Cryptography – Particular algorithmic function encoding – Public key
Reexamination Certificate
1998-02-26
2002-12-10
Hayes, Gail (Department: 2131)
Cryptography
Particular algorithmic function encoding
Public key
C380S028000, C380S285000
Reexamination Certificate
active
06493449
ABSTRACT:
BACKGROUND OF THE INVENTION
1. Field of the Invention
The invention relates to algebraic key establishment protocols for cryptographic applications.
2. Description of the Prior Art
Key Establishment Protocols
The concepts, terminology and framework for understanding cryptographic key establishment protocols is given in Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, “Handbook of Applied Cryptography,” CRC Press (1997), pages 490-491.
A ‘protocol’ is a multi-party algorithm, defined by a sequence of steps specifying the actions required of two or more parties in order to achieve a specified objective.
A ‘key establishment’ protocol is a protocol whereby a shared secret becomes available to two or more parties, for subsequent cryptographic applications.
A ‘key transport’ protocol is a key establishment protocol where one party creates or obtains a secret value, and securely transfers it to the other participating parties.
A ‘key agreement’ protocol is a key establishment protocol in which a shared secret is derived by two (or more) parties as a function of information contributed by, or associated with, each of the participating parties such that no party can predetermine the resulting value.
A ‘key-distribution’ protocol is a key establishment protocol whereby the established keys are completely determined a priori by initial keying material.
A ‘dynamic’ key establishment protocol is one whereby the key established by a fixed pair (or subset) of the participating parties varies on subsequent executions. Dynamic key establishment protocols are also referred to as ‘session’ key establishment protocols, and it is usually intended that these protocols are immune from known-key attacks.
The Diffie-Hellman key agreement protocol (also called ‘exponential key exchange’) is a fundamental algebraic protocol. It is presented in W. Diffie and M. E. Hellman, “New Directions in Cryptography,” IEEE Transaction on Information Theory vol. IT 22 (November 1976), pp. 644-654. The Diffie-Hellman key agreement protocol provided the first practical solution to the key distribution problem, allowing two parties, never having met in advance or sharing keying material, to establish a shared secret by exchanging messages over an open channel. The security rests on the intractability of the Diffie-Hellman problem and the related problem of computing discrete logarithms in the multiplicative group of the finite field GF(p) where p is a large prime, cf. Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, “Handbook of Applied Cryptography,” CRC Press (1997), page 113.
A key establishment protocol is said to have ‘perfect forward secrecy’ if compromise of long-term keys does not compromise past session keys. The idea of perfect forward security is that previous traffic is locked safely in the past. It may be provided by generating session keys by Diffie-Hellman key agreement, wherein the Diffie-Hellman exponentials are based on short term keys. If long-term secret keys are compromised, future sessions are nonetheless subject to impersonation by an active adversary (cf. Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, “Handbook of Applied Cryptography,” CRC Press (1997), page 496).
‘Point-to-point key update’ techniques based on symmetric encryption would make use of a long-term symmetric key K shared a priori by two parties A and B. The Diffie-Hellman key agreement protocol allows for the establishment of such a K. Thus, the Diffie-Hellman key agreement protocol together with the symmetric encryption system provide the primitives in specifying a key transport protocol (cf. Alfred J. Menezes, Paul C. van Oorschot, and Scott A. Vanstone, “Handbook of Applied Cryptography,” CRC Press (1997), page 497).
Combinatorial Group Theory
The definition of a monoid is given in Serge Lang, “Algebra,” Third Edition, Addison-Wesley Publishing Company Inc. (1993), page 3.
Quote
Let S be a set. A mapping S×S→S is sometimes called a law of composition (of S into itself). If x, y are elements of S, the image of the pair (x, y) under the mapping is also called their product under the law of composition, and will be denoted xy . . . . Let S be a set with a law of composition. If x, y, z are elements of S, then we may form their product in two ways: (xy)z and x(yz). If (xy)z=x(yz) for all x, y, z in S then we say that the law of composition is associative.
An element e of S such that ex=x=xe for all x &egr; S is called a unit element.
A unit element is unique, for if e′ is another unit element, we have e=ee′=e′ by assumption. In most cases, the unit element is written simply 1 (instead of e) . . . .
A monoid is a set G, with a law of composition which is associative, and having a unit element (so that in particular, G is not empty).
Unquote
The definition of a group is given in Serge Lang, “Algebra,” Third Edition, Addison-Wesley Publishing Company Inc. (1993), page 7.
Quote
A group G is a monoid, such that for every element x &egr; G there exists an element y &egr; G such that xy=yx=e. Such an element y is called an inverse for x. Such an inverse is unique. . . We denote this inverse by x
−1
.
Unquote
The basic reference for concepts, terminology, and historical framework in combinatorial group theory is the monograph by Bruce Chandler and Wilhelm Magnus, “The history of combinatorial group theory: a case study in the history of ideas,” Springer-Verlag (1982). We quote from page 3:
Quote
Combinatorial group theory may be characterized as the theory of groups which are given by generators and defining relations, or, as we would say today, by a presentation.
Unquote
The following problems were posed by M. Dehn in 1911. We quote from the monograph by Bruce Chandler and Wilhelm Magnus, “The history of combinatorial group theory: a case study in the history of ideas,” Springer-Verlag (1982), page 19.
Quote
The Word Problem (called Identitaetsproblem by Dehn) Let an arbitrary element of the group be given through its buildup in terms of the generators. Find a method to decide in a finite number of steps whether this element equals the identity element or not.
The Conjugacy Problem (called Transformationsproblem by Dehn) Any two elements S and T of the group are given. Find a method to decide whether S and T are conjugate, i.e. whether there exists an element U of the group which satisfies the relation S=UTU
−1
.
Unquote
The comparison form of the word problem can be stated as follows:
Comparison Form of the Word Problem Let u, v be any two elements of the group given. Find a method to decide in a finite number of steps whether u=v.
Assume that G is a group given by a presentation P(G). Let W(G) denote the set of all words in the generators and their inverses given in the presentation of G. The functional form of the word problem is to produce a mapping F from W(G) to W(G) such that for all u, v &egr; W(G) it follows that F(u)=F(v) if and only if u, v define the same element of G with respect to the presentation P(G). For each element u &egr; W(G) the element F(u) is termed the canonical form of u.
The functional form of the word problem requires an algorithm to produce canonical forms.
The Canonical Form Problem Let u be an arbitrary element of the given group. Specify a method to find, in a finite number of steps, a canonical form for u.
The functional form of the conjugacy problem requires, in addition, an algorithm to actually produce the conjugating element U.
Generalized Conjugacy Problem (functional form) Let S
1
, S
2
, . . . , S
n
be elements of a group G. Assume that a &egr;G is secret and the set of n pairs of elements of the group G
{
s
1
,a
−1
s
1
a},{s
2
,a
−1
s
2
a}, . . . {s
n
,a
−1
s
n
a}
are publicly announced. Find an algorithm to actually produce such an element a.
It is self evident that this problem is harder than the original conjugacy problem. It has been known for some time that there exist groups with solvable
Anshel Iris
Anshel Michael M.
Goldfeld Dorian
Arithmetica, Inc.
Hayes Gail
Seal James
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