US 8,509,426 B1Granted

XZ-elliptic curve cryptography system and method

AssigneeKing Fahd University of Petroleum and Minerals

InventorsLahouari Ghouti, Mohammad K. Ibrahim, Abdulaziz M. Al-Khoraidly

Life of this patent

Estimating the term…

Abstract

The XZ-elliptic curve cryptography system and method provides a computerized method that allows for the encryption of messages through elliptic polynomial cryptography and, particularly, with the embedding of either a symmetric secret key or a public key in the message bit string. The method of performing XZ-elliptic polynomial cryptography is based on the elliptic polynomial discrete logarithm problem. It is well known that an elliptic polynomial discrete logarithm problem is a computationally “difficult” or “hard” problem.

Claims13

Dependent claims

A computerized method of performing XZ-elliptic curve cryptography, comprising the steps of:

(a) a sending correspondent and a receiving correspondent selecting and agreeing upon an elliptic curve set EC3;
(b) the sending correspondent and the receiving correspondent further agreeing upon a random scalar k and a random shared secret key for communication Es, and agreeing upon a base point (XB,YB,ZB)εEC3;
the sending correspondent then performs the following steps:
(c) embedding a first secret message bit string into an elliptic curve message point (Xm,Ym,Zm);
(d) computing a scalar multiplication between the base point (XB,YB,ZB) and the scalar k as (XBk,YBk,ZBk)=k(XB,YB,ZB);
(e) computing a cipher point (Xc,Yc,Zc) as (Xc,Yc,Zc)=(Xm,Ym,Zm)+k(XB,YB,ZB);
(f) embedding a second secret message bit string into a data transformation index Em;
(g) transforming the cipher point coordinates Xc and Zc as {circumflex over (X)}C=XCEs and {circumflex over (Z)}C=ZCEm, respectively;
(h) calculating a cipher transformation index Ec as
E
c
=
E
s
E
m
;
(i) sending a set of appropriate bits of {circumflex over (X)}c, {circumflex over (Z)}c, and Êc to the receiving correspondent;
the receiving correspondent then performs the following steps:
(j) calculating the data transformation index Em as
E
m
=
E
s
E
c
;
(k) calculating the cipher point coordinates Xc and Zc as XC={circumflex over (X)}CEs−1 and ZC={circumflex over (Z)}CEm−1, respectively;
(l) computing a scalar multiplication between the base point (XB,YB,ZB) and the scalar k as (XBk,YBk,ZBk)=k(XB,YB,ZB);
(m) computing the elliptic curve message point (Xm,Ym,Zm) as (Xm,Ym,Zm)=(Xc,Yc,Zc)−k(XB,YB,ZB);
and
(n) retrieving the first secret message bit string from the elliptic curve message point (Xm,Ym,Zm).

30 more elements

A computerized method of performing XZ-elliptic curve cryptography with a public key, comprising the steps of:

(a) a sending correspondent and a receiving correspondent selecting and agreeing upon an elliptic curve set EC3, the sending correspondent further establishing a private key pair (kSPr,eSPr);
(b) the sending correspondent and the receiving correspondent further agreeing upon a base point (XB,YB,ZB)εEC3 and sharing a public key pair (kSPr(XB,YB,ZB),EbeSPr);
the sending correspondent then performs the following steps:
(c) calculating a shared key as (kSPr(kRPr(XB,YB,ZB)),(EbeRPr)eSPr)
(d) embedding a first secret message bit string into an elliptic curve message point (Xm,Ym,Zm);
(e) computing a cipher point (Xc,Yc,Zc) as (Xc,Yc,Zc)=(Xm,Ym,Zm)+kSPr(kRPr(XB,YB,ZB));
(f) embedding a second secret message bit string into a data transformation index Em;
(g) transforming the cipher point coordinates Xc and Zc as {circumflex over (X)}C=XC(EbeRPr)eSPr and {circumflex over (Z)}C=ZCEm, respectively;
(h) calculating a cipher transformation index Ec as
E
c
=
(
E
b
e
RPr
)
e
SPr
E
m
;
(i) sending a set of appropriate bits of {circumflex over (X)}c, {circumflex over (Z)}c and Ec to the receiving correspondent;
the receiving correspondent then performs the following steps:
(j) calculating the shared key as (kRPr(kSPr(XB,YB,ZB)),(EbeSPr)eRPr);
(k) calculating the data transformation index Em as
E
m
=
(
E
b
e
RPr
)
e
SPr
E
c
;
(l) calculating the cipher point coordinates Xc and Zc as XC={circumflex over (X)}C(EbeRPr)eSPr)−1 and ZC={circumflex over (Z)}CEm−1, respectively;
(m) computing the elliptic curve message point (Xm,Ym,Zm) as (Xm,Ym,Zm)=(Xc,Yc,Zc)−kRPr(kSPr(XB,YB,ZB));
and
(n) retrieving the first secret message bit string from the elliptic curve message point (Xm,Ym,Zm).

42 more elements

A system for performing XZ-elliptic curve cryptography, wherein a sending correspondent and a receiving correspondent select and agree upon an elliptic curve set EC3, wherein the sending correspondent further establishes a private key pair (kSPr,eSPr), the sending correspondent and the receiving correspondent further agreeing upon a base point (XB,YB,ZB)εEC and sharing a public key pair (kSPr(XB,YB,ZB),EbeSPr), the system comprising:

a processor;
computer readable memory coupled to the processor;
a user interface coupled to the processor;
a display coupled to the processor;
software stored in the memory and executable by the processor, the software having:
means for calculating a shared key as (kSPr(kRPr(XB,YB,ZB)),(EbeRPr)eSPr) at the sending correspondent;
means for embedding a first secret message bit string into an elliptic curve message point (Xm,Ym,Zm) at the sending correspondent;
means for computing a cipher point (Xc,Yc,Zc) as (Xc,Yc,Zc)=(Xm,Ym,Zm)+kSPr(kRPr(XB,YB,ZB)) at the sending correspondent;
means for embedding a second secret message bit string into a data transformation index Em, at the sending correspondent;
means for transforming the cipher point coordinates Xc and Zc as {circumflex over (X)}C=XC(EbeRPr)eSPr and {circumflex over (Z)}C=ZCEm, respectively, at the sending correspondent;
means for calculating a cipher transformation index Ec as
E
c
=
(
E
b
e
RPr
)
e
SPr
E
m
at the sending correspondent;
means for sending a set of appropriate bits of {circumflex over (X)}c, {circumflex over (Z)}c, and Ec, to the receiving correspondent;
means for calculating the shared key as (kRPr(kSPr(XB,YB,ZB)),(EbeSPr)eRPr) at the receiving correspondent;
means for calculating the data transformation index Em as
E
m
=
(
E
b
e
RPr
)
e
SPr
E
c
at the receiving correspondent;
means for calculating the cipher point coordinates Xc and Zc as XC={circumflex over (X)}C(EbeRPr)eSPr)−1 and ZC={circumflex over (Z)}CEm−1, respectively, at the receiving correspondent;
means for computing the elliptic curve message point (Xm,Ym,Zm) as (Xm,Ym,Zm)=(Xc,Yc,Zc)−kRPr(kSPr(XB,YB,ZB)) at the receiving correspondent;
and
means for retrieving the first secret message bit string from the elliptic curve message point (Xm,Ym,Zm).

43 more elements

Dependencies are inferred from the claim text.

Facts

Application: 12/958,273Filed 1 Dec 2010
Published: US 2013/0202104 A18 Aug 2013
Granted: 13 Aug 2013
Term ends: Not estimatedThe source records don’t support an estimate.

Classification4 codes

H04L 9/003
Countermeasures against attacks on cryptographic mechanisms: for power analysis, e.g. differential power analysis [DPA] or simple power analysis [SPA]
H04L 9/3013
Underlying computational problems or public-key parameters: involving the discrete logarithm problem, e.g. ElGamal or Diffie-Hellman systems
H04L 9/3066
Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy: involving…
H04L
Digital communication

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US 2013/0202104 A1Published application
US 8,509,426 B1This patent

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Abstract

Facts

Application

12/958,273Filed 1 Dec 2010

Published

US 2013/0202104 A18 Aug 2013

Granted

13 Aug 2013

Term ends

Not estimatedThe source records don’t support an estimate.

Classification4 codes

H04L 9/003
Countermeasures against attacks on cryptographic mechanisms: for power analysis, e.g. differential power analysis [DPA] or simple power analysis [SPA]
H04L 9/3013
Underlying computational problems or public-key parameters: involving the discrete logarithm problem, e.g. ElGamal or Diffie-Hellman systems
H04L 9/3066
Public key, i.e. encryption algorithm being computationally infeasible to invert or user's encryption keys not requiring secrecy: involving…
H04L
Digital communication