Summary of findings
Root(-7) was put through four empirical test suites covering output uniformity, entropy, avalanche behaviour, differential analysis, collisions and input-to-output diffusion. Where relevant it was benchmarked against SHA3‑256, SHA‑256 and SHA‑1.
p = 0.8414
Byte uniformity (χ²)
Uniform distribution not rejected at 5% or 10% significance
49.997%
Avalanche mean
127.993 of 256 bits change; ideal is 50%
0
Full 256‑bit collisions
Across one million samples
31.87 / 32
Output bytes affected
When a single input byte is changed
1 Output Uniformity Analysis Chi-square test over 32×108 output bytes 2 Entropy, Bias, Correlation & Avalanche Side-by-side comparison with SHA3‑256, SHA‑256 and SHA‑1 3 Distribution Charts 32 charts: eight measurements for each of the four algorithms 4 Differential, Position Dependency & Collision Single-bit flips, position analysis and reduced-width collisions 5 Same Prefix / Different Suffix & Byte Dependency How changes in one part of the input spread across the output
✓ Consistent with uniform distribution
Test 1: Output Uniformity Analysis of Root(-7)
The output uniformity of the Root(-7) hash function was tested using 108 random input strings. Each input produced a 32-byte (256-bit) hash value, resulting in 32×108 output bytes for analysis.
Parameter Result
Number of random input strings 108 Hash size 256 bits (32 bytes) Total output bytes analyzed 32×108 Possible byte values 256 Expected frequency per byte value 125×105 Degrees of freedom 255 Calculated chi-square value 232.4427 Critical value at 5% significance 293.2478 Critical value at 10% significance 284.3359 p-value 0.8414
The calculated chi-square value of 232.4427 is lower than the critical values at both the 5% and 10% significance levels . Therefore, the null hypothesis that the output byte values are uniformly distributed is not rejected .
The high p-value of approximately 0.8414 indicates that there is no statistically significant evidence of deviation from a uniform byte distribution.
The experiment shows that the individual output bytes of Root(-7) are statistically consistent with a uniform distribution for the tested random inputs.
✓ On par with SHA family
Test 2: Entropy, Bias, Correlation & Avalanche
Avg. bit entropy, min-entropy, avg. bit bias, byte entropy, serial correlation, bit transition rate, avalanche mean, avalanche standard deviation and avalanche mean percentage. The results are based on 106 random input strings with lengths between 8 and 100 characters . Four hash algorithms were compared: SHA3-256, SHA1, SHA256, and Root(-7).
Table 1: Entropy and Output Distribution
Algorithm Digest Bits Avg. Bit Entropy Min-Entropy Avg. Bit Bias Byte Entropy Serial Correlation
SHA3-256 256 1.00000 0.99887 0.00039 7.99999 -0.00003 SHA1 160 1.00000 0.99889 0.00039 7.99999 -0.00030 SHA256 256 1.00000 0.99886 0.00039 7.99999 -0.00004 Root(-7) 256 1.00000 0.99882 0.00041 7.99999 0.00006
All four algorithms show very high bit entropy, with an average value of 1.0 .
The min-entropy values are also very close to 1.0, indicating a highly balanced output-bit distribution in this experiment.
The average bit bias is extremely small for all algorithms.
Byte entropy is approximately 8 bits per byte for every algorithm.
Serial correlations are very close to zero, indicating little measurable correlation between consecutive output bytes.
Table 2: Avalanche and Uniqueness Results
Algorithm Bit Transition Rate Avalanche Mean Avalanche Std. Avalanche Mean (%)
SHA3-256 0.49997 128.009 bits 7.999 50.003% SHA1 0.50005 79.998 bits 6.330 49.999% SHA256 0.50002 128.015 bits 7.997 50.006% Root(-7) 0.49997 127.993 bits 8.009 49.997%
Root(-7) has an average avalanche change of 127.993 out of 256 bits , corresponding to 49.997% .
The avalanche standard deviation of Root(-7), 8.009 bits , is close to the values observed for the two 256-bit SHA algorithms.
Overall, the experiment shows that Root(-7) produces balanced and highly distributed outputs and exhibits approximately 50% bit changes in the measured avalanche test.
View distribution charts →
Distribution Charts
The charts behind Test 2, covering 106 random inputs per algorithm. Pick a measurement to compare all four algorithms side by side, and click a chart to enlarge it.
✓ No full collisions found
Test 3: Differential and Position Dependency, Differential Pair and Collision
The results are based on 106 samples, using printable ASCII inputs of length between 8 and 50 and a 256-bit Root(-7) output.
Table 1: Differential and Position Dependency Results
Test Result Short Interpretation
Differential Analysis Mean: 128.011 bits Flipping one input bit changed about 50% of the 256 output bits. Differential Std. Dev. 7.995 bits The variation is close to the expected behavior of a 256-bit random output. Minimum / Maximum 91 / 164 bits The observed avalanche distances covered a reasonable range around 128 bits. Position Dependency ~127.8–128.3 bits Different input positions produced nearly the same average output-bit changes. Output bytes affected ~99.6% Changing one input bit affected almost every output byte.
The differential results show a strong avalanche effect, with an average of 128.011 changed bits out of 256 , very close to the ideal reference of 128 bits. The position analysis also shows consistent behavior across input positions, with almost 99.6% of output bytes affected .
Table 2: Differential Pair and Collision Results
Test Result Interpretation
Differential Pair Mean 127.9999 bits Independent hash outputs differed in approximately 50% of their bits. Differential Pair Std. Dev. 8.0045 bits The distribution is close to the expected random 256-bit behavior. Minimum Pair Distance 85 bits The closest pair found still differed in 85 of 256 bits. 16-bit Collision Events 934,464 Many collisions occurred after reducing the hash to only 16 bits. Unique 16-bit Outputs 65,536 All possible 16-bit values appeared in the million samples. Full 256-bit Collisions 0 No complete 256-bit collision was observed.
The differential pair experiment produced a mean Hamming distance of almost exactly 128 bits , while the minimum observed distance was 85 bits .
In the reduced 16-bit experiment, 934,464 collision events occurred and all 65,536 possible 16-bit values were observed. No full 256-bit collision was found among the one million samples.
✓ Strong input-to-output diffusion
Test 4: Same Prefix / Different Suffix, Byte Dependency Analysis
The results are based on 106 samples, with input lengths between 8 to 50 characters and a 256-bit Root(-7) output.
Table 1: Same Prefix / Different Suffix
Measure Result
Prefix length 16 Suffix length 16 Mean output bit difference 127.9723 bits Standard deviation 8.0029 bits Mean changed output bytes 31.8745 of 32 Ideal bit difference 128 bits Ideal changed bytes 32
Changing only the suffix while keeping the prefix fixed changed almost 50% of the output bits .
An average of 127.97 out of 256 bits changed, which is very close to the ideal value of 128.
About 31.87 of 32 output bytes changed on average.
This indicates strong diffusion between different parts of the input and the final hash output.
Table 2: Byte Dependency Analysis
Measure Result
Average output bytes changed 31.87 of 32 Output-byte change probability 1.00 Input positions tested 0–49
Changing one input byte affected almost all 32 output bytes .
The average number of changed output bytes remained very consistent across input positions, generally around 31.87 .
The dependency matrix shows values close to 1.00 throughout, indicating strong diffusion from input positions to output bytes.
The small 0.99 values in the matrix represent probabilities slightly below 100%, rather than a complete lack of dependency.
The experiments show that Root(-7) has strong input-to-output diffusion: changing a small portion of the input generally affects almost the entire 256-bit output.