Systems and computer-implemented methods for generating pseudo random numbers
Described embodiments relate to systems and method for conditioning, de-biasing and/or whitening raw entropy data or for hashing data. The method comprises receiving data; determining at least a first algebraic number from the data; calculating at least one solution to one or more transcendental equations using the at least the first algebraic number as an input parameter value, wherein the one or more transcendental equations comprise a transcendental function that is capable of generating transcendental number outputs from algebraic number inputs; determining one or more sequences of pseudo random numbers based on the at least one solution; and determining an output based on the one or more sequences of pseudo random numbers. For example, the data may be received from a raw entropy source and comprise raw entropy to be transformed. Alternatively, the data may be data to be hashed and the output may comprise a hash of the data.
1. A computer implemented method comprising:
receiving data from a raw entropy source, the data comprising raw entropy to be transformed;
determining at least a first algebraic number from the data;
calculating at least one solution to one or more transcendental equations using the at least the first algebraic number as an input parameter value, wherein the one or more transcendental equations comprise a transcendental function that is capable of generating transcendental number outputs from algebraic number inputs;
determining one or more sequences of pseudo random numbers based on the at least one solution; and
determining an output based on the one or more sequences of pseudo random numbers.
2. The method of claim 1 , wherein the output comprises a whitened transformation of the data or conditioned and de-biased data.
3. The method of claim 1 , wherein the method is performed by a device comprising a True Random Number Generator (TRNG).
4. The method of claim 1 , further comprising:
determining an interruption of the raw entropy output from the raw entropy source; and
responsive to determining the interruption, providing at least a portion of the one or more sequences of random numbers as an input parameter value for the one or more transcendental equations.
5. The method of claim 4 , further comprising:
responsive to determining that the interruption has been rectified, determining the determining at least first algebraic number from the data received from the raw entropy source.
6. The method of claim 1 , wherein generating one or more sequences of pseudo random numbers comprises generating one or more sequences of decimal numbers, binary numbers and numbers that are neither decimal numbers nor binary numbers.
7. The method of claim 1 , comprising:
converting the one or more sequences of numbers into one or more sequences of integers; and
wherein determining an output based on the one or more sequences of pseudo random numbers comprises determining an output bit string based on the one or more sequences of integers.
8. The method of claim 1 , wherein determining at least a first algebraic number from the data comprises converting the data to be processed into at least a first number.
9. The method of claim 8 , wherein the data to be processed comprises a bit string and the method further comprising:
converting the bit string into a corresponding string of integers; and
converting the corresponding string of integers into the at least first number.
10. The method of claim 1 , wherein the method is performed iteratively to generating pluralities of pseudo random numbers.
11. The method of claim 1 , wherein generating the one or more sequences of pseudo random numbers comprises:
performing multiple iterations of the calculating of at least one solution to the one or more transcendental equations, wherein input parameter values for a next iteration are derived from solutions to a current iteration.
12. The method of claim 1 , wherein the one or more sequences of pseudo random numbers comprises a first sequence of numbers and determining the output based on the one or more sequences of numbers comprises:
deriving the output as a subset of numbers of the first sequence of numbers generated.
13. The method of claim 12 , wherein the first sequence of numbers is a first sequence of decimal numbers or binary numbers or numbers that are neither decimal numbers nor binary numbers and the subset of numbers comprises a subset of decimal numbers or binary numbers or numbers that are neither decimal numbers nor binary numbers.
14. The method of claim 1 , wherein the one or more sequences of pseudo random numbers comprises a first sequence of numbers and a second sequence of numbers and determining the output based on the one or more sequences of numbers comprises:
deriving the output from a first subset of numbers selected from the first sequence of numbers and a second subset of numbers selected from the second sequence of numbers.
15. The method of claim 14 ,
wherein generating the first sequence comprises assigning a first number selected from the first sequence as a first input parameter value for the one or more transcendental equations and assigning a second number selected from the from the second sequence as a second input parameter value for the one or more transcendental equations; and
wherein generating the second sequence comprises assigning the second number as the first input parameter value for the one or more transcendental equations and assigning the first number as the second input parameter value for the one or more transcendental equations.
16. The method of claim 1 , wherein the transcendental equation is:
e
-
βλ
2
=
-
αλ
Erfc
(
λ
)
where
Erfc
(
λ
)
=
1
-
(
2
/
√
π
)
∑
n
=
0
∞
[
(
-
1
)
n
λ
2
n
+
1
]
/
[
(
2
n
+
1
)
(
n
!
)
]
where
e
-
βλ
2
=
∑
n
=
0
∞
(
-
βλ
2
)
n
/
n
!
where λ is the solution;
α is a first of the at least one algebraic input parameter value and β is a second of the at least one algebraic input parameter value; and
wherein α and β are real algebraic numbers.
17. The method of claim 1 , wherein the transcendental equation is e β0 α 1 β1 . . . α n βn =exp(λ) with the following analytic solution:
β 0 +β 1 ·ln(α 1 ) . . . β n ·ln(α n )=λ
where λ is the transcendental number, β 0 is a non-zero algebraic number, β 1 . . . β n are algebraic numbers,
α 1 . . . α n are non-zero algebraic numbers, and n is an integer and is ≥1; and
wherein the at least one algebraic input parameter value comprises β 0 , β 1 . . . . β n , and α 1 . . . α n .
18. The method of claim 1 , wherein the transcendental equation is e β0 α 1 β1 =exp(λ) with the following analytic solution:
β 0 +β 1 ·ln(α 1 )=λ
where λ is the solution, β 0 is a non-zero algebraic number, β 1 is an algebraic number and α 1 is a non-zero algebraic number; and
wherein the at least one algebraic input parameter value comprises β 0 , β 1 and α 1 .
19. A non-transitory machine-readable medium storing instructions which, when executed by one or more processors, cause an electronic apparatus to perform operations comprising:
receiving data from a raw entropy source, the data comprising raw entropy to be transformed;
determining at least a first algebraic number from the data;
calculating at least one solution to one or more transcendental equations using the at least the first algebraic number as an input parameter value, wherein the one or more transcendental equations comprise a transcendental function that is capable of generating transcendental number outputs from algebraic number inputs;
determining one or more sequences of pseudo random numbers based on the at least one solution; and
determining an output based on the one or more sequences of pseudo random numbers.
20. A system comprising:
one or more processors; and
memory comprising a non-transitory machine-readable which, when executed by the one or more processors, causes the system to:
receive data from a raw entropy source, the data comprising raw entropy to be transformed;
determine at least a first algebraic number from the data;
calculate at least one solution to one or more transcendental equations using the at least the first algebraic number as an input parameter value, wherein the one or more transcendental equations comprise a transcendental function that is capable of generating transcendental number outputs from algebraic number inputs;
determine one or more sequences of pseudo random numbers based on the at least one solution; and
determine an output based on the one or more sequences of pseudo random numbers.
21. A True Random Number Generator (TRNG) comprising:
the system of claim 20 ; and
a raw entropy data source configured to provide raw entropy data to the system.
22. A computer implemented method comprising:
receiving data to be hashed;
determining at least a first algebraic number from the data;
calculating at least one solution to one or more transcendental equations using the at least the first algebraic number as an input parameter value, wherein the one or more transcendental equations comprise a transcendental function that is capable of generating transcendental number outputs from algebraic number inputs;
determining one or more sequences of pseudo random numbers based on the at least one solution; and
determining an output based on the one or more sequences of pseudo random numbers, wherein the output comprises a hash of the data.
23. The method of claim 22 , wherein the method further comprises:
converting the determined output into hexadecimal text for use as the hash; and
outputting the hexadecimal text.
24. A non-transitory machine-readable medium storing instructions which, when executed by one or more processors, cause an electronic apparatus to perform operations comprising:
receiving data to be hashed;
determining at least a first algebraic number from the data;
calculating at least one solution to one or more transcendental equations using the at least the first algebraic number as an input parameter value, wherein the one or more transcendental equations comprise a transcendental function that is capable of generating transcendental number outputs from algebraic number inputs;
determining one or more sequences of pseudo random numbers based on the at least one solution; and
determining an output based on the one or more sequences of pseudo random numbers, wherein the output comprises a hash of the data.
25. A system comprising:
one or more processors; and
memory comprising a non-transitory machine-readable which, when executed by the one or more processors, causes the system to:
receive data to be hashed;
determine at least a first algebraic number from the data;
calculate at least one solution to one or more transcendental equations using the at least the first algebraic number as an input parameter value, wherein the one or more transcendental equations comprise a transcendental function that is capable of generating transcendental number outputs from algebraic number inputs;
determine one or more sequences of pseudo random numbers based on the at least one solution; and
determine an output based on the one or more sequences of pseudo random numbers, wherein the output comprises a hash of the data.
26. A hashing system, comprising:
the system of claim 25 ; and
a data source configured to provide data to be hashed to the system.