Questions tagged [notation]

Questions on the meaning, history, and usage of symbols and notation in cryptography. Please remember to mention where (book, paper, webpage, etc.) you encountered any notation you are asking about.

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Carter-Wegman MAC different notations

While studying Carter-Wegman message authentication codes MAC, I got two different notations for the broader concept and have problems with understanding the difference, if any exist. Let $K, K_1, K_2$...
0 votes
0 answers
58 views

Does this protocol description and design look sound?

I am working on a tool that uses password derived keys for AES and a selectable modes of operation to encrypt (and later decrypt) text for storage on an insecure media. The tool is constrained to ...
0 votes
1 answer
59 views

Cartesian Product Notation for Encryption and Decryption

I'm learning about encryption $(E)$ and decryption $(D)$ algorithms and am wondering what the following notation means, specifically the cartesian product for $E$ and $D$. $k$ represents key, $P$ ...
1 vote
1 answer
70 views

Question about the notation $H\leftarrow\mathcal{H}_{\mathcal{K}}$ in an IND-CCA security experiment

In the paper "Hybrid Key Encapsulation Mechanisms and Authenticated Key Exchange" by Bindel et al., I looked at the IND-CCA security experiment (figure 3). In the first line of the game, ...
3 votes
1 answer
181 views

Understanding $\mathbb{Z}_q[X]/(X^N + 1)$ notation in cryptography paper

I can't seem to understand what $A_{N,q}$ represents in this paper (Section 2.1 - Notations). More specifically, I'm struggling to grasp the meaning of $\mathbb{Z}_q[X]/(X^N + 1)$. Could someone ...
0 votes
1 answer
69 views

On the construction and notation of a certain OWF

I am studying a little bit about one-way functions from Dr Goyal's notes. He constructs the following OWF: Let $D=\Bbb Z_2^{n^3}$ and $R=\Bbb Z_2^{2 n}$. Given $x \in D$, $f$ interprets $x$ as a set $...
0 votes
1 answer
35 views

Confusing notation in signature scheme

In the paper Efficient and Secure Pairing-Free Certificateless Aggregate Signature Scheme for Healthcare Wireless Medical Sensor Networks, on the signature generation part (Page 5), there is an ...
0 votes
1 answer
41 views

Why would be the use of such hash function definition? What would be the input of these functions?

$ G \space is \space an \space elliptic \space curve \space group \space G \space with \space order \space q$ and three hash functions are defined as this: $$ H_1: \{0,1\}^*\times G \rightarrow Z^*_q ...
7 votes
2 answers
620 views

Probability conventions in cryptography

I am working on Victor Shoup's tutorial on game-based security proof and want to figure out some notions from the perspective of probability theory. Consider the following PRF advantage defined on ...
0 votes
1 answer
96 views

Perp notation question

What does the symbol $\perp$ mean? I saw this on a paper where some entity outputs $\perp$.
1 vote
0 answers
33 views

what is the notation of nth power of 1? [duplicate]

when reading theoretical materials, it is often seen that "a common input 1^n..." what does it mean? Are there any similar questions and answers previously in StackExchange?
1 vote
2 answers
169 views

Can you identify these cryptographic algorithmic symbols?

I am mostly self taught and have limited knowledge of the symbology in the image below. It is a diagram of a sub block the RIPEMD-160 algorithm: Are such symbols standardized? Is there a good ...
0 votes
1 answer
133 views

Notation question: Dividing 2 Elliptic Curve Points producing a third point

I'm working my way through some papers and ran across what seems to be division of two points that produce a third point. I'm new to ECC and am having a terrible time trying to figure out what this ...
1 vote
1 answer
219 views

Superscript vs subscript notation in cryptographic formulation

I'm currently reading this paper [PDF]. On page 4, I bumped into these notations : \begin{equation} \text { Experiment } \operatorname{Exp}_{\mathcal{F} \mathcal{E}, A}^{\text {ind-mode }}(k) \text { :...
1 vote
1 answer
96 views

Understanding this notation for the probability distribution of order preserving encryption

I'm reading this PDF: https://link.springer.com/content/pdf/10.1007/978-3-642-01001-9_13.pdf about order preserving encryption functions and there's this on page 9 (or 232): It's describing the ...
0 votes
1 answer
551 views

Discrete Logarithm Notation

I'm studying for my final, and some of the practice problems that I have found have discrete logarithms in this notation $L_n(p)$. What does this notation equate to? For example:
1 vote
0 answers
75 views

In DES : Matsui's notation(LC) vs Biham's notation(DC)

I'm doing linear attack for 8 rounds DES. And I just wondering about the difference between Matsui's notation and Biham's notation. In Matsui's paper, numbers bits from 0 to 63 from right to left. In ...
0 votes
0 answers
85 views

For a given security parameter $\kappa$. what does $poly(\kappa)$ mean?

Let ($Gen,Enc,Dec$) be an LHE scheme with security parameter $\kappa$ and message space $M$. Assume that a multiplication operation exists in $M$, i.e., is a finite ring. Let $F : \{0, 1\}^s × L → M$ ...
1 vote
1 answer
236 views

One way function existence

Let $x = (x_1, x_2,...,x_n)\in\{0,1\}^n$ for $n\in\mathbb{N}$. Prove that if one-way functions (OWFs) exist, then there exists a one-way function $f$ such that for every bit $i\in[1,n]$ there exists ...
1 vote
1 answer
260 views

Big-Omega and small-Omega question

I found this on page 538 (resp. 576) of Introduction to Modern Cryptography 2nd (resp 3rd) edition by Katz and Lindell. The third bullet point under Example A.6 says: $$\begin{array}{l}\text{Let }f(n)=...
0 votes
1 answer
222 views

How does this generate a permutation?

In the process of trying to answer this question, I ended up getting stuck. I found a paper which seems to solve their issue: its authors define a process which yields a pseudorandom permutation ...
1 vote
1 answer
198 views

What does $\text{PK}=\text{pk}\oplus\text{H}(\text{opk})$ represent in this paper?

In Lu, Au and Zhang's 2019 paper (Linkable) Ring Signature from Hash-Then-One-Way Signature, they …extend [M. Abe, M. Ohkubo, and K. Suzuki's 2019 1-out-of-n signatures from a variety of keys] ...
2 votes
1 answer
157 views

What does a vertical bar mean in this context?

I'm doing a few exercises regarding Schnorr's identification scheme. I have the exercise starting off like this, with the values defined: Let $p = 311$ and $r = 31\ |\ (p - 1)$. Let $g = 169$, which ...
22 votes
3 answers
4k views

Why does key generation take an input $1^k$, and how do I represent it in practice?

In my lecture, the lecturer said: Let $K$ be the key generation algorithm. Given a security parameter represented in unary, $1^k$, $K(1^k)$ will output a keypair $(pk; sk)$, known as the public ...
2 votes
0 answers
536 views

What does it mean for “1^n”(i.e., the security parameter written in unary)? [duplicate]

As it is mentioned above, I am confused about “Gen takes as input $1^n$ and outputs a key“. Does it mean that the input of Gen is 1111....111 with length of n?
5 votes
2 answers
16k views

what is meaning of operator " := " in cryptography?

I've seen := used in several cryptography papers where the this symbol has been used instead of assignment operator ($ \leftarrow $ ) or equal sign (=), but never with an accompanying explanation. It'...
0 votes
1 answer
76 views

What does this Big-O notation mean based on appendix?

I am reading the following book: Introduction to Modern Cryptography Second Edition by Jonathan Katz and Yehuda Lindell. I am going through page 533 where they list what some of the notation means, ...
0 votes
1 answer
119 views

Notation for commitment schemes

I was reading this set of lecture notes on commitment schemes, where they define a commitment scheme $\text{Com}(b, r) = f(r), h(r) \oplus b$ as a secure commitment scheme. In this case, $f : \{0,1\}^...
0 votes
1 answer
1k views

What's the difference between these hash functions H, H1 and H2?

These are hash functions, but I don't understand their notations, can anyone tell me what's the difference between H, H1 and H2? $\mathbb{G}_1$ is an additive group and $\mathbb{G}_2$ is a ...
14 votes
1 answer
1k views

What the X stands for in the front of Elliptic curve names like X25519

I have seen Curve25519 and X25519, Curve448 and X448. I've seen a small note in this answer (Historical note: Originally, X25519 was called Curve25519, but now Curve25519 just means the elliptic ...
9 votes
1 answer
504 views

What does "$<\!\!<\!\!<$" mean?

What does the symbol $<\!\!<\!\!<$ mean? It is normally followed by a number. An example: It would be appreciated if you could explain it in fairly basic terms.
0 votes
1 answer
168 views

What does the "x" operator stands for in cryptography?

I am studying commitment scheme and in of the notes from the class this statement comes up. I'd like to know exactly what the X = M x R mean, since I don't seem to understand how the "x" operator ...
4 votes
1 answer
552 views

Is there a difference between $\mathbb{Z}_2^b$, $\text{GF}(2^b)$ and $\text{GF}(2)^b$?

I have read a lot about Galois Fields (GF). They are also presented in The The Design of Rijndael: AES - The Advanced Encryption Standard on pages 13 and 14. In computer memory, the polynomials in $...
2 votes
1 answer
174 views

What is the meaning of the notation $\langle a, b\rangle$ in certain cryptography papers?

I'm trying to understand a paper from Brakerski and Vaikuntanathan. In it, they use the notation $\langle a,b \rangle$ but they don't explain what that means. The answers on this question say that it'...
0 votes
1 answer
120 views

How to interpret a specific mathematical notation in "FIPS 202" (SHA-3 / Keccak)

tl;dr What does "d <= |Z|" mean in the context of bit array manipulations? As a learning exercise I'm working on an implementation of SHA-3. I am not a cryptography specialist in any way but I ...
2 votes
1 answer
119 views

Meaning of pseudocode "$(C, S):=$"

I've seen pseudocode of this form in the Montgomery multiplication related theses: $(C, S) := a[j]*b[0]+C$. What's the meaning of "$(C, S):=$", or what is the real code that was converted from this ...
29 votes
2 answers
22k views

What exactly is a negligible (and non-negligible) function?

The mathematical definition of neglible and non-neglible functions is fairly clear-cut, but why they are important and how they are used in cryptography?
-1 votes
1 answer
277 views

Meaning of "trans" in crypto algorithm

The paper Snarky signatures: Minimal Signatures of Knowledge from Simulation-Extractable SNARKs by Jens Groth contains this line (on page 16): $$\eta \in (\mathbb{Z})^{\left|Q\right|} \leftarrow \...
4 votes
1 answer
271 views

SIMON Cryptanalysis

I'm reading Cryptanalysis of the SIMON Family of Block Ciphers. In Section 3.1, it says: For SIMON, consider an n-bit input difference $\alpha= x\oplus x'$ to $F$ of Hamming weight one. As the ...
1 vote
1 answer
153 views

What does the〈 〉notation mean in this situation?

I was looking at solutions for a problem but I noticed it said $\langle2\rangle$ and was unsure of what that meant. From Homework 5 (PDF)
0 votes
1 answer
77 views

RIBE Scheme Implementation - How to encrypt message

This question is regarding "A Fully Secure Revocable ID-Based Encryption in the Standard Model" - Tung-Tso TSAI, Yuh-Min TSENG, Tsu-Yang WU The encryption method specifies part of the encryption ...
7 votes
1 answer
1k views

Why is the BSI not using powers of two?

In their Technical Guideline TR-02102-2 Cryptographic Mechanisms: Recommendations and Key Lengths the BSI is giving minimal key lengths for - e.g. the TLS handshake protocol. All of these are not ...
17 votes
2 answers
3k views

How did || come to be used in crypto texts to represent concatenation?

In RFC5647, NIST SP 800-38D, etc., || is used to denote concatenation. How did that come to be? In most programming languages || ...
2 votes
2 answers
139 views

Rational exponents on group generators

In elementary concepts, mostly scalar exponents shows up in group operations: $g^x$ As one may encounter in more advanced papers, there are rational exponents over generators. Simply seems like: $g^...
6 votes
1 answer
1k views

What are the curly brackets in this cryptographic hash function definition?

Reading the Schnorr signature Wikipedia page, I stumbled upon the following statement: All users agree on a cryptographic hash function $H:\{0,1\}^*\to\mathbb{Z}_q$. What do these curly brackets ...
6 votes
1 answer
647 views

"Power of one" as input to functions of a cryptosystem

What does $1^\lambda$ mean when you pass it as a parameter to the functions of a cryptosystem. The cryptosystem in question is this and a picture reference is this. I have been told it signifies the ...
11 votes
1 answer
665 views

How can I instantiate a generalized hash function?

I've come across a bunch of "strange" hash function notations, such as the following ones and now I don't know how to choose / instantiate them. Can you please explain me what this notation means and ...
2 votes
1 answer
197 views

local collisions in SHA 0

I have found interesting for me paper about constructing collisions in SHA 0. On page 60 the author determines M variable which is a concatenation of 32-bits vectors $M^{(i)}$, but I can't find any ...
0 votes
1 answer
70 views

A file chunk in a group/field

I am reading a paper by Shacham and Waters on Compact Proofs of Retrievability. In this paper on page number 3 under section 1.1 in the very first line the author states that an encoded file is broken ...
2 votes
1 answer
1k views

How to select elements randomly from a multiplicative group Zn*

I am working on a problem in which I have two large safe primes say p and q randomly selected each of 512 bits. I have generated ...