{"id":23657,"date":"2023-05-05T06:09:58","date_gmt":"2023-05-05T06:09:58","guid":{"rendered":"https:\/\/www.booksofall.com\/cn\/?post_type=product&p=23657"},"modified":"2023-05-05T06:09:59","modified_gmt":"2023-05-05T06:09:59","slug":"crypto-101","status":"publish","type":"product","link":"https:\/\/www.booksofall.com\/cn\/crypto-101\/","title":{"rendered":"Crypto 101"},"content":{"rendered":"
\n

5 – Exclusive or<\/h2>\n

5.1 Description<\/h4>\n

Exclusive or<\/a>, often called \u201cXOR\u201d, is a Boolean binary operator that is true when either the first input or the second input, but not both, are true.<\/p>\n

Another way to think of XOR is as something called a \u201cprogrammable inverter\u201d: one input bit decides whether to invert the other input bit, or to just pass it through unchanged. \u201cInverting\u201d bits is colloquially called \u201cflipping\u201d bits, a term we\u02bcll use often throughout the book.<\/p>\n

\"\"<\/p>\n

In mathematics and cryptography papers, exclusive or is generally represented by a cross in a circle:\u00a0\u2295<\/i>. We\u02bcll use the same notation in this book:<\/p>\n

\"\"<\/p>\n<\/div>\n

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The inputs and output here are named as if we\u02bcre using XOR as an encryption operation. On the left, we have the plaintext bit\u00a0Pi<\/i>. The\u00a0i\u00a0<\/i>is just an index, since we\u02bcll usually deal with more than one such bit. On top, we have the key bit\u00a0ki<\/i>, that decides whether or not to invert\u00a0Pi<\/i>. On the right, we have the ciphertext bit,\u00a0Ci<\/i>, which is the result of the XOR operation.<\/p>\n

5.2 A few properties of XOR<\/h4>\n

Since we\u02bcll be dealing with XOR extensively during this book, we\u02bcll take a closer look at some of its properties. If you\u02bcre already familiar with how XOR works, feel free to skip this section.<\/p>\n

We saw that the output of XOR is 1 when one input or the other (but not both) is 1:<\/p>\n

\"\"<\/p>\n

There are a few useful arithmetic tricks we can derive from that.<\/p>\n

    \n
  1. You can apply XOR in any order: a\u2295<\/i>(b\u2295c<\/i>) = (a\u2295b<\/i>)\u2295c<\/i><\/li>\n
  2. You can flip the operands<\/a> around: a\u2295 b\u00a0<\/i>=\u00a0b\u2295 a<\/i><\/li>\n
  3. Any bit XOR itself is 0: a \u2295 a\u00a0<\/i>= 0. If\u00a0a\u00a0<\/i>is 0, then it s\u0313 0\u2295\u00a0<\/i>0 = 0; if\u00a0a\u00a0<\/i>is 1, then it s\u0313 1\u2295\u00a0<\/i>1 = 0.<\/li>\n
  4. Any bit XOR 0 is that bit again: a\u2295\u00a0<\/i>0 =\u00a0a<\/i>. If\u00a0a\u00a0<\/i>is 0, then it s\u0313 0\u2295\u00a0<\/i>0 = 0; if\u00a0a\u00a0<\/i>is 1, then it s\u0313 1\u2295\u00a0<\/i>0 = 1.<\/li>\n<\/ol>\n<\/div>\n
    <\/div>\n
    \n

    These rules also imply\u00a0a\u2295 b\u2295 a\u00a0<\/i>=\u00a0b<\/i>:<\/p>\n

    \"\"<\/p>\n

    We\u02bcll use this property often when using XOR for encryption<\/a>; you can think of that first XOR with\u00a0a\u00a0<\/i>as encrypting, and the second one as decrypting.<\/p>\n

    \n

    5.3 Bitwise XOR<\/h4>\n

    XOR, as we\u02bcve just defined it, operates only on single bits or Boolean values. Since we usually deal with values comprised of many bits, most programming languages provide a \u201cbitwise XOR<\/a>\u201d operator: an operator that performs XOR on the respective bits in a value.<\/p>\n

    Python, for example, provides the ^ (caret<\/a>) operator that performs bitwise XOR on integers. It does this by first expressing those two integers in binary<\/a>, and then performing XOR on their respective bits. Hence the name, bitwise\u00a0<\/i>XOR.<\/p>\n

    \"\"<\/p>\n<\/div>\n<\/div>\n","protected":false},"excerpt":{"rendered":"