Summary

with Rapid Miner:

cipherTime = - 0.000115096427166367 + 0.000000163201822481 * length, Coefficient = 0.998140067270979300
decipherTime = - 0.000310405294140850 + 0.000000163601687315 * length, Coefficient = 0.997187435292768000

The Coefficient values indicate how well the model fits the data, they are close to 1.0, indicating that we have accounted for almost all of the variability of cipherTime (0.998140067270979300) and decipherTime (0.997187435292768000) with bit stream length variable.

with Excel:

cipherTime = - 0,000115096427165798 + 0,000000163201822481 * length, Multiple R = 0,998140067270974
decipherTime = - 0,000310405294141294 + 0,000000163601687315 * length, Multiple R = 0,997187435292772

Which confirms RapidMiner results with a negligible variation.

With these results we'll be able to establish cipher, decipher algorithm relations with random number tests and hash functions (all components of DiceLock).

Data calculation

We compute "cipherTime" and "decipherTime" in seconds depending of the length of the stream being ciphered and deciphered.

Computing "cipherTime" comprises the stages:
- initializing cipher algorithm,
- setting symmetric key,
- setting initialization vector, and
- cipher plain text stream.

Computing "decipherTime" comprises the stages:
- initializing cipher algorithm,
- setting symmetric key,
- setting initialization vector, and
- decipher cipher text stream.

In order to establish the relation between encrypted and decrypted stream length with execution time we have performed the following execution:

Execution output:

We can see the last plain streams encrypted and decrypted, the time in seconds it took and the number of correctly ciphered and deciphered text sequences, 31219 sequences.

At the end plaintext is ciphered with key and initiallization vector set to 0's. It's shown that expected ciphertext matches computed ciphertext. Finally, deciphered text sequence matches KAT plain text.

OFB-AES256.csv created file with stream length, cipher time and decipher time

RapidMiner plots

With OFB-AES256.csv we can analyze if it's possible to establish some strong relationship between stream length and execution time in seconds. We've worked with RapidMiner software to see if we can extract some conclusion.

Cipher & decipher plot

Once loaded OFB-AES256.csv with RapidMiner we can see the plot of "cipherTime" and "decipherTime" (on Y axis) regarding stream "length" (on X axis).

It's obvious that cipher and decipher operation execution time has a clear relationship with stream length and that linear regressions can be obtained.

Cipher plot with RapidMiner

The next step performed has been to plot just "cipherTime" (Y axis) against stream "length" (X axis). Although the majority of points are over a line, some points are outliers (they are above the line). We can think about these outliers as system interruptions.

With this information we are able to establish a relationship between "cipherTime" and stream "length". A linear regression equation can be established to get a final relationship between cipher algorithm and random number tests.

Decipher plot with RapidMiner

The plot of "decipherTime" (Y axis) against stream "length" (X axis). Although the majority of points are over a line, some points are outliers (they are above the line). We can think about these outliers as system interruptions too.

With this information we are able to establish a relationship between "decipherTime" and stream "length". A linear regression equation can be established to get a final relationship between decipher algorithm and random number tests.

LINEAR REGRESSION

RapidMiner software allows to get models on data and establish relationaships between them.

We have taken a two ways approach, establilsh the linear regression with RapidMiner and verify such linear regression with Microsoft Excel.

RapidMiner linear regression analysis

Cipher linear regression

With the data regarding "cipherTime" and stream "length" we have perfomed the linear regression of "cipherTime" depending on stream "length".

The result obtained is the following one:

That is:

intercept = - 0.000115096427166367,

and the slope = 0.000000163201822481,

with a Standard Coefficient of:

The Coefficient value indicates how well the model fits the data, it's close to 1.0, indicating that we have accounted for almost all of the variability of cipherTime (0.998140067270979300) with stream length variable.

Decipher linear regression

With the data regarding "decipherTime" and stream "length" we have perfomed the linear regression of "decipherTime" depending on stream "length".

The result obtained is the following one:

That is:

intercept = - 0.000310405294140850,

and the slope = 0.000000163601687315,

with a Standard Coefficient of:

The Coefficient value indicates how well the model fits the data, it's close to 1.0, indicating that we have accounted for almost all of the variability of decipherTime (0.997187435292768000) with stream length variable.

Excel linear regression analysis

Microsoft Excel provides for data analysis. We perform the same Linear Regression of the same OFB-AES256.csv file in order to check that we get the same results.

Cipher linear regression

In Excel we analyze all 32.219 "cipherTime" and stream "length" data samples with Regression and we get the linear regression and ANOVA:

That is:

intercept = - 0,000115096427165798, (Excel result)

intercept = - 0.000115096427166367, (RapidMiner result, negligible difference)

slope = 0,000000163201822481, (Excel result)

slope = 0.000000163201822481 (equals RapidMiner result)

And the important statistical information

The Coefficient is close to 1.0, indicating that we have accounted for almost all of the variability of cipherTime (0,998140067270974) with stream length variable.

coefficient = 0,998140067270974, (Excel result)

coefficient = 0.998140067270979300, (RapidMiner result, negligible difference)

Cipher fit plot

The fit plot of calculated "estimated values" with the linear regression formula (pink line) and the sample data (blue points) shows us that the linear regression formula adjust quite well to real values.

The linear regression formula can be used to estimate time relationships between cipher AES 256 with OFB operation mode with random number tests.

Decipher linear regression

In Excel we analyze all 32.219 "decipherTime" and stream "length" data samples with Regression and we get the linear regression and ANOVA:

That is:

intercept = - 0,000310405294141294, (Excel result)

intercept = - 0.000310405294140850, (RapidMiner result, negligible difference)

slope = 0,000000163601687315, (Excel result)

slope = 0.000000163601687315 (equals RapidMiner result)

And the important statistical information

The Coefficient is close to 1.0, indicating that we have accounted for almost all of the variability of decipherTime (0,997187435292772) with stream length variable.

coefficient = 0,997187435292772, (Excel result)

coefficient = 0.997187435292768000, (RapidMiner result, negligible difference)

Decipher fit plot

The fit plot of calculated "estimated values" with the linear regression formula (pink line) and the sample data (blue points) shows us that the linear regression formula adjust quite well to real values.

The linear regression formula can be used to estimate time relationships between decipher AES 256 with OFB operation mode with random number tests.

Relevant C++ Source Code used

baseSymmetricCipher.h
baseSymmetricCipher.cpp
baseBlockCipher.h
baseBlockCipher.cpp
baseAES.h
baseAES.cpp
AES256.h
AES256.cpp
baseSymmetricCipher.h
baseSymmetricCipher.cpp
baseSymmetricStreamer.h
baseSymmetricStreamer.cpp
baseBlockCipherOperationMode.h
baseBlockCipherOperationMode.cpp
OFB_Mode.h
OFB_Mode.cpp
counterTime.h
counterTime.cpp

Execution environment:

Hardware:
Dell Precision M65
Intel Core 2 CPU T7200 @ 2.00GHz 2.00 GHZ
2,00 GB RAM

Software:
Windows Vista Ultimate Service Pack 2
32 bits Operating System

Developed with:
Microsoft Visual Studio 2008 Version 9.0.21022.8 RTM
Microsoft .NET Framework Version 3.5 SP1
Installed Edition: Professional Microsoft Visual C++ 2008 91605-130-0262317-60659
DiceLock release version: Optimization: Maximize Speed (/O2), Favor Fast Code (/Ot)