From ecdff8dd435420805c146fa8df0a7f5125351432 Mon Sep 17 00:00:00 2001 From: sinu <65924192+sinui0@users.noreply.github.com> Date: Wed, 6 Sep 2023 11:23:24 -0700 Subject: [PATCH 01/10] update mpc-tls top level --- src/protocol/mpc-tls/README.md | 6 +++--- 1 file changed, 3 insertions(+), 3 deletions(-) diff --git a/src/protocol/mpc-tls/README.md b/src/protocol/mpc-tls/README.md index d0f9e89..c3710d0 100644 --- a/src/protocol/mpc-tls/README.md +++ b/src/protocol/mpc-tls/README.md @@ -1,9 +1,9 @@ # MPC-TLS -During the MPC-TLS Phase the `Prover` and the `Notary` work together to generate an authenticated `Transcript` of a TLS session with a `Server`. +During the MPC-TLS Phase the `Prover` and the `Verifier` work together to generate an authenticated `Transcript` of a TLS session with a `Server`. Listed below are some key points regarding this process: - - The identity of the `Server` is not revealed to the `Notary`, but the `Prover` is capable of proving the `Server` identity to a `Verifier` later. - - The `Notary` only ever sees the *encrypted* application data of the TLS session. + - The identity of the `Server` is can be hidden from the `Verifier`, while the `Prover` is still capable of proving the `Server` identity later. + - The `Verifier` only ever sees the *encrypted* application data of the TLS session. - The protocol guarantees that the `Prover` is not solely capable of constructing requests, nor can they forge responses from the `Server`. From 35b2257f6cb531960076d0bf53d90c9972d2cb52 Mon Sep 17 00:00:00 2001 From: sinu <65924192+sinui0@users.noreply.github.com> Date: Wed, 6 Sep 2023 11:23:36 -0700 Subject: [PATCH 02/10] update handshake --- src/protocol/mpc-tls/handshake.md | 10 +++++----- 1 file changed, 5 insertions(+), 5 deletions(-) diff --git a/src/protocol/mpc-tls/handshake.md b/src/protocol/mpc-tls/handshake.md index 2c5d50b..80cfffb 100644 --- a/src/protocol/mpc-tls/handshake.md +++ b/src/protocol/mpc-tls/handshake.md @@ -1,12 +1,12 @@ # TLS Handshake -A TLS handshake is the first step in establishing a TLS connection between a `User` and a `Server`. In TLSNotary the `User` is the one who starts the TLS handshake and physically communicates with the `Server`, but all cryptographic TLS operations are performed together with the `Notary` using MPC. +A TLS handshake is the first step in establishing a TLS connection between a `Prover` and a `Server`. In TLSNotary the `Prover` is the one who starts the TLS handshake and physically communicates with the `Server`, but all cryptographic TLS operations are performed together with the `Verifier` using MPC. -The `User` and `Notary` use a series of MPC protocols to compute the TLS session key in such a way that both only have their share of the key and never learn the full key. Both parties then proceed to complete the TLS handshake using their shares of the key. +The `Prover` and `Verifier` use a series of MPC protocols to compute the TLS session key in such a way that both only have their share of the key and never learn the full key. Both parties then proceed to complete the TLS handshake using their shares of the key. -With the shares of the session key computed and the TLS handshake completed, the parties now proceed to the next MPC protocol where they use their session key shares to jointly generate encrypted requests and decrypt server responses while keeping the plaintext of both the requests and responses private from the `Notary`. +With the shares of the session key computed and the TLS handshake completed, the parties now proceed to the next MPC protocol where they use their session key shares to jointly generate encrypted requests and decrypt server responses while keeping the plaintext of both the requests and responses private from the `Verifier`. -> Note: to a third party observer, the `User`'s connection to the server appears like a regular TLS connection and the security guaranteed by TLS remains intact for the `User`. +> Note: to a third party observer, the `Prover`'s connection to the server appears like a regular TLS connection and the security guaranteed by TLS remains intact for the `Prover`. > -> The only exception is that since the `Notary` is a party to the MPC TLS, the security for the `User` against a malicious `Notary` is guaranteed by the underlying MPC protocols and not by the TLS. \ No newline at end of file +> The only exception is that since the `Verifier` is a party to the MPC TLS, the security for the `Prover` against a malicious `Verifier` is guaranteed by the underlying MPC protocols and not by TLS. \ No newline at end of file From 63ee71406b1801cc1fe4235042f9f7ce9ed43f82 Mon Sep 17 00:00:00 2001 From: sinu <65924192+sinui0@users.noreply.github.com> Date: Wed, 6 Sep 2023 12:17:04 -0700 Subject: [PATCH 03/10] update encryption and decryption --- diagrams/mpc-encryption.drawio | 1 + src/png-diagrams/mpc-decryption.png | Bin 0 -> 93235 bytes src/png-diagrams/mpc-encryption.png | Bin 0 -> 95214 bytes src/protocol/mpc-tls/encryption.md | 22 +++++++++------------- 4 files changed, 10 insertions(+), 13 deletions(-) create mode 100644 diagrams/mpc-encryption.drawio create mode 100644 src/png-diagrams/mpc-decryption.png create mode 100644 src/png-diagrams/mpc-encryption.png diff --git a/diagrams/mpc-encryption.drawio b/diagrams/mpc-encryption.drawio new file mode 100644 index 0000000..f0855ac --- /dev/null +++ b/diagrams/mpc-encryption.drawio @@ -0,0 +1 @@ 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z)qlovOQtIm{JOBwICY8D`>dEe)u_|B7ooyi-oK%$jH70pHXkMgxJezVFlj^CHcFVQI&1Qe$mqxc=*xihb}ZfkYQ4wizKPm?S%OjR&qW5_DiqdBW(?q|8$ zbx<{aX4t1A^Wo`r`Y#)>(;r+uUIYSRE^=5vUa>1*;Rh8LI|L-{1X-Cevt&&Bc-!i4 zBmppJmI;Hngrd3eTxC0V0SY%zqLbq2X5s&!MwbXvyoo5Ath>BDn-rAXK0JJiCdIpbeG$a>i3rJRA97BK zJUHQpX#iAxC07)824Hw*_fK?}fp7V11Fq-%%T?qF+=M6dW6yJu{u5;fr#phM#NI>a zJDLeWFOk3!6hP_H~2GRBZ{B+YsnbYWu zpej=<0lgvfq5Lb3?HX;8rumB_ezC{kWRn7*EZ_ewH Date: Wed, 6 Sep 2023 12:33:48 -0700 Subject: [PATCH 04/10] rename signing section, move it, and update it --- src/SUMMARY.md | 9 ++++----- src/protocol/mpc-tls/commitment.md | 7 ------- src/protocol/mpc-tls/signing.md | 11 ----------- src/protocol/notarization.md | 15 +++++++++++++++ 4 files changed, 19 insertions(+), 23 deletions(-) delete mode 100644 src/protocol/mpc-tls/commitment.md delete mode 100644 src/protocol/mpc-tls/signing.md create mode 100644 src/protocol/notarization.md diff --git a/src/SUMMARY.md b/src/SUMMARY.md index d4102b5..167aa43 100644 --- a/src/SUMMARY.md +++ b/src/SUMMARY.md @@ -6,13 +6,12 @@ # Protocol -1. [MPC-TLS](./protocol/mpc-tls/README.md) +- [MPC-TLS](./protocol/mpc-tls/README.md) - [TLS Handshake](./protocol/mpc-tls/handshake.md) - [Encryption and Decryption](./protocol/mpc-tls/encryption.md) - - [Commitment](./protocol/mpc-tls/commitment.md) - - [Signing](./protocol/mpc-tls/signing.md) -2. [Verification](./protocol/verification.md) -3. [Selective Disclosure]() +- [Notarization](./protocol/notarization.md) +- [Verification](./protocol/verification.md) +- [Selective Disclosure]() # MPC diff --git a/src/protocol/mpc-tls/commitment.md b/src/protocol/mpc-tls/commitment.md deleted file mode 100644 index eeacefb..0000000 --- a/src/protocol/mpc-tls/commitment.md +++ /dev/null @@ -1,7 +0,0 @@ -# Commitment - -As part of the TLSNotary protocol, the `User` creates authenticated commitments to the plaintext and the `Notary` signs those commitments without ever seeing the plaintext. This offers a way for the `User` to selectively prove the authenticity of arbitrary portions of the plaintext to the `Verifier`. - -A naive approach of creating such authenticated commitments is to extend the `Encryption and Decryption` steps to also compute a commitment (e.g. a blake3 hash) to the plaintext using MPC and have the `Notary` sign that commitment. Unfortunately, such MPC approach is too resource-intensive, prompting us to provide a more lightweight commitment scheme. - -The high-level idea is that the `User` creates a commitment to the encodings from the MPC protocol used for `Encryption and Decryption`. Since those encodings are chosen by the `Notary` and are not known to the `User` at the time when she makes a commitment, they can be thought of as "authenticated plaintext". \ No newline at end of file diff --git a/src/protocol/mpc-tls/signing.md b/src/protocol/mpc-tls/signing.md deleted file mode 100644 index 75efa6e..0000000 --- a/src/protocol/mpc-tls/signing.md +++ /dev/null @@ -1,11 +0,0 @@ -# Signing the Session Header - -At the end of the TLSNotary protocol, the `Notary` signs an artifact known as a `Session Header`, thereby attesting to the authenticity of the plaintext from a TLS session. A `Session Header` contains a `User`'s commitment to the plaintext and a `User`'s commitment to TLS-specific data which uniquely identifies the server. - -The `User` can later use the signed `Session Header` to prove data provenance to a third-party `Verifier`. - -It's important to highlight that throughout the entire TLSNotary protocol, including this signing stage, the `Notary` does not gain knowledge of either the plaintext or the identity of the server with which the `User` communicated. - - - - diff --git a/src/protocol/notarization.md b/src/protocol/notarization.md new file mode 100644 index 0000000..5b5eee9 --- /dev/null +++ b/src/protocol/notarization.md @@ -0,0 +1,15 @@ +# Notarization + +As part of the TLSNotary protocol, the `Prover` can create authenticated commitments to the plaintext and have the `Notary` sign them without ever seeing the plaintext. This offers a way for the `Prover` to selectively prove the authenticity of arbitrary portions of the plaintext to a different `Verifier`. + +A naive approach of creating such authenticated commitments is to extend the `Encryption and Decryption` steps to also compute a commitment (e.g. BLAKE3 hash) to the plaintext using MPC and have the `Notary` sign that commitment. Unfortunately, such an approach is too resource-intensive, prompting us to provide a more lightweight commitment scheme. + +The high-level idea is that the `Prover` creates a commitment to the encodings from the MPC protocol used for `Encryption and Decryption`. Since those encodings are chosen by the `Notary` and are not known to the `Prover` at the time when she makes a commitment, they can be thought of as "authenticated plaintext". + +## Signing the Session Header + +The `Notary` signs an artifact known as a `Session Header`, thereby attesting to the authenticity of the plaintext from a TLS session. A `Session Header` contains a `Prover`'s commitment to the plaintext and a `Prover`'s commitment to TLS-specific data which uniquely identifies the server. + +The `Prover` can later use the signed `Session Header` to prove data provenance to a third-party `Verifier`. + +It's important to highlight that throughout the entire TLSNotary protocol, including this signing stage, the `Notary` does not gain knowledge of either the plaintext or the identity of the server with which the `Prover` communicated. From 72b1adb1ffeee85b622503781bd491ff2e9e7f72 Mon Sep 17 00:00:00 2001 From: sinu <65924192+sinui0@users.noreply.github.com> Date: Wed, 6 Sep 2023 12:36:11 -0700 Subject: [PATCH 05/10] typo --- src/protocol/mpc-tls/README.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/protocol/mpc-tls/README.md b/src/protocol/mpc-tls/README.md index c3710d0..0c37b1b 100644 --- a/src/protocol/mpc-tls/README.md +++ b/src/protocol/mpc-tls/README.md @@ -4,6 +4,6 @@ During the MPC-TLS Phase the `Prover` and the `Verifier` work together to genera Listed below are some key points regarding this process: - - The identity of the `Server` is can be hidden from the `Verifier`, while the `Prover` is still capable of proving the `Server` identity later. + - The identity of the `Server` can be hidden from the `Verifier`, while the `Prover` is still capable of proving the `Server` identity later. - The `Verifier` only ever sees the *encrypted* application data of the TLS session. - The protocol guarantees that the `Prover` is not solely capable of constructing requests, nor can they forge responses from the `Server`. From aa218c41184cefe33f7ef9e4b92bdc434ab8595c Mon Sep 17 00:00:00 2001 From: sinu <65924192+sinui0@users.noreply.github.com> Date: Wed, 6 Sep 2023 12:38:14 -0700 Subject: [PATCH 06/10] remove redundant tls --- src/SUMMARY.md | 2 +- src/protocol/mpc-tls/handshake.md | 2 +- 2 files changed, 2 insertions(+), 2 deletions(-) diff --git a/src/SUMMARY.md b/src/SUMMARY.md index 167aa43..5c21606 100644 --- a/src/SUMMARY.md +++ b/src/SUMMARY.md @@ -7,7 +7,7 @@ # Protocol - [MPC-TLS](./protocol/mpc-tls/README.md) - - [TLS Handshake](./protocol/mpc-tls/handshake.md) + - [Handshake](./protocol/mpc-tls/handshake.md) - [Encryption and Decryption](./protocol/mpc-tls/encryption.md) - [Notarization](./protocol/notarization.md) - [Verification](./protocol/verification.md) diff --git a/src/protocol/mpc-tls/handshake.md b/src/protocol/mpc-tls/handshake.md index 80cfffb..ab427eb 100644 --- a/src/protocol/mpc-tls/handshake.md +++ b/src/protocol/mpc-tls/handshake.md @@ -1,4 +1,4 @@ -# TLS Handshake +# Handshake A TLS handshake is the first step in establishing a TLS connection between a `Prover` and a `Server`. In TLSNotary the `Prover` is the one who starts the TLS handshake and physically communicates with the `Server`, but all cryptographic TLS operations are performed together with the `Verifier` using MPC. From 1a32e6e0595e7ee3327c15756ac26227471fbb17 Mon Sep 17 00:00:00 2001 From: sinu <65924192+sinui0@users.noreply.github.com> Date: Wed, 6 Sep 2023 12:41:05 -0700 Subject: [PATCH 07/10] add reference to ke section --- src/protocol/mpc-tls/handshake.md | 7 ++++--- 1 file changed, 4 insertions(+), 3 deletions(-) diff --git a/src/protocol/mpc-tls/handshake.md b/src/protocol/mpc-tls/handshake.md index ab427eb..e38a588 100644 --- a/src/protocol/mpc-tls/handshake.md +++ b/src/protocol/mpc-tls/handshake.md @@ -4,9 +4,10 @@ A TLS handshake is the first step in establishing a TLS connection between a `Pr The `Prover` and `Verifier` use a series of MPC protocols to compute the TLS session key in such a way that both only have their share of the key and never learn the full key. Both parties then proceed to complete the TLS handshake using their shares of the key. -With the shares of the session key computed and the TLS handshake completed, the parties now proceed to the next MPC protocol where they use their session key shares to jointly generate encrypted requests and decrypt server responses while keeping the plaintext of both the requests and responses private from the `Verifier`. - +See our section on [Key Exchange](../../mpc/key_exchange.md) for more details of how this is done. > Note: to a third party observer, the `Prover`'s connection to the server appears like a regular TLS connection and the security guaranteed by TLS remains intact for the `Prover`. > -> The only exception is that since the `Verifier` is a party to the MPC TLS, the security for the `Prover` against a malicious `Verifier` is guaranteed by the underlying MPC protocols and not by TLS. \ No newline at end of file +> The only exception is that since the `Verifier` is a party to the MPC TLS, the security for the `Prover` against a malicious `Verifier` is provided by the underlying MPC protocols and not by TLS. + +With the shares of the session key computed and the TLS handshake completed, the parties now proceed to the next MPC protocol where they use their session key shares to jointly generate encrypted requests and decrypt server responses while keeping the plaintext of both the requests and responses private from the `Verifier`. \ No newline at end of file From 77231b586fc053e34596f24df03ea946388b7bad Mon Sep 17 00:00:00 2001 From: sinu <65924192+sinui0@users.noreply.github.com> Date: Wed, 6 Sep 2023 12:41:54 -0700 Subject: [PATCH 08/10] remove verification link until updated --- src/SUMMARY.md | 2 +- 1 file changed, 1 insertion(+), 1 deletion(-) diff --git a/src/SUMMARY.md b/src/SUMMARY.md index 5c21606..23e55b6 100644 --- a/src/SUMMARY.md +++ b/src/SUMMARY.md @@ -10,7 +10,7 @@ - [Handshake](./protocol/mpc-tls/handshake.md) - [Encryption and Decryption](./protocol/mpc-tls/encryption.md) - [Notarization](./protocol/notarization.md) -- [Verification](./protocol/verification.md) +- [Verification]() - [Selective Disclosure]() # MPC From 6d59a0aeac729cb856f7000547b1b23c7d7de907 Mon Sep 17 00:00:00 2001 From: sinu <65924192+sinui0@users.noreply.github.com> Date: Wed, 6 Sep 2023 14:23:14 -0700 Subject: [PATCH 09/10] update key exchange --- diagrams/key_exchange.drawio | 2 +- src/css/global.css | 4 ++ src/mpc/key_exchange.md | 61 ++++++++++++++++++++---------- src/png-diagrams/key_exchange.png | Bin 248795 -> 184481 bytes 4 files changed, 47 insertions(+), 20 deletions(-) diff --git a/diagrams/key_exchange.drawio b/diagrams/key_exchange.drawio index ba55fdf..5338e31 100644 --- a/diagrams/key_exchange.drawio +++ b/diagrams/key_exchange.drawio @@ -1 +1 @@ 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\ No newline at end of file diff --git a/src/css/global.css b/src/css/global.css index bfde080..7cac5b5 100644 --- a/src/css/global.css +++ b/src/css/global.css @@ -1,3 +1,7 @@ :root { --content-max-width: 1000px; } + +.aligncenter { + text-align: center; +} diff --git a/src/mpc/key_exchange.md b/src/mpc/key_exchange.md index 5712a5e..4655b2b 100644 --- a/src/mpc/key_exchange.md +++ b/src/mpc/key_exchange.md @@ -1,33 +1,54 @@ # Key Exchange -In TLS, the first step towards obtaining TLS session keys is to compute a shared secret between the client and the server by running the [ECDH protocol](https://en.wikipedia.org/wiki/Elliptic-curve_Diffie–Hellman). The resulting shared secret in TLS terms is called the pre-master secret `PMS`. +In TLS, the first step towards obtaining TLS session keys is to compute a shared secret between the client and the server by running the [ECDH protocol](https://en.wikipedia.org/wiki/Elliptic-curve_Diffie–Hellman). The resulting shared secret in TLS terms is called the pre-master secret. - +

-Using the notation from Wikipedia, below is the 3-party ECDH protocol between the `Server` the `Requester` and the `Notary`, enabling the `Requester` and the `Notary` to arrive at shares of `PMS`. +Below we will denote: +- $P$ as the `Prover` +- $V$ as the `Verifier` +- $S$ as the `Server` +- $\mathbb{pms}$ as the TLS pre-master secret. -1. `Server` sends its public key $Q_b$ to `Requester`, and `Requester` forwards it to `Notary` -2. `Requester` picks a random private key share $d_c$ and computes a public key share $Q_c = d_c * G$ -3. `Notary` picks a random private key share $d_n$ and computes a public key share $Q_n = d_n * G$ -4. `Notary` sends $Q_n$ to `Requester` who computes $Q_a = Q_c + Q_n $ and sends $Q_a$ to `Server` -5. `Requester` computes an EC point $(x_p, y_p) = d_c * Q_b$ -6. `Notary` computes an EC point $(x_q, y_q) = d_n * Q_b$ -7. Addition of points $(x_p, y_p)$ and $(x_q, y_q)$ results in the coordinate $x_r$, which is `PMS`. (The coordinate $y_r$ is not used in TLS) +## ECDH +Using the notation from Wikipedia, below is the 3-party ECDH protocol between $S$, $P$ and $V$, enabling $P$ and $V$ to arrive at shares of $\mathbb{pms}$. + + +1. $S$ sends its public key $Q_b$ to $P$, and $P$ forwards it to $V$ +2. $P$ picks a random private key share $d_c$ and computes a public key share $Q_c = d_c * G$ +3. $V$ picks a random private key share $d_n$ and computes a public key share $Q_n = d_n * G$ +4. $V$ sends $Q_n$ to $P$ who computes $Q_a = Q_c + Q_n $ and sends $Q_a$ to $S$ +5. $P$ computes an EC point $(x_p, y_p) = d_c * Q_b$ +6. $V$ computes an EC point $(x_q, y_q) = d_n * Q_b$ + +Now our goal is to compute additive shares of $\mathbb{pms}$, which we'll redenote as $x_r$, using elliptic curve point addition -Using the notation from [here](https://en.wikipedia.org/wiki/Elliptic_curve_point_multiplication#Point_addition), our goal is to compute $$ x_r = (\frac{y_q-y_p}{x_q-x_p})^2 - x_p - x_q $$ + in such a way that -1. Neither party learns the other party's $x$ value -2. Neither party learns $x_r$, only their respective shares of $x_r$. -We will use two maliciously secure protocols described on p.25 in the paper [Efficient Secure Two-Party Exponentiation](https://www.cs.umd.edu/~fenghao/paper/modexp.pdf): +- Neither party learns the other party's share. +- Neither party learns $x_r$, only their respective shares of $x_r$. + +To do this we will need two functionalities defined below: + +### A2M -- `A2M` protocol, which converts additive shares into multiplicative shares, i.e. given shares `a` and `b` such that `a + b = c`, it converts them into shares `d` and `e` such that `d * e = c` -- `M2A` protocol, which converts multiplicative shares into additive shares +$\mathsf{A2M}$ converts additive shares into multiplicative shares. -We apply `A2M` to $y_q + (-y_p)$ to get $A_q * A_p$ and also we apply `A2M` to $x_q + (-x_p)$ to get $B_q * B_p$. Then the above can be rewritten as: +For example, given additive shares $a$ and $b$ such that $a + b = c$, invoking $\mathsf{A2M}$ gives $(d, e) \larr (a, b)_{\mathsf{A2M}}$ such that $d * e = c$ + +### M2A + +$\mathsf{M2A}$, which converts multiplicative shares into additive shares. + +For example, given multiplicative shares $d$ and $e$ such that $d * e = c$, invoking $\mathsf{M2A}$ gives $(a, b) \larr (d, e)_{\mathsf{M2A}}$ such that $a + b = c$ + +### Deriving additive shares of the pre-master secret + +We apply $\mathsf{A2M}$ to $y_q + (-y_p)$ to get $A_q * A_p$ and also we apply $\mathsf{A2M}$ to $x_q + (-x_p)$ to get $B_q * B_p$. Then the above can be rewritten as: $$x_r = (\frac{A_q}{B_q})^2 * (\frac{A_p}{B_p})^2 - x_p - x_q $$ @@ -35,6 +56,8 @@ Then the first party locally computes the first factor and gets $C_q$, the secon $$x_r = C_q * C_p - x_p - x_q $$ -Now we apply `M2A` to $C_q * C_p$ to get $D_q + D_p$, which leads us to two final terms each of which is the share of $x_r$ of the respective party: +Now we apply $\mathsf{M2A}$ to $C_q * C_p$ to get $D_q + D_p$, which leads us to two final terms each of which is the share of $x_r$ of the respective party: + +$$x_r = (D_q - x_q) + (D_p - x_p)$$ -$$x_r = (D_q - x_q) + (D_p - x_p)$$ \ No newline at end of file +Now each party holds their respective additive shares of the TLS pre-master secret. \ No newline at end of file diff --git a/src/png-diagrams/key_exchange.png b/src/png-diagrams/key_exchange.png index 48db9562ae3fbc6db9cb81d70c9fc835fa0c4743..632556579cec9c509e947cfef778245660e91ca7 100644 GIT binary patch literal 184481 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a/src/mpc/key_exchange.md b/src/mpc/key_exchange.md index 4655b2b..9ca8bc3 100644 --- a/src/mpc/key_exchange.md +++ b/src/mpc/key_exchange.md @@ -4,16 +4,16 @@ In TLS, the first step towards obtaining TLS session keys is to compute a shared

-Below we will denote: +## ECDH + +We will denote: - $P$ as the `Prover` - $V$ as the `Verifier` - $S$ as the `Server` - $\mathbb{pms}$ as the TLS pre-master secret. -## ECDH - -Using the notation from Wikipedia, below is the 3-party ECDH protocol between $S$, $P$ and $V$, enabling $P$ and $V$ to arrive at shares of $\mathbb{pms}$. +Below is the 3-party ECDH protocol between $S$, $P$ and $V$, enabling $P$ and $V$ to arrive at shares of $\mathbb{pms}$. 1. $S$ sends its public key $Q_b$ to $P$, and $P$ forwards it to $V$