Patent ID: 12250310

DETAILED DESCRIPTION

In their paper “A Study of the Digits of π, e and Certain Other Irrational Numbers”, Dr. Johnson and Dr. Leeming discuss academic research conducted by the University of Victoria which demonstrates that irrational numbers are known to have a uniform distribution of digits (0-9) in their mantissas. This means that each digit is equally probable to occur at a given index in the mantissa, which makes it much more difficult to accurately predict.

The inventive subject matter takes advantage of this phenomenon to provide apparatus, systems and methods in which elliptical curve cryptography utilizes an elliptic curve consistent with the formula pxmod q=r, where x is a private key having an irrational number component. The irrational component is employed to provide much greater entropy than would be achieved where x is a prime number. Essentially, every mod or every outcome is equally possible and therefore less predictable.

Example 1

FIG.3is a prior art elliptic curve of Formula 1, y2=x3+17x+12, depicted in a Cartesian coordinate system. The corresponding modular equivalent form is 3xmod 17=12.

For any prime number x>2, 3xmod 24 is in the mod 3 position, which gives the following equation:

3^x=24⁢y+3=17⁢z+123^x=24⁢y+324⁢y+3=1,TagBox[",", "NumberComma", Rule[SyntaxForm, "0"]]594,TagBox[",", "NumberComma", Rule[SyntaxForm, "0"]]323y=66,TagBox[",", "NumberComma", Rule[SyntaxForm, "0"]]4303^x=17⁢z+1217⁢z+12=1,TagBox[",", "NumberComma", Rule[SyntaxForm, "0"]]594,TagBox[",", "NumberComma", Rule[SyntaxForm, "0"]]323z=93,TagBox[",", "NumberComma", Rule[SyntaxForm, "0"]]783

Therefore, knowing that x is prime number, it is relatively easy to establish that x=13.

(3^13)⁢mod⁢17=12

However, if for example x were the sqrt of 14, then

(3^(sqrt⁡(14))⁢mod⁢17=9.98517397828401148467582645364861592086987958061166

Since (3{circumflex over ( )}(sqrt(14)) mod 17 lands on an irrational position, it is uniform in a mod 24 configuration and is more difficult to predict. This is due to the fact that sqrt(14) is an irrational number instead of a prime number.

Example 2

FIG.4is a prior art elliptic curve of Formula 2, y2=x3+19x+6, depicted in a Cartesian coordinate system. The corresponding modular equivalent form is 5× mod 19=6.

For any prime number x>2, 5xmod 24 is in the mod 3 position, which gives the following equation.

5^x=24⁢y+5=19⁢z+65^x=24⁢y+524⁢y+5=48,TagBox[",", "NumberComma", Rule[SyntaxForm, "0"]]828,TagBox[",", "NumberComma", Rule[SyntaxForm, "0"]]125y=2,TagBox[",", "NumberComma", Rule[SyntaxForm, "0"]]034,TagBox[",", "NumberComma", Rule[SyntaxForm, "0"]]5055^x=19⁢z+619⁢z+6=48,TagBox[",", "NumberComma", Rule[SyntaxForm, "0"]]828,TagBox[",", "NumberComma", Rule[SyntaxForm, "0"]]125z=256,TagBox[",", "NumberComma", Rule[SyntaxForm, "0"]]990

Therefore, knowing that x is prime number, it is relatively easy to establish that x=11

(5^11)⁢mod⁢19=6

However, if for example x were the sqrt of 14, then

(5^(sqrt⁡(15))⁢mod⁢19=15.4472078099179837335353661644128682985263055535097

Since (5{circumflex over ( )}(sqrt(15)) mod 19 lands on an irrational position, it is uniform in a mod 24 configuration and is more difficult to predict. This is due to the fact that sqrt(15) is an irrational number instead of a prime number.

Example 3

FIG.5is a prior art elliptic curve of Formula 3, y2=x3+29x+24, depicted in a Cartesian coordinate system. The corresponding modular equivalent form is 7xmod 29=24.

For any prime number x>2, 7xmod 24 is in the mod 3 position, which gives the following equation.

7^x=24⁢y+7=29⁢z+247^x=24⁢y+724⁢y+7=232630513987207y=96929380828007^x=29⁢z+2429⁢z+24=232630513987207z=8021741861627Therefore,x=17(7^17)⁢mod⁢29=24

However, if for example x were the sqrt of 20, then

(7^(sqrt⁡(20))⁢mod⁢29=14.1845295892212562582618852640758417051932397541661

Since (7{circumflex over ( )}(sqrt(20)) mod 19 lands on an irrational position, it is uniform in a mod 24 configuration and is more difficult to predict. This is due to the fact that sqrt(20) is an irrational number instead of a prime number.

FIG.6is a flowchart describing a method of decrypting encrypted data using elliptical curve cryptography. In general, the method600includes:Step610—initializing one or more memory locations to instantiate an elliptical curve algorithm consistent with the formula, pxmod q=r, where x is an irrational component;Step620—operating a digital logic circuitry to apply a series of one or more mathematical operations to the elliptic curve to produce a private key from a public key; andStep630—using the private key to de-crypt the encrypted data.

In preferred embodiments, the irrational component comprises a portion of a root of a non-perfect square, for example portion of a square root of 2 or a square root of 5. Also in preferred embodiments, p in the formula pxmod q=r is 3, 5, or 7.

FIG.7is a flowchart describing a method of increasing entropy in elliptical curve cryptography, comprising:Step710—initializing one or more memory locations to instantiate an elliptical curve algorithm consistent with the formula, px mod q=r, where x is an irrational component instead of a prime number;Step720—using a public key to encrypt the data;Step730—operating digital logic circuitry to apply a series of one or more mathematical operations to the elliptic curve to produce a private key from the public key; andStep740—providing the private key to an entity for us in decrypting the data.

In the above discussion, references are made regarding memories and digital logic circuitry. It should be appreciated that the use of such terms is deemed to include servers, services, interfaces, portals, platforms, or other systems formed from computing devices having at least one processor configured to execute software instructions stored on a computer readable tangible, non-transitory medium. For example, a server can include one or more computers operating as a web server, database server, or other type of computer server in a manner to fulfill described roles, responsibilities, or functions.

Also, as used in the description above, and throughout the claims that follow, the meaning of “a,” “an,” and “the” includes plural reference unless the context clearly dictates otherwise. Also, as used in the description herein, the meaning of “in” includes “in” and “on” unless the context clearly dictates otherwise.

Still further, all methods described herein can be performed in any suitable order unless otherwise indicated herein or otherwise clearly contradicted by context. The use of any and all examples, or exemplary language (e.g. “such as”) provided with respect to certain embodiments herein is intended merely to better illuminate the invention and does not pose a limitation on the scope of the invention otherwise claimed. No language in the specification should be construed as indicating any non-claimed element essential to the practice of the invention. Unless a contrary meaning is explicitly stated, all ranges are inclusive of their endpoints, and open-ended ranges are to be interpreted as bounded on the open end by commercially feasible embodiments.

It should be apparent to those skilled in the art that many more modifications besides those already described are possible without departing from the inventive concepts herein. The inventive subject matter, therefore, is not to be restricted except in the spirit of the appended claims. Moreover, in interpreting both the specification and the claims, all terms should be interpreted in the broadest possible manner consistent with the context.