Assessing cryptographic entropy
Systems, methods, software, and combinations thereof for evaluating entropy in a cryptography system are described. In some aspects, sample values are produced by an entropy source system. A typicality can be determined for each of the sample values. A grading is determined for preselected distributions based on the typicalities of the sample values. A subset of the preselected distributions are selected based on the gradings. An entropy of the entropy source system is calculated based on the subset of the plurality of distributions.
1. A method for assessing entropy in a cryptography system, the method comprising:
obtaining a sample value produced by an entropy source system;
determining a typicality grading for each of a preselected plurality of probability distributions, the typicality gradings providing a comparison among the plurality of probability distributions, the typicality grading for each individual probability distribution being determined based on the cumulative probability of values that are, according to that individual probability distribution, less probable than the sample value;
wherein the typicality grading g k (x,p) for each individual probability distribution is determined according to:
g
k
(
x
,
p
)
=
(
∑
y
:
P
p
(
y
)
<
P
p
(
x
)
P
p
(
y
)
)
+
k
(
∑
y
:
P
p
(
y
)
=
P
p
(
x
)
P
p
(
y
)
)
,
where k ε [0,1] represents an inclusivity level between zero and one inclusive, P p (x) represents the probability of the sample value x according to the individual probability distribution p, and y: P p (y) <P p (x) represents the values that are less probable than the sample value;
selecting a subset of the plurality of probability distributions based on the typicality gradings; and
calculating an entropy of the entropy source system based on the subset of the plurality of probability distributions.
2. The method of claim 1 , wherein determining the typicality grading for an individual probability distribution comprises determining a typicality of the sample value for that individual probability distribution.
3. The method of claim 2 , wherein the typicality of the sample value indicates a probability that a randomly-selected value from the entropy source system is at most as probable as the sample value.
4. The method of claim 1 , further comprising:
determining whether the entropy source system has adequate entropy to achieve a specified security level in the cryptography system; and
modifying the entropy source system based on a determination that the entropy source system has inadequate entropy.
5. The method of claim 1 , further comprising:
seeding a pseudorandom number generator based on a sample value produced by the entropy source system; and
obtaining a cryptographic secret based on an output of the seeded pseudorandom number generator.
6. The method of claim 1 , comprising:
determining, based on the sample value, a typicality and a likelihood for each of the preselected plurality of probability distributions; and
computing the typicality gradings based on the typicalities and likelihoods.
7. The method of claim 1 , wherein selecting a subset of the plurality of probability distributions based on the typicality gradings includes at least one of:
selecting one or more probability distributions having a maximum typicality grading; or
selecting one or more probability distributions having a typicality grading above a specified threshold typicality grading.
8. The method of claim 1 , wherein calculating the entropy includes:
identifying side channel information and computational resources available to an adversary;
determining, for each probability distribution in the subset of probability distributions, a logarithm of a maximal probability of the adversary guessing the cryptographic secret using the side channel information and the resources available to the adversary; and
identifying as the entropy the minimal value of the logarithm determined for the subset of probability distributions.
9. A method for assessing entropy in a cryptography system, the method comprising:
obtaining a sample value produced by an entropy source system;
determining a typicality grading for each of a preselected plurality of probability distributions, the typicality gradings providing a comparison among the plurality of probability distributions, the typicality grading for each individual probability distribution being determined based on the cumulative probability of values that are, according to that individual probability distribution, less probable than the sample value;
wherein the cumulative probability of values that are less probable than the sample value comprises the sum:
(
∑
y
:
P
p
(
y
)
<
P
p
(
x
)
P
p
(
y
)
)
,
where P p (x) represents the probability of the sample value x according to the individual probability distribution p, and y: P p (y) <P p (x) represents the values that are less probable than the sample value;
selecting a subset of the plurality of probability distributions based on the typicality gradings; and
calculating an entropy of the entropy source system based on the subset of the plurality of probability distributions.
10. The method of claim 9 , wherein determining the typicality grading for an individual probability distribution comprises determining a typicality of the sample value for that individual probability distribution.
11. The method of claim 10 , wherein the typicality of the sample value indicates a probability that a randomly-selected value from the entropy source system is at most as probable as the sample value.
12. The method of claim 9 , further comprising:
determining whether the entropy source system has adequate entropy to achieve a specified security level in the cryptography system; and
modifying the entropy source system based on a determination that the entropy source system has inadequate entropy.
13. The method of claim 9 , further comprising:
seeding a pseudorandom number generator based on a sample value produced by the entropy source system; and
obtaining a cryptographic secret based on an output of the seeded pseudorandom number generator.
14. The method of claim 9 , comprising;
determining, based on the sample value, a typicality and a likelihood for each of the preselected plurality of probability distributions; and
computing the typicality gradings based on the typicalities and likelihoods.
15. The method of claim 9 , wherein selecting a subset of the plurality of probability distributions based on the typicality gradings includes at least one of:
selecting one or more probability distributions having a maximum typicality grading; or
selecting one or more probability distributions having a typicality grading above a specified threshold typicality grading.
16. The method of claim 9 , wherein calculating the entropy includes:
identifying side channel information and computational resources available to an adversary;
determining, for each probability distribution in the subset of probability distributions, a logarithm of a maximal probability of the adversary guessing the cryptographic secret using the side channel information and the resources available to the adversary; and
identifying as the entropy the minimal value of the logarithm determined for the subset of probability distributions.