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:

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

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.

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 Gács-Körner Common Information variable.

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:

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

One-Way Communication¶

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

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

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.

Lower Intrinsic Mutual Information¶

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

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

Secrecy Capacity¶

Next is the secrecy capacity:

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

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

Necessary Intrinsic Mutual Information¶

A tighter bound is given by the necessary_intrinsic_mutual_information() [GGunluK17], which is the maximum of the two one-way secret key agreement rates:

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

Interactive Intrinsic Mutual Information¶

$\begin{split}\I{X : Y \uparrow\uparrow\uparrow\uparrow Z} = \max \sum_{i \textrm{even}} \I{U_i : Y | U_{0 \ldots i}} - \I{U_i : Z | U_{0 \ldots i}} + \\ \sum_{i \textrm{odd}} \I{U_i : X | U_{0 \ldots i}} - \I{U_i : Z | U_{0 \ldots i}}\end{split}$

Upper Bounds¶

Upper Intrinsic Mutual Information¶

The secret key agreement rate is trivially upper bounded by:

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

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

The intrinsic_mutual_information() [MW97] 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 $$\I{X : Y | Z}$$.

Reduced Intrinsic Mutual Information¶

This bound can be improved, producing the reduced_intrinsic_mutual_information() [RSW03]:

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

This bound improves upon the 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 $$U$$.

Minimal Intrinsic Mutual Information¶

The Reduced Intrinsic Mutual Information can be further reduced into the minimal_intrinsic_total_correlation() [GA17]:

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

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{split}\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}\end{split}

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 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:

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


First, we consider the distribution intrinsic_1:

In [2]: In [3]: print(intrinsic_1)
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
----> 1 print(intrinsic_1)

NameError: name 'intrinsic_1' is not defined

In [3]: Class:          Distribution

In [4]: Alphabet:       ('0', '1', '2', '3') for all rvs
...: Base:           linear
...: Outcome Class:  str
...: Outcome Length: 3
...: RV Names:       None
...:
File "<ipython-input-4-f997bcd614eb>", line 1
Alphabet:       ('0', '1', '2', '3') for all rvs
^
SyntaxError: invalid syntax


With upper bounds:

In [5]: In [4]: upper_intrinsic_mutual_information(intrinsic_1, [[0], [1]], [2])
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
----> 1 upper_intrinsic_mutual_information(intrinsic_1, [[0], [1]], [2])

NameError: name 'intrinsic_1' is not defined

In [6]: 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 intrinsic_mutual_information(), however can detect this:

In [7]: In [5]: intrinsic_mutual_information(intrinsic_1, [[0], [1]], [2])
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-7-cf75a74ddcab> in <module>
----> 1 intrinsic_mutual_information(intrinsic_1, [[0], [1]], [2])

NameError: name 'intrinsic_1' is not defined

In [8]: Out[5]: 0.0


Next, let’s consider the distribution intrinsic_2:

In [9]: In [7]: print(intrinsic_2)
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-9-094f23e395aa> in <module>
----> 1 print(intrinsic_2)

NameError: name 'intrinsic_2' is not defined

In [10]: Class:          Distribution

In [11]: Alphabet:       (('0', '1', '2', '3'), ('0', '1', '2', '3'), ('0', '1'))

In [12]: Base:           linear
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-12-822bc640d06e> in <module>
----> 1 Base:           linear

NameError: name 'linear' is not defined

In [13]: Outcome Class:  str
....: Outcome Length: 3
....: RV Names:       None
....:
File "<ipython-input-13-ab3546a5787a>", line 1
Outcome Class:  str
^
SyntaxError: invalid syntax


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:

In [14]: In [8]: intrinsic_mutual_information(intrinsic_2, [[0], [1]], [2])
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-14-066b11243fe9> in <module>
----> 1 intrinsic_mutual_information(intrinsic_2, [[0], [1]], [2])

NameError: name 'intrinsic_2' is not defined

In [15]: 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:

In [16]: In [9]: minimal_intrinsic_mutual_information(intrinsic_2, [[0], [1]], [2], bounds=(3,))
---------------------------------------------------------------------------
NameError                                 Traceback (most recent call last)
<ipython-input-16-523cc5b9b003> in <module>
----> 1 minimal_intrinsic_mutual_information(intrinsic_2, [[0], [1]], [2], bounds=(3,))

NameError: name 'intrinsic_2' is not defined

In [17]: Out[9]: 1.0