is the keys $X$ and $Y$, so you can concatenate those to get the PRK for HKDF-Expand. An output key would then be $\operatorname{HMAC}(X||Y, \text{info} || \text{0x01})$, if the size of the HMAC is long ...


4

The curve equation $Y^2=X^3+AX+B$ is traditional because it greatly simplifies a lot of theory. I like to use it for teaching. But all of these curve equations are in a sense equivalent, and for any smooth cubic curve, you can usually find an isomorphic curve of desired form. However, a long time ago, people realized that different curve equations have ...


4

One common pitfall when implementing HMAC(key, data) is mishandling the case when key is longer than the underlying hash block. In your case salt is 80 octets, which is longer that SHA-256 "block" (64 octets) so the salt have to be run through SHA-256 before being XOR'ed with i_padin the HMAC. Without seeing any actual code, and provided that the test ...


4

There might be better ways to do this, but I wanted to do it with only primitives found in VIFF (why? because it is the MPC framework I am most familiar with). There could be specialized protocols which are better. In VIFF, we have access a primitive >= which returns 0 or 1 (false or true). We can do the comparison you seek using that plus some simple ...


4

Let's see: AES CTR + MAC: still good advice. His recommendation of 256-bit keys clashes with Schneier's (also 2009) recommendation of 128-bit due to the weaker key schedule with 256-bit keys. Neither choice is broken, however. HMAC-SHA-256 as MAC: still good advice. SHA-3 is still not finalized. 256-bit random UIDs: 256-bits isn't going to risk collisions, ...


4

See “format-preserving encryption” at WikiPedia. Depending on the size of the message space, one can get such a scheme by: sorting pseudorandom values, see section 4.1 of “Format preserving encryption”, or using this arbitrary-size scheme described in “Perfect Block Ciphers With Small Blocks”, or using swap-or-not as described in “An Enciphering Scheme ...


4

Adi Shamir's secret database of all primes is to cryptography venues what the Dahu is to French summer camps. For why, see the answers to this related question. The three other future work items in the quoted presentation are in the same vein (Breaking RSA-1024 with Fermat factoring; Breaking RSA-1024 using $1024 = 2*2*2*2*2*2*2*2*2*2$; Breaking RSA-1024 ...


4

Like the other answers say, it does not always have to be the case. One other case where it is often not stored is when you have a single use key, for example as part of some hybrid encryption scheme. Then there is no need to use a nonce at all and it is usually taken to have zero value.


3

That depends entirely on the size of $p$ and $q$. Given a factorization of $N = pq$, an attacker can compute $g^u \bmod p$ and $g^v \bmod p$, and then attempt to solve the CDH problem modulo $p$, giving him $g^{uv} \bmod p$. Then, he can then compute $g^u \bmod q$ and $g^v \bmod q$, and then attempt to solve the CDH problem modulo $q$, giving him $g^{uv} ...


3

T' method was introduced in the paper Cryptanalysis of Block Ciphers with Overdefined Systems of Equations by Nicolas Courtois and Josef Pieprzyk (see section 6.1 and Appendix E). It is a part of XSL attack on block ciphers (such as Rijndael and Serpent). During XSL attack cipher is represented as a system of multivariate quadratic polynomials and the goal ...


3

A conceivable attack is inspired by this extract of LUKS On-Disk Format Specification Version 1.1.1, section 1: A partition can have as many user passwords as there are key slots. To access a partition, the user has to supply only one of these passwords. If a password is changed, the old copy of the master key encrypted by the old password must be ...


3

The catch how ever is that if a small part of the file is given along with the location of that bytes from the beginning of the file we should be able to decrypt just that piece. Normal CTR mode encryption allows one to decrypt any block of the file independent of the rest, so no need to invent your own mode. With AES the block size is always 128 bits, ...


3

The main difference is that secp256r1 is a prime field curve, while secp256k1 is a Koblitz curve. Koblitz curves are known to be a few bits weaker than prime field curves, but since we are talking about 256-bit curves, neither is broken in "5-10 years" unless there's a breakthrough. The other difference is how the parameters have been chosen. In secp256r1 ...


3

The correction question you should ask about why various operations in RC4 (or, for that matter, any other cipher) are there would be "if I were to remove that, what would the impact be? Would this weaken the cipher in some way?" At your current state of knowledge, that may be a rather imponderable question, but it is still the correct one. I can try to ...


3

I don't believe he is answering the right question. You essentially asked "why are public keys so much larger than symmetric keys", and after his first sentence (which started to address the question, but was a bit vague), he tried to answer the distinct question "why are public key operations so much slower" (not that he got the details of that correct; ...



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