BaseNcoding

There are well-known algorithms for binary data to string encoding exist, such as algorithms with alphabet length with radix of power 2 (base32, base64) and not of power 2 (base85 and base91).

This library contains an implementation of algorithm for general case, that is a custom-length alphabet can be used. Here is online BaseNcoding DEMO.

The idea of the developed algorithm is based on base85 encoding, except block size is not constant, but it's calculation depends on an alphabet length.

Steps of algorithm

1. Calculation of block size in bits and chars.
2. Conversation of an input string to the byte array (using UTF8 Encoding).
3. Splitting byte array on n-bit groups.
4. Conversation of every group to radix n.
5. Tail bits processing.

Mathematical justification

For optimal block size calculation the following considerations has been used:

In this system:

• a — Length of alphabet A.
• k — Count of encoding chars.
• b — Digit radix (2 in most cases).
• n — Number of bits in radix b for representing k chars of alphabet A.
• r_values — Compression ratio (less is better).
• r_bits — Bits per char count (greater is better).
• mbc — Block max bits count.
• ⌊x⌋ — The largest integer not greater than x (floor).
• ⌈x⌉ — The smallest integer not less than x (ceiling).

See implementation in GetOptimalBitsCount method.

Tail bits processing

The most difficult problem in this project which nevertheless has been solved with only integer calculations.

Calculation of total and tail number of bits and chars during encoding

int mainBitsLength = (data.Length * 8 / BlockBitsCount) * BlockBitsCount;
int tailBitsLength = data.Length * 8 - mainBitsLength;
int totalBitsLength = mainBitsLength + tailBitsLength;
int mainCharsCount = mainBitsLength * BlockCharsCount / BlockBitsCount;
int tailCharsCount = (tailBitsLength * BlockCharsCount + BlockBitsCount - 1) / BlockBitsCount;
int totalCharsCount = mainCharsCount + tailCharsCount;
int iterationCount = mainCharsCount / BlockCharsCount;

Calculation of total and tail number of bits and chars during decoding

int totalBitsLength = ((data.Length - 1) * BlockBitsCount / BlockCharsCount + 8) / 8 * 8;
int mainBitsLength = totalBitsLength / BlockBitsCount * BlockBitsCount;
int tailBitsLength = totalBitsLength - mainBitsLength;
int mainCharsCount = mainBitsLength * BlockCharsCount / BlockBitsCount;
int tailCharsCount = (tailBitsLength * BlockCharsCount + BlockBitsCount - 1) / BlockBitsCount;
BigInteger tailBits = CharsToBits(data, mainCharsCount, tailCharsCount);
if (tailBits >> tailBitsLength != 0)
{
	totalBitsLength += 8;
	mainBitsLength = totalBitsLength / BlockBitsCount * BlockBitsCount;
	tailBitsLength = totalBitsLength - mainBitsLength;
	mainCharsCount = mainBitsLength * BlockCharsCount / BlockBitsCount;
	tailCharsCount = (tailBitsLength * BlockCharsCount + BlockBitsCount - 1) / BlockBitsCount;
}
int iterationCount = mainCharsCount / BlockCharsCount;

Experiment

Diagram of optimal block size and alphabet length dependence has been calculated with help of system above for mbc = 64:

<details> <summary>Diagram values</summary>

| a | n | k | r_bits | r_values | |----|----|----|---------|-----------| | 2 | 1 | 1 | 1 | 1 | | 3 | 19 | 12 | 1.5833 | 1.0136 | | 4 | 2 | 1 | 2 | 1 | | 5 | 58 | 25 | 2.32 | 1.034 | | 6 | 31 | 12 | 2.5833 | 1.0136 | | 7 | 14 | 5 | 2.8 | 1.0258 | | 8 | 3 | 1 | 3 | 1 | | 9 | 19 | 6 | 3.1667 | 1.0136 | | 10 | 63 | 19 | 3.3158 | 1.0842 | | 11 | 38 | 11 | 3.4545 | 1.038 | | 12 | 43 | 12 | 3.5833 | 1.0136 | | 13 | 37 | 10 | 3.7 | 1.0031 | | 14 | 19 | 5 | 3.8 | 1.0258 | | 15 | 39 | 10 | 3.9 | 1.0489 | | 16 | 4 | 1 | 4 | 1 | | 17 | 49 | 12 | 4.0833 | 1.0349 | | 18 | 25 | 6 | 4.1667 | 1.0136 | | 19 | 55 | 13 | 4.2308 | 1.1672 | | 20 | 56 | 13 | 4.3077 | 1.1369 | | 21 | 57 | 13 | 4.3846 | 1.0719 | | 22 | 49 | 11 | 4.4545 | 1.038 | | 23 | 9 | 2 | 4.5 | 1.0332 | | 24 | 55 | 12 | 4.5833 | 1.0136 | | 25 | 51 | 11 | 4.6364 | 1.0588 | | 26 | 47 | 10 | 4.7 | 1.0031 | | 27 | 19 | 4 | 4.75 | 1.0136 | | 28 | 24 | 5 | 4.8 | 1.0258 | | 29 | 34 | 7 | 4.8571 | 1.0041 | | 30 | 49 | 10 | 4.9 | 1.0489 | | 31 | 64 | 13 | 4.9231 | 1.3237 | | 32 | 5 | 1 | 5 | 1 | | 33 | 5 | 1 | 5 | 1.0313 | | 34 | 61 | 12 | 5.0833 | 1.0349 | | 35 | 41 | 8 | 5.125 | 1.024 | | 36 | 31 | 6 | 5.1667 | 1.0136 | | 37 | 26 | 5 | 5.2 | 1.0333 | | 38 | 47 | 9 | 5.2222 | 1.1739 | | 39 | 58 | 11 | 5.2727 | 1.1015 | | 40 | 53 | 10 | 5.3 | 1.1642 | | 41 | 16 | 3 | 5.3333 | 1.0517 | | 42 | 43 | 8 | 5.375 | 1.1008 | | 43 | 27 | 5 | 5.4 | 1.0953 | | 44 | 60 | 11 | 5.4545 | 1.038 | | 45 | 60 | 11 | 5.4545 | 1.329 | | 46 | 11 | 2 | 5.5 | 1.0332 | | 47 | 61 | 11 | 5.5455 | 1.0721 | | 48 | 39 | 7 | 5.5714 | 1.0679 | | 49 | 28 | 5 | 5.6 | 1.0523 | | 50 | 62 | 11 | 5.6364 | 1.0588 | | 51 | 17 | 3 | 5.6667 | 1.012 | | 52 | 57 | 10 | 5.7 | 1.0031 | | 53 | 63 | 11 | 5.7273 | 1.005 | | 54 | 23 | 4 | 5.75 | 1.0136 | | 55 | 52 | 9 | 5.7778 | 1.0226 | | 56 | 29 | 5 | 5.8 | 1.0258 | | 57 | 64 | 11 | 5.8182 | 1.1187 | | 58 | 41 | 7 | 5.8571 | 1.0041 | | 59 | 47 | 8 | 5.875 | 1.0433 | | 60 | 59 | 10 | 5.9 | 1.0489 | | 61 | 59 | 10 | 5.9 | 1.2375 | | 62 | 59 | 10 | 5.9 | 1.456 | | 63 | 59 | 10 | 5.9 | 1.7086 | | 64 | 6 | 1 | 6 | 1 | | 65 | 6 | 1 | 6 | 1.0156 | | 66 | 6 | 1 | 6 | 1.0313 | | 67 | 6 | 1 | 6 | 1.0469 | | 68 | 6 | 1 | 6 | 1.0625 | | 69 | 61 | 10 | 6.1 | 1.0609 | | 70 | 49 | 8 | 6.125 | 1.024 | | 71 | 43 | 7 | 6.1429 | 1.034 | | 72 | 37 | 6 | 6.1667 | 1.0136 | | 73 | 37 | 6 | 6.1667 | 1.1011 | | 74 | 31 | 5 | 6.2 | 1.0333 | | 75 | 56 | 9 | 6.2222 | 1.042 | | 76 | 56 | 9 | 6.2222 | 1.1739 | | 77 | 25 | 4 | 6.25 | 1.0476 | | 78 | 25 | 4 | 6.25 | 1.1031 | | 79 | 63 | 10 | 6.3 | 1.0266 | | 80 | 63 | 10 | 6.3 | 1.1642 | | 81 | 19 | 3 | 6.3333 | 1.0136 | | 82 | 19 | 3 | 6.3333 | 1.0517 | | 83 | 51 | 8 | 6.375 | 1.0002 | | 84 | 51 | 8 | 6.375 | 1.1008 | | 85 | 32 | 5 | 6.4 | 1.0331 | | 86 | 32 | 5 | 6.4 | 1.0953 | | 87 | 45 | 7 | 6.4286 | 1.0722 | | 88 | 58 | 9 | 6.4444 | 1.098 | | 89 | 58 | 9 | 6.4444 | 1.2155 | | 90 | 58 | 9 | 6.4444 | 1.3441 | | 91 | 13 | 2 | 6.5 | 1.0109 | | 92 | 13 | 2 | 6.5 | 1.0332 | | 93 | 13 | 2 | 6.5 | 1.0558 | | 94 | 13 | 2 | 6.5 | 1.0786 |

</details>

One can see that known base64, base85, base91 encodings have been developed in good points (minimal block size with good compression ratio).

Big numbers (C# BitInteger and JavaScript BigInt) are used for some calculations.

The algorithm also has a parallel version.

License

BaseN algorithm is licensed under the Apache 2.0 License.

More detail explanation available on Russian.