No addresses, No amounts!

Provides Privacy, Scalability and Fungibility

First implementation by

A Blockchain protocol relying on Homomorphic Commitments

Hides amounts using Pedersen Commitments

Each output on Grin blockchain is a Pedersen Commitment

Pedersen Commitments are homomorphic, perfectly hiding and computationally binding

For an amount $a \in \{0,1,\dots,2^{64}-1\}$ and blinding factor $k \in \mathbb{F}_q$

where $G,H \in \mathbb{G}$ such that DL relation between them is unknown

Given an output $P \in \mathbb{G}$ it is infeasible to find the amount it commits to

Each output comes with a range proof proving $a \in \{0,1,\dots,2^{64}-1\}$

Dandelion

Cut-through

Block added to Blockchain!

RTO

$$\sum_{i=1,2,4}O_i+ \left(\sum_{i=1,2} f_i\right) H - \sum_{i=1}^{4}I_i = \sum_{i=1,2}X_i + k_{\text{off}}G$$

A block contains $n$ kernels, $n =$ #Transactions

Each kernel contains fee and a kernel excess

Coinbase fee $f_{\text{cb}} = 0$, mining reward $r = 60$ grin

Each kernel also contains a Schnorr signature  proving that $X_i = x_iG$ for some $x_i \in \mathbb{F}_q$ 

Block validation check:

General strategy: Compute number of donor coinbase outputs! 

We define a directed graph $G = (V,E)$ such that 

Nodes $V = V_{\text{bl}} \cup V_{\text{cb}},$ where $V_{\text{bl}}$ are blocks and $V_{\text{cb}}$ are coinbase outputs

Edges $E = E_1 \cup E_2$ where 

$E_1 = (v_1, v_2) \in V_{\text{cb}} \times V_{\text{bl}}$ if coinbase output $v_1$ is spent in block $v_2$

$E_2 = (v_1, v_2) \in V_{\text{bl}}^2$ if at least one RTO in block $v_1$ is spent in block $v_2$

$16$

$1493$

$18$

$1489$

$1514$

$1504$

$h_1$

$h_1$

$h_2$

$h_2$

$h_3$

$h_3$

A vertex $c \in V_{\text{cb}}$ in $G$ is called a donor of a block $b \in V_{\text{bl}}$ if there is a directed path from $c$ to $b$ in $G$.

$1499$

$16$

$1482$

$1469$

$1458$

$1481$

$1489$

$1495$

$1493$

$18$

$1479$

$38$

$33$

$9$

$5$

$7$

Subgraph for $h=1499$, $G^{(h)} = (V^{(h)}, E^{(h)})$ where $V^{(h)} = V^{(h)}_{\text{bl}} \cup V^{(h)}_{\text{cb}}$

$$\therefore \ \mathcal{A}(O^{h}) \le  7r + \sum_{b \in V_{\text{cb}}^{(h)}} f_b - \sum_{b \in V_{\text{bl}}^{(h)}} f_b$$

Analysis for RTOs in 612,102 blocks (till March 17th, 2020)

$\text{Flow ratio of RTO (FR)} = \frac{\text{Flow upper bound of RTO}}{\text{Trivial upper bound of RTO}}$

For gauging effectiveness of flow upper bounds, we compute and plot

$\text{Block height}$

$88\%$ blocks have $FR > 0.9$, 

$6.6\%$ blocks with $h>10^5$ have $FR < 0.5$

Unspent RTOs depict the current state of the Blockchain (Fig. 2)

$\text{Block height}$

Jagged pattern in Flow ratio is observed in Fig. 1, Why?

$983$ URTOs have upper bound less that $1800$

$\text{Flow ratio}$

$95\%$ of $110,149$ URTOs have $FR > 0.9$ 

Figure 1

Figure 2

Amounts in very few RTOs found to be in a narrow range

Confidentiality of most URTOs is preserved, however...

Transaction structure could reveal some information about amounts inspite of perfectly hiding commitments

Transaction volume increase might strengthen amount confidentiality

Linkability in inputs and outputs could be leveraged for tighter bounds

Would be interesting to design such analysis for Beam, Monero,...

Listening to ~600 peer nodes, transactions could be traced to their origin before they are aggregated 

Ivan Bogatty claimed to have traced 96% of all Grin transactions 

Image credits: https://github.com/bogatyy/grin-linkability

A. Kumar et al. demonstrated 3 attacks on traceability of inputs in Monero transactions, showing that In $87\%$ of cases, the real output being redeemed can be identified!

Idea#1: $65\%$ transactions have 0 mix-ins as of Feb, 2017!

Idea#2: An input being spent in a ring is the one with the highest block height, where it appeared as a TXO.

Image credits: https://eprint.iacr.org/2017/338.pdf

M$\ddot{o}$ser et al. presented traceability analysis of Monero similar and concurrent to that of Kumar et al's work

Proposed a novel Binned Mixin Sampling strategy as a counter-measure

Characterised Monero usage based on user-behaviour

https://arxiv.org/pdf/1704.04299.pdf

A. Poelstra, "MimbleWimble" [Online], Available: 

link

T. P. Pedersen, "Non-Interactive and Information-Theoretic Secure Verifiable Secret Sharing", in Advances in Cryptology - CRYPTO '91, Springer, 1992, pp. 129-140.

M. Möser, et al. “An Empirical Analysis of Traceability in the Monero Blockchain”. Proceedings on Privacy Enhancing Technologies (2018)

"Linking 96% of Grin transactions" [Online], Available: 

link

A. Kumar, C. Fischer, S. Tople and P. Saxena, "A traceability analysis of Monero’s blockchain", European Symposium on Research in Computer Security – ESORICS 2017, pp. 153-173, 2017.

Thank you!

Happy to answer any questions!