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.. py:module:: dit.multivariate.secret_key_agreement



## Secret Key Agreement

One of the only methods of encrypting a message from Alice to Bomb such that no third party (Eve) can possibly decrypt it is a one-time pad. This technique requires that Alice and Bob have a secret sequence of bits, S, which Alice then encrypts by computing the exclusive-or of it with the plaintext, P, to produce the cyphertext, C: C = S \oplus P. Bob can then decrypt by xoring again: P = S \oplus C.

In order to pull this off, Alice and Bob need to construct S out of some sort joint randomness, p(x, y, z), and public communication, V, which is assumed to have perfect fidelity. The maximum rate at which S can be constructed in the secret key agreement rate.

### Background

Given N IID copies of a joint distribution governed by p(x, y, z), let X^N denote the random variables observed by Alice, Y^N denote the random variables observed by Bob, and Z^N denote the random variables observed by Even. A secret key agreement scheme consists of functions f and g, as well as a protocol for public communication (producing V), and is considered R-achievable if:

S_X = f(X^N, V) \\
S_Y = g(Y^N, V) \\
p(S_X = S_Y = S) \geq 1 - \epsilon \\
\I{S : V Z^N} \leq \epsilon \\
\frac{1}{N} \H{S} \geq R - \epsilon


The maximum rate R such that there exists a R-achievable scheme is known as the secret key agreement rate. Intuitively, this means there exists some procedure such that, for every N observations, Alice and Bob can publicly converse and then construct S bits which agree almost surely, and are almost surely independent of everything Eve has access to. S is then known as a secret key.

There are three general classes of secret key agreement rates, depending on which parties are permitted to communicate. We discuss them below.

.. py:module:: dit.multivariate.secret_key_agreement.no_communication


### No Communication

In the case that neither Alice nor Bob are permitted communication, the no-communication secret key agreement rate is given by:

\operatorname{S}[X : Y || Z] = \I{X \meet Y | Z}


where X \meet Y is the :ref:Gács-Körner Common Information variable.

.. py:module:: dit.multivariate.secret_key_agreement.secrecy_capacity


#### Secrecy Capacity

Consider the situation that no party is allowed to communication, but rather than passively observing p(X, Y, Z) Alice has full access to driving the channel p(Y, Z | X). In this case we arrive at a maximum secret-key agreement rate known as the secrecy capacity, which is given by:

\operatorname{S_C}[X \rightarrow Y || Z] = \displaystyle \max_{U - X - YZ} \I{U : Y} - \I{U : Z}

.. py:module:: dit.multivariate.secret_key_agreement.one_way_skar


### One-Way Communication

If only Alice is allowed to publicly broadcast information, the secret key agreement rate is given by:

\operatorname{S}[X \rightarrow Y || Z] = \displaystyle \max_{V - U - X - YZ} \I{U : Y | V} - \I{U : Z | V}

.. py:module:: dit.multivariate.secret_key_agreement.two_way_skar


### Two-Way Communication

When both Alice and Bob are permitted communication, the secret key agreement rate, \operatorname{S}[X \leftrightarrow Y || Z], is much more difficult to compute, and in fact only upper and lower bounds on this rate are known.

#### Lower Bounds

The first few lower bounds on two-way secret key agreement rate are simply symmetrized forms of the more restricted secret key agreement rates.

.. py:module:: dit.multivariate.secret_key_agreement.trivial_bounds

##### Lower Intrinsic Mutual Information

The first lower bound on the secret key agreement rate is known in dit as the :py:func:lower_intrinsic_mutual_information, and is given by:

\I{X : Y \uparrow Z} = \max
\begin{cases}
\I{X : Y} - \I{X : Z} \\
\I{X : Y} - \I{Y : Z} \\
0
\end{cases}

.. py:module:: dit.multivariate.secret_key_agreement.skar_lower_bounds

##### Secrecy Capacity

Next is the secrecy capacity:

\I{X : Y \uparrow\uparrow Z} = \max
\begin{cases}
\displaystyle \max_{U - X - YZ} \I{U : Y} - \I{U : Z} \\
\displaystyle \max_{U - Y - XZ} \I{U : X} - \I{U : Z}
\end{cases}


This gives the secret key agreement rate when communication is not allowed.

##### Necessary Intrinsic Mutual Information

A tighter bound is given by the :py:func:necessary_intrinsic_mutual_information :cite:gohari2017achieving, which is the maximum of the two one-way secret key agreement rates:

\I{X : Y \uparrow\uparrow\uparrow Z} = \max
\begin{cases}
\displaystyle \max_{V - U - X - YZ} \I{U : Y | V} - \I{U : Z | V} \\
\displaystyle \max_{V - U - Y - XZ} \I{U : X | V} - \I{U : Z | V}
\end{cases}

.. py:module:: dit.multivariate.secret_key_agreement.interactive_intrinsic_mutual_informations

##### Interactive Intrinsic Mutual Information
\I{X : Y \uparrow\uparrow\uparrow\uparrow Z} = \max
\sum_{i \textrm{even}} \I{U_i : Y | U_{0 \ldots i}} - \I{U_i : Z | U_{0 \ldots i}} + \\
\sum_{i \textrm{odd}}  \I{U_i : X | U_{0 \ldots i}} - \I{U_i : Z | U_{0 \ldots i}}


#### Upper Bounds

.. py:module:: dit.multivariate.secret_key_agreement.trivial_bounds

##### Upper Intrinsic Mutual Information

The secret key agreement rate is trivially upper bounded by:

\min\{ \I{X : Y}, \I{X : Y | Z} \}

.. py:module:: dit.multivariate.secret_key_agreement.intrinsic_mutual_informations


Intrinsic Mutual Information *****************&******

The :py:func:intrinsic_mutual_information :cite:maurer1997intrinsic is defined as:

\I{X : Y \downarrow Z} = \min_{p(\overline{z} | z)} \I{X : Y | \overline{Z}}


It is straightforward to see that p(\overline{z} | z) being a constant achieves \I{X : Y}, and p(\overline{z} | z) being the identity achieves \I{X : Y | Z}.

.. py:module:: dit.multivariate.secret_key_agreement.reduced_intrinsic_mutual_informations

##### Reduced Intrinsic Mutual Information

This bound can be improved, producing the :py:func:reduced_intrinsic_mutual_information :cite:renner2003new:

\I{X : Y \downarrow\downarrow Z} = \min_{U} \I{X : Y \downarrow ZU} + \H{U}


This bound improves upon the :ref:Intrinsic Mutual Information when a small amount of information, U, can result in a larger decrease in the amount of information shared between X and Y given Z and U.

.. py:module:: dit.multivariate.secret_key_agreement.minimal_intrinsic_mutual_informations

##### Minimal Intrinsic Mutual Information

The :ref:Reduced Intrinsic Mutual Information can be further reduced into the :py:func:minimal_intrinsic_total_correlation :cite:gohari2017comments:

\I{X : Y \downarrow\downarrow\downarrow Z} = \min_{U} \I{X : Y | U} + \I{XY : U | Z}

##### Two-Part Intrinsic Mutual Information
\I{X : Y \downarrow\downarrow\downarrow\downarrow Z} = inf_{J} min_{V - U - XY - ZJ} \I{X : Y | J} + \I{U : J | V} - \I{U : Z | V}


#### All Together Now

Taken together, we see the following structure:

\begin{align}
&\min\{ \I{X : Y}, \I{X : Y | Z} \} \\
&\quad \geq \I{X : Y \downarrow Z} \\
&\quad\quad \geq \I{X : Y \downarrow\downarrow Z} \\
&\quad\quad\quad \geq \I{X : Y \downarrow\downarrow\downarrow Z} \\
&\quad\quad\quad\quad \geq \I{X : Y \downarrow\downarrow\downarrow\downarrow Z} \\
&\quad\quad\quad\quad\quad \geq S[X \leftrightarrow Y || Z] \\
&\quad\quad\quad\quad\quad\quad \geq \I{X : Y \uparrow\uparrow\uparrow\uparrow Z} \\
&\quad\quad\quad\quad\quad\quad\quad \geq \I{X : Y \uparrow\uparrow\uparrow Z} \\
&\quad\quad\quad\quad\quad\quad\quad\quad \geq \I{X : Y \uparrow\uparrow Z} \\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad \geq \I{X : Y \uparrow Z} \\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \geq S[X : Y || Z] \\
&\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad\quad \geq 0.0
\end{align}


#### Generalizations

Most of the above bounds have straightforward multivariate generalizations. These are not necessarily bounds on the multiparty secret key agreement rate. For example, one could compute the :py:func:minimal_intrinsic_dual_total_correlation:

\B{X_0 : \ldots : X_n \downarrow\downarrow\downarrow Z} = \min_{U} \B{X_0 : \ldots : X_n | U} + \I{X_0, \ldots, X_n : U | Z}


#### Examples

Let us consider a few examples:

.. ipython::

In [1]: from dit.multivariate.secret_key_agreement import *

In [2]: from dit.example_dists.intrinsic import intrinsic_1, intrinsic_2, intrinsic_3



First, we consider the distribution intrinsic_1:

.. ipython::

In [3]: print(intrinsic_1)
Class:          Distribution
Alphabet:       ('0', '1', '2', '3') for all rvs
Base:           linear
Outcome Class:  str
Outcome Length: 3
RV Names:       None

x     p(x)
000   1/8
011   1/8
101   1/8
110   1/8
222   1/4
333   1/4



With upper bounds:

.. ipython::

@doctest float
In [4]: upper_intrinsic_mutual_information(intrinsic_1, [[0], [1]], [2])
Out[4]: 0.5



We see that the trivial upper bound is 0.5, because without conditioning on Z, X and Y can agree when the observe either a 2 or a 3, which results in \I{X : Y} = 0.5. Given Z, however, that information is no longer private. But, given Z, a conditional dependence is induced between X and Y: Z knows that if she is a 0 that X and Y agree, and if she is a 1 they disagree. This results \I{X : Y | Z} = 0.5. In either case, however, X and Y can not agree upon a secret key: in the first case the eavesdropper knows their correlation, while in the second they are actually independent.

The :py:func:intrinsic_mutual_information, however can detect this:

.. ipython::

@doctest float
In [5]: intrinsic_mutual_information(intrinsic_1, [[0], [1]], [2])
Out[5]: 0.0



Next, let's consider the distribution intrinsic_2:

.. ipython::

In [7]: print(intrinsic_2)
Class:          Distribution
Alphabet:       (('0', '1', '2', '3'), ('0', '1', '2', '3'), ('0', '1'))
Base:           linear
Outcome Class:  str
Outcome Length: 3
RV Names:       None

x     p(x)
000   1/8
011   1/8
101   1/8
110   1/8
220   1/4
331   1/4



In this case, Z no longer can distinguish between the case where X and Y can agree on a secret bit, and when they can not, because she can not determine when they are in the 01 regime or in the 23 regime:

.. ipython::

@doctest float
In [8]: intrinsic_mutual_information(intrinsic_2, [[0], [1]], [2])
Out[8]: 1.5



This seems to imply that X and Y can adopt a scheme such as: if they observe either a 0 or a 1, write down 0, and if they observe either a 2 or a 3, write that down. This has a weakness, however: what if Z were able to distinguish the two regimes? This costs her 1 bit, but reduces the secrecy of X and Y to nil. Thus, the secret key agreement rate is actually only 1 bit:

.. ipython::

@doctest float
In [9]: minimal_intrinsic_mutual_information(intrinsic_2, [[0], [1]], [2], bounds=(3,))
Out[9]: 1.0