47

In perfectly secret schemes like the one-time pad, the probability of success does not improve with greater computational power. However, in modern cryptographic schemes, we generally do not try to achieve perfect secrecy(yes governments may use the one time pad, but this is generally not practical for the average user). In fact, given unbounded ...


32

The $1^k$ is a formalism that's only there to make the theoreticians happy. You can safely ignore it. When you actually implement the cryptosystem, you don't try to pass the string $1^k$; instead, you pass $k$, the security parameter (a representation of how much cryptographic strength is desired from the key generation algorithm). I wish I could leave it ...


22

OK, there seems to be some confusion with regards to terminology, so let's try to clean that up. I'll try and define things myself, but also provide the more formal Wikipedia definitions. Encryption. Encryption usually is the process of concealing information solely based on the secrecy of some smaller value, which is called "a key" most of the time. Modern ...


18

The example is using a shorthand notation for the rotors that somewhat obscures the way they actually work. For example, the first rotor in your example, BDFHJLCPRTXVZNYEIWGAKMUSQO, actually applies the following permutation of the alphabet: ABCDEFGHIJKLMNOPQRSTUVWXYZ ↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓↓ BDFHJLCPRTXVZNYEIWGAKMUSQO Applying this rotor in the reverse ...


17

The notation $c=\oplus~c_i$ is (terrible) shorthand for $$c=\bigoplus_{i \in I(c)} c_i$$ where the sum sign should be replaced by the big xor sign which could also be written as $$ c=\sum_{i \in I(c)} c_i,$$where $\sum$ denotes vector addition modulo 2. An example of this decomposition (for length 8 vectors) is $$c=(1,0,1,0,0,0,1,0)=$$ which is nonzero in ...


16

As for the leading zero, I believe the tools are just displaying what's in the ASN.1 as is; the BER/DER encoding rules will insist on a leading 00 byte in some cases. Specifically, if you encode a positive integer, the msbit of the value stored must be 0 (if it is a 1, the encoded value is assumed to be negative); if the msbyte of the value you want to ...


15

The origin is set theory and not programming languages. In the context of cryptography, I could describe a set that is $$x_1 \parallel x_2 \parallel \dots \parallel x_n$$ as a concatenation of the series described by $$\parallel_{i=1}^n x_i.$$ Furthermore, it's worth noting that + to a mathematician would suggest that it is a commutative, which might not ...


15

It is an open standard by IETF.org We can find the details in the mail archive of IETF, D. J. Bernstein's response; It has become increasingly common for "Curve25519" to refer to an elliptic curve, while the original paper defined "Curve25519" as an X-coordinate DH system using that curve. "Ed25519" unambiguously refers to an ...


14

$$x<\!\!<\!\!<k$$ normally means cyclic rotation of a bit string $x$ to the left by $k$ bits.


12

My guess is that it is stated a round decimal number (e.g. 2000) of bits in order not to disqualify solutions using keys that can be up to the next round binary number (e.g. 2048) of bits, but are occasionally slightly less. In particular, in RSA, when we make the product of two 1024-bit primes, the result is 2047 or 2048-bit. This scenario happens with ...


11

The symbol of the circle with the + in it is one of many symbols for exclusive-or. XOR, EOR, EXOR, ⊻, ⊕, ↮, and ≢. Binary OR is true when either input is true; binary XOR is true when exactly one input is true. If both inputs are true, the XOR result is false. One property of this is that if either input bit flips, the output bit will also flip. That's sort ...


11

$Z_2^5$ means that you are working in $GF(2)^5$. $GF(2)$ is the Finite Field with two elements: 0 and 1 with the addition and multiplications defined: $0 + 0 = 0\\ 0 + 1 = 1\\ 1 + 0 = 1\\ 1 + 1 = 0$ It is equivalent to XOR. $0 \times 0 = 0\\ 0 \times 1 = 0\\ 1 \times 0 = 0\\ 1 \times 1 = 1$ It is equivalent to AND. the $ ^5$ is the dimension of the space ...


10

As you probably know $f(\lambda)=O(\lambda^4)$ means that $|f|$ asymptotically upper bounded by some constant times $\lambda^4$. The notation $f(\lambda)=\Omega(\lambda^4)$ corresponds to an asymptotic lower-bound. Now, the $\tilde O$ and $\tilde \Omega$ are closely related notations, where we not only ignore constants but also values which are polynomial ...


10

I'll review the standard mathematical notations used for $H_1:\{0,1\}^*\times\mathbb Z_p^∗\to\mathbb Z_q^∗$ , going from the bottom up. Hopefully, that will make the rest evident. $\{0,1\}$ is the set with the two elements $0$ and $1$, known as Booleans. $\{0,1\}^k$ (for some non-negative integer $k$ ) is the set of tuples with $k$ Booleans, or ...


9

While this is a very good explanation, I would like to add that you will see negligible functions also in other proofs. One example are peusdorandom strings. If an attacker looks at a string, he should only be able to decide if this string is pseudo-random or "real" random" with probability (distribution) $$½ + \mathit{negl}(n)$$ He can always toss a coin (...


9

The notation $1^\lambda$ means a string with $\lambda$ characters all of them equal to 1. For instance, if $\lambda = 3$, then $1^\lambda$ is $111$. And yes, it typically stands to the security parameter, from which the probability of "breaking" the system is measured (as well as the resources needed to do so and also to execute the cryptosystem's ...


9

Some languages like PL/I and Oracle Database SQL indeed use || for string concatenation. One reason is maybe that + might be confusing when talking about fundamental cryptography, since there is a lot of math involved. The mathematical notation for 'OR' would be reversed caret $\lor$ and the exclusive 'OR', better known as 'XOR' is a circled plus $\oplus$. ...


8

As one of the authors of the paper, let me give you an answer. The operation $F$ is indeed applied to both $x$ and $x'$. By stating that $\oplus$ is invariant under rotation, we mean that if you first rotate $x$ and $x'$ and take the difference with $\oplus$, you get the same result as if you first take the difference with $\oplus$ and then rotate the ...


8

Usually, $^\top$ denotes the transpose of a matrix or vector. Common variants include $^t$ and $^T$. Hence $(c_1,c_2)^\top$ is just $\begin{pmatrix}c_1\\c_2\end{pmatrix}$. Such notation is often used to fit column vectors neatly into text.


8

Lindell and Katz use; $\leftarrow$ as possibly probabilistic process assignment. Some other uses $\stackrel r\gets$ or $\stackrel\$\gets$ $:=$ for deterministic process assignment. $=$ for equality $\stackrel{def}{=}$ for defining a variable.


7

There is not a single standard for pseudocode. The := operator is the assignment operator from Pascal, a programming language which was in widespread usage in the 1970s and 1980s, especially for teaching purposes. Many academics have thus been exposed to Pascal and remember it. In Pascal, the equality comparison is =, which matches mathematical practice. By ...


7

What exactly does $0...0$ and $1...1$ mean usually? This simply means a (more or less) long string of $0$s or $1$s or more clearly $000000...000000$ and $111111...111111$. Related notiational notes, you may have to use soon: Sometimes the notation $0^n$ and $1^n$ is also used for these strings with exactly $n$ zeroes and ones. Even more generally it's ...


7

Can you please explain me what this notation means? Of course, $f:A\times B\times C\rightarrow D$ is fancy mathematican's language for saying: "a function f, that takes an element from A, B and C (in this order) and maps this to an element of D" (arbitrarily extend this explanation to as many arguments as you wish). In this particular instance, the authors ...


6

The image description page for the larger image describes it pretty well. Specifically, the line at the top of the figure: shows the 4-ary Boolean function $f(x_1, x_2, x_3, x_4) = x_1 x_2 + x_3 x_4$ in a graphical form. Specifically, interpreting each possible input as a 4-bit binary number (e.g. $(0, 1, 0, 1) \mapsto 0101_2 = 5$), the corresponding ...


6

This example is correct. The inversed versions are the inverse permutation; that is, if the forward direction is the permutation $P$, then the inverse permutation $P^{-1}$ has the property that $P^{-1}(P(X)) = X$ for all $X$. That is, if $X$ is a plaintext letter, and we run it through in the forward direction (giving us $P(X)$), and then run it through in ...


6

Typically that means a string of either $n$ zeros or $n$ ones.


6

To quote yyyyyyy from the comments: The $_R$ has nothing to do with the field — it is associated to $\in$! To quote your first link: "For a set $S$, by $a\in_R S$, we mean that $a$ is randomly chosen from $S$." and to quote SEJPM from the comments: If $p\in \mathbb P$ (with $\mathbb P$ being the set of all primes) then the notations $GF(p);\...


6

It means concatenation. Z, Counter, and SharedInfo are three bitstrings which are to be concatenated before hashing. The [ ] around SharedInfo means it may be absent in which case you would use an empty string instead. (Since concatenating an empty string to the end yields the same result as not concatenating anything.)


6

This has little to do with cryptography or hash functions. It's slightly abused standard mathematical notation. $\{0,1\}$ is the set consisting of $0$ and $1$, so the set of all single bits. For any set $S$, $S^n$ for any natural number $n$ refers to the set of $n$-tuples of Elements from $S$, e.g., $S^2 = S \times S$. So strictly speaking $\{0,1\}^n$ ...


5

If you got an expression that resembles $\{1\}^n$ (or $1^n$) at a place in a surrounding expression where you would expect an $n$-bit bit string to be, the $\{1\}^n$ expression means a string of $n$ bits each with the bit value $1$. Conversely, $\{0\}^n$ means a string of $n$ zero valued bits, and $\{0,1\}^n$ just means any bit string of length $n$. In ...


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