EUROCRYPT '96 IACR Distinguished Lecture)

Signal & Information Processing Laboratory

Swiss Federal Institute of Technology

ETH Zentrüm

CH-8092 Zürich, Switzerland

massey@isi.ee.ethz.ch

July 17, 1996

Transparency 2 indicates the kind of provable
security that I would like to see eventually reached in cryptography. By way
of contrast, transparency 3
indicates the kind of provable security that most
people talk about today. From a complexity viewpoint, transparency 2 deals
with a * non-uniform* complexity measure while
transparency 3 deals
with a * uniform* complexity measure (i.e., the same algorithm must
compute all instances of the function). In the former case, it makes sense to
talk about a function (i.e., one and only one instance of a ``function")
being difficult, whereas in the latter case one must always talk about the
difficulty of an infinite sequence of functions. My lecture was aimed at the
former kind of difficulty.

Transparency 3
gives the definition of a one-way function (* a la*
Diffie and Hellman). Here I suppose function to mean one and only one
``function" so that the underlying notion of difficulty relates to some
non-uniform complexity measure. The concept of a one-way function is so
fundamental in modern cryptography that it seems to me essential that we face
head-on the question of whether such a thing exists.

On transparencies 6 and 7, I review the essentials of Cauchy's theory of
permutations on **n** objects. Cauchy noted that a permutation is fully
described by the * cycles* that it induces. Every permutation can be
written as the composition of disjoint cycles in which, because of the
disjointness, the order of the cycles is arbitrary. A * transposition*
is a permutation that swaps two objects but leaves the other **n - 2**
objects unchanged. Every permutation can be written as a composition of
transpositions. Thus, it seems natural to define the * transposition
complexity* of a permutation as the smallest number of transpositions
whose composition realizes that permutation.

As shown on transparency 7, an **m**-cycle
can be realized with **m - 1** transpositions. Hence, as shown on transparency 8, a permutation that is the composition
of **c** disjoint cycles has transposition complexity at most **n - c**.
It seems almost obvious that **n - c** is exactly the transposition
complexity, but I indicate at the bottom of transparency 8 the questions that must be answered
before one can conclude that this is the case. I digressed on transparency 9 to give the simple proof that a *
permuation on n objects and its inverse have the same transposition
complexity*. Hence,

On transparencies 13 and 14, I use Shannon's famous counting argument [1] for underbounding the gate complexity of the
most difficult boolean function of n variables. One simply counts (more
precisely, one overbounds) the number of different acyclic gate networks of
**K** gates. If this number is less that the number
2^{2n} of different boolean functions of **n**
variables, then some such function has gate complexity greater than **K**.
Using apparently very crude bounds, I show on these transparencies that for
every **n>7** there are boolean functions of **n**
variables whose gate complexity is at least 2^{n}/(2n) (and indeed
virtually all such functions have at least this gate complexity). Standard
synthesis arguments can be used to show that a number of gates greater than
this lower bound by only a small factor suffices to realize any such function.
On transparency 15, I summarize the upper and
lower bounds of this type on gate complexity that were obtained by my former
doctoral student, Alain Hiltgen [2]. It seems
to me quite remarkable that, for all n>27, the most difficult boolean
function of n variables has gate complexity that we are sure lies between
2^{n}/n and 2×2^{n}/n and that * virtually all*
such functions have gate complexity in this same narrow range. It may seem
quite surprising then, as mentioned on transparency
16, that no one has exhibited a constructive function of **n**
variables whose gate complexity exceeds **3n - 3**, cf. [3] for this construction. This becomes less
surprising when one notes that the strongest general tool available today for
proving lower bounds on the gate complexity of a specific boolean function is
the quite trivial argument that this complexity must be at least the number of
non-idle variables less one.

Transparency 17 shifts the discussion to the
gate complexity of * invertible functions* from **n** bits to
**n** bits, i.e., to functions in the set P_{n} of permutations on
the set {0,1}^{n}, which is where cryptographers are especially
interested in finding one-way functions (should they exist). On transparency 18, I give the results of an exhaustive
search made about six years ago by two of my doctoral students, Alain Hiltgen
and Jürg Ganz, for such functions that are more difficult to invert than
to compute [4]. There are obviously no such
functions in P_{1} or in P_{2}, but we were very surprised to
find that there were also none in P_{3} even though there are
functions in P_{3} whose gate complexity is 7. The first such
computationally asymmetric function showed up in P_{4}, requiring 5
gates to compute while its inverse requires 6. This function is given on
transparency 18 and is probably the first computationally asymmetric function
ever found.

On transparency 19, I briefly consider the
more general set B_{nm} of functions from **n** bits to **m**
bits and mention the almost trivial Lamagne-Savage lower bound [5] on their gate complexity, which is the strongest
tool available today for proving lower bounds on the gate complexity of a
specific function in B_{nm}. On transparency
20, I show the form of certain binary [i.e., over the field GF(2) of two
elements] matrices introduced by Boppana and Lagarias [6]. Alain Hiltgen [7]
showed that the linear functions in P_{n} described by these matrices
are * feebly one-way* for all A n>4 in the sense that the function
has gate complexity **n + 1** while its inverse has gate complexity equal
to the integer part of 3(n-1)/2. His simple proof is given on transparency 21. What interests me here is that the
lower bound on complexity obtained in the proof results from the two rather
trivial arguments mentioned above, viz. the idle variable bound and the
Lamagna-Savage bound.

Hiltgen [7], with considerably more effort,
modified the Boppana-Lagarias matrices to obtain, for every n>3, a
feebly-one-way function with gate complexity **n + 2** whose inverse has
gate complexity 2(n-1). These functions in P_{n}, whose inverse is
about twice as complex as the function itself, still hold the world record for
computational asymmetry.

On transparency 22, I begin the demonstration
that constitutes the climax of this lecture. Again I make use of Shannon's
counting argument to show that, for all n>5, virtually all functions in
P_{n} have gate complexity greater than 2^{n}/5. Then I
overbounded the gate complexity of all functions in P_{n} simply by
applying Hiltgen's upper bound of 2×2^{n}/n on the gate
complexity of every boolean function of **n** variables (see transparency 15) to each of the **n** one-bit
output functions of a function in P_{n}. This gives an upper bound of
2×2^{n} on the gate complexity of all functions in
P_{n}. But the inverse of a function in P_{n} is again a
function in P_{n} so this establishes the proposition given on transparency 24, viz.

Proposition:For all n>27, virtually all functions in P_{n}have gate complexity different by a factor of less than 10 from the gate complexity of their inverse function.

The inescapable conclusion is that

The final transparency 24 gives the solution of the cryptographic puzzle on transparency 5.

**1**- C. E. Shannon, ``The Synthesis of Two-Terminal Switching
Circuits,"
*Bell System Tech. J.*, vol. 28, pp. 59-98, 1949. **2**- A. P. L. Hiltgen,
*Cryptographically Relevant Contributions to Combinatorial Complexity Theory*, (ETH-Zürich Dissertation). Konstanz: Hartung-Gorre Verlag, 1994. **3**- N. Blum, ``A Boolean Function Requiring 3
**n**Network Size,"*Theoretical Comp. Sci.*, vol. 28, pp. 337-345, 1984. **4**- A. P. Hiltgen and J. Ganz,
*On the Existence of Specific Complexity--Asymmetric Permutations*, Technical Rept., Signal and Information Proc. Lab., ETH-Zürich, 1992. **5**- E. A. Lamagna and J. E. Savage, ``Combinational Complexity of
Some Monotone Functions,"
*Proc. 15th Ann. Symp. Switching and Automata Th.*, pp. 140-144, 1974. **6**- R. B. Boppana and J. C. Lagarias, ``One-Way Functions and
Circuit
Complexity,"
*Information and Computation*, vol. 74, pp. 226-2240, 1987. **7**- A. P. Hiltgen, ``Constructions of Feebly-One-Way Families of
Permutations," in
*Advances in Cryptology--AUSCRYPT '92*(Eds. J. Seberry and Y. Zheng), Lecture Notes in Computer Science No. 718. New York: Springer, 1993, pp. 422-434.

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