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/*
 * Licensed to the Apache Software Foundation (ASF) under one or more
 * contributor license agreements.  See the NOTICE file distributed with
 * this work for additional information regarding copyright ownership.
 * The ASF licenses this file to You under the Apache License, Version 2.0
 * (the "License"); you may not use this file except in compliance with
 * the License.  You may obtain a copy of the License at
 *
 *      http://www.apache.org/licenses/LICENSE-2.0
 *
 * Unless required by applicable law or agreed to in writing, software
 * distributed under the License is distributed on an "AS IS" BASIS,
 * WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
 * See the License for the specific language governing permissions and
 * limitations under the License.
 */

package org.apache.commons.codec.binary;

/**
 * Provides Base32 encoding and decoding as defined by RFC 4648.
 *
 * 

* The class can be parameterized in the following manner with various constructors: *

*
    *
  • Whether to use the "base32hex" variant instead of the default "base32"
  • *
  • Line length: Default 76. Line length that aren't multiples of 8 will still essentially end up being multiples of * 8 in the encoded data. *
  • Line separator: Default is CRLF ("\r\n")
  • *
*

* This class operates directly on byte streams, and not character streams. *

*

* This class is thread-safe. *

* * @see RFC 4648 * * @since 1.5 */ public class Base32 extends BaseNCodec { /** * BASE32 characters are 5 bits in length. * They are formed by taking a block of five octets to form a 40-bit string, * which is converted into eight BASE32 characters. */ private static final int BITS_PER_ENCODED_BYTE = 5; private static final int BYTES_PER_ENCODED_BLOCK = 8; private static final int BYTES_PER_UNENCODED_BLOCK = 5; /** * Chunk separator per RFC 2045 section 2.1. * * @see RFC 2045 section 2.1 */ private static final byte[] CHUNK_SEPARATOR = {'\r', '\n'}; /** * This array is a lookup table that translates Unicode characters drawn from the "Base32 Alphabet" (as specified * in Table 3 of RFC 4648) into their 5-bit positive integer equivalents. Characters that are not in the Base32 * alphabet but fall within the bounds of the array are translated to -1. */ private static final byte[] DECODE_TABLE = { // 0 1 2 3 4 5 6 7 8 9 A B C D E F -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, // 00-0f -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, // 10-1f -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, // 20-2f -1, -1, 26, 27, 28, 29, 30, 31, -1, -1, -1, -1, -1, -1, -1, -1, // 30-3f 2-7 -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, // 40-4f A-O 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, // 50-5a P-Z -1, -1, -1, -1, -1, // 5b - 5f -1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, // 60 - 6f a-o 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, // 70 - 7a p-z/**/ }; /** * This array is a lookup table that translates 5-bit positive integer index values into their "Base32 Alphabet" * equivalents as specified in Table 3 of RFC 4648. */ private static final byte[] ENCODE_TABLE = { 'A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S', 'T', 'U', 'V', 'W', 'X', 'Y', 'Z', '2', '3', '4', '5', '6', '7', }; /** * This array is a lookup table that translates Unicode characters drawn from the "Base32 Hex Alphabet" (as * specified in Table 4 of RFC 4648) into their 5-bit positive integer equivalents. Characters that are not in the * Base32 Hex alphabet but fall within the bounds of the array are translated to -1. */ private static final byte[] HEX_DECODE_TABLE = { // 0 1 2 3 4 5 6 7 8 9 A B C D E F -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, // 00-0f -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, // 10-1f -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, -1, // 20-2f 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, -1, -1, -1, -1, -1, -1, // 30-3f 2-7 -1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, // 40-4f A-O 25, 26, 27, 28, 29, 30, 31, // 50-56 P-V -1, -1, -1, -1, -1, -1, -1, -1, -1, // 57-5f Z-_ -1, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, // 60-6f `-o 25, 26, 27, 28, 29, 30, 31 // 70-76 p-v }; /** * This array is a lookup table that translates 5-bit positive integer index values into their * "Base32 Hex Alphabet" equivalents as specified in Table 4 of RFC 4648. */ private static final byte[] HEX_ENCODE_TABLE = { '0', '1', '2', '3', '4', '5', '6', '7', '8', '9', 'A', 'B', 'C', 'D', 'E', 'F', 'G', 'H', 'I', 'J', 'K', 'L', 'M', 'N', 'O', 'P', 'Q', 'R', 'S', 'T', 'U', 'V', }; /** Mask used to extract 7 bits, used when decoding final trailing character. */ private static final long MASK_7BITS = 0x7fL; /** Mask used to extract 6 bits, used when decoding final trailing character. */ private static final long MASK_6BITS = 0x3fL; /** Mask used to extract 5 bits, used when encoding Base32 bytes */ private static final int MASK_5BITS = 0x1f; /** Mask used to extract 4 bits, used when decoding final trailing character. */ private static final long MASK_4BITS = 0x0fL; /** Mask used to extract 3 bits, used when decoding final trailing character. */ private static final long MASK_3BITS = 0x07L; /** Mask used to extract 2 bits, used when decoding final trailing character. */ private static final long MASK_2BITS = 0x03L; /** Mask used to extract 1 bits, used when decoding final trailing character. */ private static final long MASK_1BITS = 0x01L; // The static final fields above are used for the original static byte[] methods on Base32. // The private member fields below are used with the new streaming approach, which requires // some state be preserved between calls of encode() and decode(). /** * Place holder for the bytes we're dealing with for our based logic. * Bitwise operations store and extract the encoding or decoding from this variable. */ /** * Convenience variable to help us determine when our buffer is going to run out of room and needs resizing. * {@code decodeSize = {@link #BYTES_PER_ENCODED_BLOCK} - 1 + lineSeparator.length;} */ private final int decodeSize; /** * Decode table to use. */ private final byte[] decodeTable; /** * Convenience variable to help us determine when our buffer is going to run out of room and needs resizing. * {@code encodeSize = {@link #BYTES_PER_ENCODED_BLOCK} + lineSeparator.length;} */ private final int encodeSize; /** * Encode table to use. */ private final byte[] encodeTable; /** * Line separator for encoding. Not used when decoding. Only used if lineLength > 0. */ private final byte[] lineSeparator; /** * Creates a Base32 codec used for decoding and encoding. *

* When encoding the line length is 0 (no chunking). *

* */ public Base32() { this(false); } /** * Creates a Base32 codec used for decoding and encoding. *

* When encoding the line length is 0 (no chunking). *

* @param pad byte used as padding byte. */ public Base32(final byte pad) { this(false, pad); } /** * Creates a Base32 codec used for decoding and encoding. *

* When encoding the line length is 0 (no chunking). *

* @param useHex if {@code true} then use Base32 Hex alphabet */ public Base32(final boolean useHex) { this(0, null, useHex, PAD_DEFAULT); } /** * Creates a Base32 codec used for decoding and encoding. *

* When encoding the line length is 0 (no chunking). *

* @param useHex if {@code true} then use Base32 Hex alphabet * @param pad byte used as padding byte. */ public Base32(final boolean useHex, final byte pad) { this(0, null, useHex, pad); } /** * Creates a Base32 codec used for decoding and encoding. *

* When encoding the line length is given in the constructor, the line separator is CRLF. *

* * @param lineLength * Each line of encoded data will be at most of the given length (rounded down to nearest multiple of * 8). If lineLength <= 0, then the output will not be divided into lines (chunks). Ignored when * decoding. */ public Base32(final int lineLength) { this(lineLength, CHUNK_SEPARATOR); } /** * Creates a Base32 codec used for decoding and encoding. *

* When encoding the line length and line separator are given in the constructor. *

*

* Line lengths that aren't multiples of 8 will still essentially end up being multiples of 8 in the encoded data. *

* * @param lineLength * Each line of encoded data will be at most of the given length (rounded down to nearest multiple of * 8). If lineLength <= 0, then the output will not be divided into lines (chunks). Ignored when * decoding. * @param lineSeparator * Each line of encoded data will end with this sequence of bytes. * @throws IllegalArgumentException * The provided lineSeparator included some Base32 characters. That's not going to work! */ public Base32(final int lineLength, final byte[] lineSeparator) { this(lineLength, lineSeparator, false, PAD_DEFAULT); } /** * Creates a Base32 / Base32 Hex codec used for decoding and encoding. *

* When encoding the line length and line separator are given in the constructor. *

*

* Line lengths that aren't multiples of 8 will still essentially end up being multiples of 8 in the encoded data. *

* * @param lineLength * Each line of encoded data will be at most of the given length (rounded down to nearest multiple of * 8). If lineLength <= 0, then the output will not be divided into lines (chunks). Ignored when * decoding. * @param lineSeparator * Each line of encoded data will end with this sequence of bytes. * @param useHex * if {@code true}, then use Base32 Hex alphabet, otherwise use Base32 alphabet * @throws IllegalArgumentException * The provided lineSeparator included some Base32 characters. That's not going to work! Or the * lineLength > 0 and lineSeparator is null. */ public Base32(final int lineLength, final byte[] lineSeparator, final boolean useHex) { this(lineLength, lineSeparator, useHex, PAD_DEFAULT); } /** * Creates a Base32 / Base32 Hex codec used for decoding and encoding. *

* When encoding the line length and line separator are given in the constructor. *

*

* Line lengths that aren't multiples of 8 will still essentially end up being multiples of 8 in the encoded data. *

* * @param lineLength * Each line of encoded data will be at most of the given length (rounded down to nearest multiple of * 8). If lineLength <= 0, then the output will not be divided into lines (chunks). Ignored when * decoding. * @param lineSeparator * Each line of encoded data will end with this sequence of bytes. * @param useHex * if {@code true}, then use Base32 Hex alphabet, otherwise use Base32 alphabet * @param pad byte used as padding byte. * @throws IllegalArgumentException * The provided lineSeparator included some Base32 characters. That's not going to work! Or the * lineLength > 0 and lineSeparator is null. */ public Base32(final int lineLength, final byte[] lineSeparator, final boolean useHex, final byte pad) { super(BYTES_PER_UNENCODED_BLOCK, BYTES_PER_ENCODED_BLOCK, lineLength, lineSeparator == null ? 0 : lineSeparator.length, pad); if (useHex) { this.encodeTable = HEX_ENCODE_TABLE; this.decodeTable = HEX_DECODE_TABLE; } else { this.encodeTable = ENCODE_TABLE; this.decodeTable = DECODE_TABLE; } if (lineLength > 0) { if (lineSeparator == null) { throw new IllegalArgumentException("lineLength " + lineLength + " > 0, but lineSeparator is null"); } // Must be done after initializing the tables if (containsAlphabetOrPad(lineSeparator)) { final String sep = StringUtils.newStringUtf8(lineSeparator); throw new IllegalArgumentException("lineSeparator must not contain Base32 characters: [" + sep + "]"); } this.encodeSize = BYTES_PER_ENCODED_BLOCK + lineSeparator.length; this.lineSeparator = new byte[lineSeparator.length]; System.arraycopy(lineSeparator, 0, this.lineSeparator, 0, lineSeparator.length); } else { this.encodeSize = BYTES_PER_ENCODED_BLOCK; this.lineSeparator = null; } this.decodeSize = this.encodeSize - 1; if (isInAlphabet(pad) || isWhiteSpace(pad)) { throw new IllegalArgumentException("pad must not be in alphabet or whitespace"); } } /** *

* Decodes all of the provided data, starting at inPos, for inAvail bytes. Should be called at least twice: once * with the data to decode, and once with inAvail set to "-1" to alert decoder that EOF has been reached. The "-1" * call is not necessary when decoding, but it doesn't hurt, either. *

*

* Ignores all non-Base32 characters. This is how chunked (e.g. 76 character) data is handled, since CR and LF are * silently ignored, but has implications for other bytes, too. This method subscribes to the garbage-in, * garbage-out philosophy: it will not check the provided data for validity. *

*

* Output is written to {@link org.apache.commons.codec.binary.BaseNCodec.Context#buffer Context#buffer} as 8-bit * octets, using {@link org.apache.commons.codec.binary.BaseNCodec.Context#pos Context#pos} as the buffer position *

* * @param input byte[] array of ascii data to Base32 decode. * @param inPos Position to start reading data from. * @param inAvail Amount of bytes available from input for decoding. * @param context the context to be used * */ @Override void decode(final byte[] input, int inPos, final int inAvail, final Context context) { // package protected for access from I/O streams if (context.eof) { return; } if (inAvail < 0) { context.eof = true; } for (int i = 0; i < inAvail; i++) { final byte b = input[inPos++]; if (b == pad) { // We're done. context.eof = true; break; } final byte[] buffer = ensureBufferSize(decodeSize, context); if (b >= 0 && b < this.decodeTable.length) { final int result = this.decodeTable[b]; if (result >= 0) { context.modulus = (context.modulus+1) % BYTES_PER_ENCODED_BLOCK; // collect decoded bytes context.lbitWorkArea = (context.lbitWorkArea << BITS_PER_ENCODED_BYTE) + result; if (context.modulus == 0) { // we can output the 5 bytes buffer[context.pos++] = (byte) ((context.lbitWorkArea >> 32) & MASK_8BITS); buffer[context.pos++] = (byte) ((context.lbitWorkArea >> 24) & MASK_8BITS); buffer[context.pos++] = (byte) ((context.lbitWorkArea >> 16) & MASK_8BITS); buffer[context.pos++] = (byte) ((context.lbitWorkArea >> 8) & MASK_8BITS); buffer[context.pos++] = (byte) (context.lbitWorkArea & MASK_8BITS); } } } } // Two forms of EOF as far as Base32 decoder is concerned: actual // EOF (-1) and first time '=' character is encountered in stream. // This approach makes the '=' padding characters completely optional. if (context.eof && context.modulus >= 2) { // if modulus < 2, nothing to do final byte[] buffer = ensureBufferSize(decodeSize, context); // we ignore partial bytes, i.e. only multiples of 8 count switch (context.modulus) { case 2 : // 10 bits, drop 2 and output one byte validateCharacter(MASK_2BITS, context); buffer[context.pos++] = (byte) ((context.lbitWorkArea >> 2) & MASK_8BITS); break; case 3 : // 15 bits, drop 7 and output 1 byte validateCharacter(MASK_7BITS, context); buffer[context.pos++] = (byte) ((context.lbitWorkArea >> 7) & MASK_8BITS); break; case 4 : // 20 bits = 2*8 + 4 validateCharacter(MASK_4BITS, context); context.lbitWorkArea = context.lbitWorkArea >> 4; // drop 4 bits buffer[context.pos++] = (byte) ((context.lbitWorkArea >> 8) & MASK_8BITS); buffer[context.pos++] = (byte) ((context.lbitWorkArea) & MASK_8BITS); break; case 5 : // 25bits = 3*8 + 1 validateCharacter(MASK_1BITS, context); context.lbitWorkArea = context.lbitWorkArea >> 1; buffer[context.pos++] = (byte) ((context.lbitWorkArea >> 16) & MASK_8BITS); buffer[context.pos++] = (byte) ((context.lbitWorkArea >> 8) & MASK_8BITS); buffer[context.pos++] = (byte) ((context.lbitWorkArea) & MASK_8BITS); break; case 6 : // 30bits = 3*8 + 6 validateCharacter(MASK_6BITS, context); context.lbitWorkArea = context.lbitWorkArea >> 6; buffer[context.pos++] = (byte) ((context.lbitWorkArea >> 16) & MASK_8BITS); buffer[context.pos++] = (byte) ((context.lbitWorkArea >> 8) & MASK_8BITS); buffer[context.pos++] = (byte) ((context.lbitWorkArea) & MASK_8BITS); break; case 7 : // 35 = 4*8 +3 validateCharacter(MASK_3BITS, context); context.lbitWorkArea = context.lbitWorkArea >> 3; buffer[context.pos++] = (byte) ((context.lbitWorkArea >> 24) & MASK_8BITS); buffer[context.pos++] = (byte) ((context.lbitWorkArea >> 16) & MASK_8BITS); buffer[context.pos++] = (byte) ((context.lbitWorkArea >> 8) & MASK_8BITS); buffer[context.pos++] = (byte) ((context.lbitWorkArea) & MASK_8BITS); break; default: // modulus can be 0-7, and we excluded 0,1 already throw new IllegalStateException("Impossible modulus "+context.modulus); } } } /** *

* Encodes all of the provided data, starting at inPos, for inAvail bytes. Must be called at least twice: once with * the data to encode, and once with inAvail set to "-1" to alert encoder that EOF has been reached, so flush last * remaining bytes (if not multiple of 5). *

* * @param input * byte[] array of binary data to Base32 encode. * @param inPos * Position to start reading data from. * @param inAvail * Amount of bytes available from input for encoding. * @param context the context to be used */ @Override void encode(final byte[] input, int inPos, final int inAvail, final Context context) { // package protected for access from I/O streams if (context.eof) { return; } // inAvail < 0 is how we're informed of EOF in the underlying data we're // encoding. if (inAvail < 0) { context.eof = true; if (0 == context.modulus && lineLength == 0) { return; // no leftovers to process and not using chunking } final byte[] buffer = ensureBufferSize(encodeSize, context); final int savedPos = context.pos; switch (context.modulus) { // % 5 case 0 : break; case 1 : // Only 1 octet; take top 5 bits then remainder buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 3) & MASK_5BITS]; // 8-1*5 = 3 buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea << 2) & MASK_5BITS]; // 5-3=2 buffer[context.pos++] = pad; buffer[context.pos++] = pad; buffer[context.pos++] = pad; buffer[context.pos++] = pad; buffer[context.pos++] = pad; buffer[context.pos++] = pad; break; case 2 : // 2 octets = 16 bits to use buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 11) & MASK_5BITS]; // 16-1*5 = 11 buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 6) & MASK_5BITS]; // 16-2*5 = 6 buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 1) & MASK_5BITS]; // 16-3*5 = 1 buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea << 4) & MASK_5BITS]; // 5-1 = 4 buffer[context.pos++] = pad; buffer[context.pos++] = pad; buffer[context.pos++] = pad; buffer[context.pos++] = pad; break; case 3 : // 3 octets = 24 bits to use buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 19) & MASK_5BITS]; // 24-1*5 = 19 buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 14) & MASK_5BITS]; // 24-2*5 = 14 buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 9) & MASK_5BITS]; // 24-3*5 = 9 buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 4) & MASK_5BITS]; // 24-4*5 = 4 buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea << 1) & MASK_5BITS]; // 5-4 = 1 buffer[context.pos++] = pad; buffer[context.pos++] = pad; buffer[context.pos++] = pad; break; case 4 : // 4 octets = 32 bits to use buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 27) & MASK_5BITS]; // 32-1*5 = 27 buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 22) & MASK_5BITS]; // 32-2*5 = 22 buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 17) & MASK_5BITS]; // 32-3*5 = 17 buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 12) & MASK_5BITS]; // 32-4*5 = 12 buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 7) & MASK_5BITS]; // 32-5*5 = 7 buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 2) & MASK_5BITS]; // 32-6*5 = 2 buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea << 3) & MASK_5BITS]; // 5-2 = 3 buffer[context.pos++] = pad; break; default: throw new IllegalStateException("Impossible modulus "+context.modulus); } context.currentLinePos += context.pos - savedPos; // keep track of current line position // if currentPos == 0 we are at the start of a line, so don't add CRLF if (lineLength > 0 && context.currentLinePos > 0){ // add chunk separator if required System.arraycopy(lineSeparator, 0, buffer, context.pos, lineSeparator.length); context.pos += lineSeparator.length; } } else { for (int i = 0; i < inAvail; i++) { final byte[] buffer = ensureBufferSize(encodeSize, context); context.modulus = (context.modulus+1) % BYTES_PER_UNENCODED_BLOCK; int b = input[inPos++]; if (b < 0) { b += 256; } context.lbitWorkArea = (context.lbitWorkArea << 8) + b; // BITS_PER_BYTE if (0 == context.modulus) { // we have enough bytes to create our output buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 35) & MASK_5BITS]; buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 30) & MASK_5BITS]; buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 25) & MASK_5BITS]; buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 20) & MASK_5BITS]; buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 15) & MASK_5BITS]; buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 10) & MASK_5BITS]; buffer[context.pos++] = encodeTable[(int)(context.lbitWorkArea >> 5) & MASK_5BITS]; buffer[context.pos++] = encodeTable[(int)context.lbitWorkArea & MASK_5BITS]; context.currentLinePos += BYTES_PER_ENCODED_BLOCK; if (lineLength > 0 && lineLength <= context.currentLinePos) { System.arraycopy(lineSeparator, 0, buffer, context.pos, lineSeparator.length); context.pos += lineSeparator.length; context.currentLinePos = 0; } } } } } /** * Returns whether or not the {@code octet} is in the Base32 alphabet. * * @param octet * The value to test * @return {@code true} if the value is defined in the the Base32 alphabet {@code false} otherwise. */ @Override public boolean isInAlphabet(final byte octet) { return octet >= 0 && octet < decodeTable.length && decodeTable[octet] != -1; } /** * Validates whether decoding the final trailing character is possible in the context * of the set of possible base 32 values. * *

The character is valid if the lower bits within the provided mask are zero. This * is used to test the final trailing base-32 digit is zero in the bits that will be discarded. * * @param emptyBitsMask The mask of the lower bits that should be empty * @param context the context to be used * * @throws IllegalArgumentException if the bits being checked contain any non-zero value */ private static void validateCharacter(final long emptyBitsMask, final Context context) { // Use the long bit work area if ((context.lbitWorkArea & emptyBitsMask) != 0) { throw new IllegalArgumentException( "Last encoded character (before the paddings if any) is a valid base 32 alphabet but not a possible value. " + "Expected the discarded bits to be zero."); } } }





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