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 multiplicative group. Both groups have the same prime order p and their bilinear mapping is $e: \mathbb{G}_1 \times \mathbb{G}_1 \to \mathbb{G}_2$
Source:https://www.cs.nmsu.edu/~misra/papers/T-ITS-2011.pdf
\begin{align}
H&amp;:   \mathbb G_1\to\{0,1\}^n\\
H_1&amp;:  \mathbb G_2\to\{0,1\}^n\\
H_2&amp;:  \{0,1\}^n\times\{0,1\}^n\to\{0,1\}^n
\end{align}
The notation $H:\mathbb G_1\to\{0,1\}^n$ means that $H$ takes as input an element of $\mathbb G_1$, and produces as output an $n$-bit bistring.
The notation $H_1:\mathbb G_2\to\{0,1\}^n$ similarly means that $H_1$ takes as input an element of $\mathbb G_2$, and produces as output an $n$-bit bistring.
The notation $H_2:\{0,1\}^n\times\{0,1\}^n\to\{0,1\}^n$ means that $H_2$ takes as input a pair of $n$-bit bistrings, and produces as output an $n$-bit bistring.
If you ask me, the names are not well chosen. It would be more mnemonic to use $H_1$ (resp. $H_2$, $H$) where there's $H$ (resp $H_1$, $H_2$).
But that's far from the worse there is in this paper. Things like

The ECDLP had been proved to be a hard problem [19].

are wrong (what's cited to support this falsehood actually makes ECDLP tractable on some curves). I hope studying this paper was not recommended as part of a course in cryptography. If so, I fear there's something deeply flawed in this recommendation process.

Addition per comment: I do not see that the authors explain how they build their hashes, except for a mention of SHA-1 in section VI. A reasonable implementation with $n=160$ based on SHA-1 could use
$$\begin{align}
H(g)&amp;\underset{\text{def}}=\operatorname{SHA-1}(c_0\mathbin\|\hat g)\\
H_1(g)&amp;\underset{\text{def}}=\operatorname{SHA-1}(c_1\mathbin\|\hat g)\\
H_2(u,v)&amp;\underset{\text{def}}=\operatorname{SHA-1}(c_2\mathbin\|u\mathbin\|v)\end{align}$$
where $\hat g$ is a bitstring uniquely representing (as a fixed number of bits) an element $g$ of $\mathbb G_1$ or $\mathbb G_2$, and $c_0$, $c_1$, $c_2$ are public distinct 32-bit bitstrings tasked with making the three hash functions somewhat independent from the perspective of security analysys.
Note: SHA-256 or better should be used rather than SHA-1, which is obsolete and quite practically broken as a collision-resistant hash when operating on messages sizably larger than 768-bit. But as far as we know, SHA-1 remains 80-bit secure in the above constructions, because what's hashed is short.
Note: This shall not be construed as an endorsement of the article. I do not understand enough about functional requirements of a Pseudonymous Authentication-Based Conditional Privacy Protocol for VANETs to tell if they are met by the cryptography outlined; and only understand some of the crypto involved. The article does not tell enough to convince a cryptographer about the security of the implementation. In particular, I have doubt that the BLS signature based on &quot;the Type-A curve defined in the PBC library with the default parameters&quot; could be fast, 154-bit as suggested (the exact size is not stated), and secure to the 80-bit security level targeted.