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<\/p>\n\n\n\nThe most spectacular successes of quantum information processing tech-<\/p>\n\n\n\n
niques have been in the \ufb01eld of cryptography, the science of secret message<\/p>\n\n\n\n
exchange.<\/p>\n\n\n\n
1<\/p>\n\n\n\n
The reason why Shor\u2019s algorithm shot into prominence and major<\/p>\n\n\n\n
players started funding quantum computing research was the challenge it of-<\/p>\n\n\n\n
fered to currently trusted schemes of data encryption, particularly the RSA<\/p>\n\n\n\n
scheme that is the basis of almost all current public encryption systems, your<\/p>\n\n\n\n
online banking transactions or purchases for instance!<\/p>\n\n\n\n
Encrypt<\/p>\n\n\n\n
Decrypt<\/p>\n\n\n\n
FIGURE 9.4: Communication scenario for cryptography.<\/p>\n\n\n\n
In this section we will provide a quick birds-eye view of major crypto-<\/p>\n\n\n\n
graphic paradigms and where quantum information processing steps in to<\/p>\n\n\n\n
make things better. For a delightful survey of the history and current trends<\/p>\n\n\n\n
in cryptography I urge you to read the book by Simon Singh [66]. The progress<\/p>\n\n\n\n
in quantum cryptography is comprehensively dealt with in the review article<\/p>\n\n\n\n
by Gisin et. al. [38].<\/p>\n\n\n\n
1<\/p>\n\n\n\n
The word \u201ccryptography\u201d is derived from the Greek language: crypto=\u201csecret,\u201d gra-<\/p>\n\n\n\n
phy=\u201cwriting.\u201d It is actually one part of the science of \u201ccryptology,\u201d the second part being<\/p>\n\n\n\n
\u201ccryptanalysis,\u201d which is the art of decoding an encryption. These two go hand-in-hand: to<\/p>\n\n\n\n
test the success of any cryptographic scheme a thorough cryptanalysis is important.<\/p>\n\n\n\n
<\/p>\n\n\n\n
184 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
9.3.1 Basic cryptographic paradigms<\/p>\n\n\n\n
Almost ever since mankind used language for communication, need was<\/p>\n\n\n\n
felt for secrecy in that communication, as a protection of personal or national<\/p>\n\n\n\n
interests. The basic scheme (Figure 9.4) is the conversion of a natural lan-<\/p>\n\n\n\n
guage into a secret form, i.e., encryption, before transmission, and this needs<\/p>\n\n\n\n
to be tested against di\ufb00erent eavesdropping techniques. Several interesting<\/p>\n\n\n\n
cryptographic schemes have evolved as our mathematical and logical prowess<\/p>\n\n\n\n
increased. For instance, many of us may have played as children by exchanging<\/p>\n\n\n\n
secret notes in which the text was encoded by replacing each letter by another<\/p>\n\n\n\n
shifted down the alphabet by a few letters. This in fact was an ancient cipher<\/p>\n\n\n\n
system attributed to Julius Caesar! The receiver then decodes the message by<\/p>\n\n\n\n
shifting the letters back by the same amount.<\/p>\n\n\n\n
Example 9.3.1. The Caesar Cipher: suppose you decide to encode by shifting<\/p>\n\n\n\n
each alphabet by 5 letters:<\/p>\n\n\n\n
Plain: A B C D E F G H I J . . . Y Z<\/p>\n\n\n\n
Cipher: F G H I J K L M N O . . . D E<\/p>\n\n\n\n
then the message \u201cTHIS IS A SECRET\u201d would be encoded as \u201cYMNX NX<\/p>\n\n\n\n
FJHWT.\u201d The sender and receiver both agree as to what scheme of encoding<\/p>\n\n\n\n
they\u2019ll use. The danger in such messaging is that if the message is intercepted,<\/p>\n\n\n\n
then a clever cryptanalyst can \ufb01gure out the scheme used and easily translate<\/p>\n\n\n\n
any further messages sent by the same scheme. One can try to devise more<\/p>\n\n\n\n
complicated translation schemes, but as long as they are one-to-one, statistical<\/p>\n\n\n\n
methods such as the average frequency of letters in the English language may<\/p>\n\n\n\n
be used to break the code.<\/p>\n\n\n\n
The basic paradigm for any secret communication has two requisites:<\/p>\n\n\n\n
1. An eavesdropper should not be able to decrypt the message<\/p>\n\n\n\n
2. The sender should not be impersonated.<\/p>\n\n\n\n
The process of encryption is basically a mathematical transformation E on<\/p>\n\n\n\n
the input message m, which converts the plain text into a coded ciphertext c.<\/p>\n\n\n\n
The set of symbols used for the cypher is known as the alphabet. The physical<\/p>\n\n\n\n
analog is placing the message in a box that is then locked. The locked box<\/p>\n\n\n\n
is then transported to Bob, who opens the box, i.e., decodes the message,<\/p>\n\n\n\n
and obtains the plain text again. In this raw form of the protocol, security<\/p>\n\n\n\n
is minimal since the message can always be intercepted by an eavesdropper<\/p>\n\n\n\n
(Eve), who can then try to break the code (or recreate the key). The only<\/p>\n\n\n\n
way this threat can be met is that each time Alice wishes to send a message,<\/p>\n\n\n\n
she uses a new box (or a new algorithm for encryption) and that is wasteful.<\/p>\n\n\n\n
Instead, what she opts for is to change the lock and therefore the key K<\/p>\n\n\n\n
A<\/p>\n\n\n\n
used<\/p>\n\n\n\n
for locking the box. Bob then unlocks using his key K<\/p>\n\n\n\n
B<\/p>\n\n\n\n
, which has to be the<\/p>\n\n\n\n
![\"\"](\"https:\/\/learning.oreilly.com\/api\/v2\/epubs\/urn:orm:book:9781482238129\/files\/bgd2.png\")
<\/p>\n\n\n\n
Information and Communication 185<\/p>\n\n\n\n
correct one for the lock Alice used. The problem of keeping the message secret<\/p>\n\n\n\n
now reduces to keeping the keys secure.<\/p>\n\n\n\n
FIGURE 9.5: Private key cryptography.<\/p>\n\n\n\n
Mathematically, we represent the process of cryptography by<\/p>\n\n\n\n
m \u2192 E(m, K<\/p>\n\n\n\n
A<\/p>\n\n\n\n
) = c (encryption) (9.3)<\/p>\n\n\n\n
c \u2192 D(c, K<\/p>\n\n\n\n
B<\/p>\n\n\n\n
) = m (decryption). (9.4)<\/p>\n\n\n\n
The functions E and D need to be inverses of each other in this sense:<\/p>\n\n\n\n
D(E) = . (9.5)<\/p>\n\n\n\n
The encrypting function can be kept simple, so as to be computationally<\/p>\n\n\n\n
e\ufb03cient, and can be publicly known. This is the modern principle of cryptog-<\/p>\n\n\n\n
raphy, sometimes known as Kerckho\ufb00 \u2019s principle. The weak point of an<\/p>\n\n\n\n
encryption system should be easily changed if it falls into the enemy\u2019s hands.<\/p>\n\n\n\n
The choice then is for the key system, whether the encoding key is kept secret<\/p>\n\n\n\n
or not. This results in two sets of schemes:<\/p>\n\n\n\n
1. Symmetric or private key cryptography: Alice and Bob share the same<\/p>\n\n\n\n
key K (Figure 9.5). This is like the box with a lock, whose key is shared<\/p>\n\n\n\n
by both parties. The sharing must be done in an e\ufb03cient and secret<\/p>\n\n\n\n
way. The catch is in this step. If A and B are far separated, how can<\/p>\n\n\n\n
one transmit the key to the other in a secure way? This is the problem<\/p>\n\n\n\n
of secure key distribution.<\/p>\n\n\n\n
2. Public key cryptography: here the same key is not used by both parties.<\/p>\n\n\n\n
The sender uses a public or insecure key K<\/p>\n\n\n\n
A<\/p>\n\n\n\n
to encrypt the message<\/p>\n\n\n\n
(Figure 9.6). The decryption process is achieved by a private, secure<\/p>\n\n\n\n
keyK<\/p>\n\n\n\n
B<\/p>\n\n\n\n
. This process is like a ballot box that is locked and given to the<\/p>\n\n\n\n
sender, who posts the message in it. The receiver alone can unlock it<\/p>\n\n\n\n
with his secret key. Here the E and D processes are asymmetric, and<\/p>\n\n\n\n
the problem of distribution of keys doesn\u2019t arise. The security of the<\/p>\n\n\n\n
protocol lies in the di\ufb03culty of operating D without the knowledge of<\/p>\n\n\n\n
K<\/p>\n\n\n\n
B<\/p>\n\n\n\n
.<\/p>\n\n\n\n
<\/p>\n\n\n\n
186 Introduction to Quantum Physics and Information Processing<\/p>\n\n\n\n
FIGURE 9.6: Public key cryptography<\/p>\n\n\n\n
Due to the di\ufb03culty in sharing truly secure keys, especially when a large<\/p>\n\n\n\n
number of parties are involved, modern cryptographic schemes are usually<\/p>\n\n\n\n
of the second kind. One of the most widely used protocols for public-key<\/p>\n\n\n\n
encryption is a two-step process due to Di\ufb03e and Hellman [28], and by Merkle<\/p>\n\n\n\n
[47]. Known as the D\u2013H protocol, Alice and Bob each have a private key<\/p>\n\n\n\n
denoted L and a public key denoted K.<\/p>\n\n\n\n
Encryption (Alice): c<\/p>\n\n\n\n
1<\/p>\n\n\n\n
= E(m, L<\/p>\n\n\n\n
A<\/p>\n\n\n\n
); c = E(c<\/p>\n\n\n\n
1<\/p>\n\n\n\n
, K<\/p>\n\n\n\n
B<\/p>\n\n\n\n
)<\/p>\n\n\n\n
Decryption (Bob): m<\/p>\n\n\n\n
1<\/p>\n\n\n\n
= D(c, L<\/p>\n\n\n\n
B<\/p>\n\n\n\n
); m = D(m<\/p>\n\n\n\n
1<\/p>\n\n\n\n
, K<\/p>\n\n\n\n
A<\/p>\n\n\n\n
). (9.6)<\/p>\n\n\n\n
Exercise 9.2. Show that in the two-way public key cryptosystem of Equation 9.6,<\/p>\n\n\n\n
E and D are indeed inverses of each other.<\/p>\n\n\n\n
Example 9.3.2. Private key cryptography: the Vernam cipher or one-time pad.<\/p>\n\n\n\n
A and B agree on a common encryption system and share a common<\/p>\n\n\n\n
secret key K. One example of such an encryption is encoding the message<\/p>\n\n\n\n
in N symbols and performing a bitwise addition mod N with the key. The<\/p>\n\n\n\n
inverse is performed using the same key.<\/p>\n\n\n\n
c = m + k mod N, m = c \u2212 K mod N.<\/p>\n\n\n\n
\u2022 The key is a one-time use only. This is because it can be easily recon-<\/p>\n\n\n\n
structed from the cipher if it is intercepted.<\/p>\n\n\n\n
\u2022 The advantage of this technique is that the process is computationally<\/p>\n\n\n\n
simple.<\/p>\n\n\n\n
\u2022 C. Shannon has proved that this system is truly unbreakable as long as<\/p>\n\n\n\n
the key is secret and is of the same length as the message<\/p>\n","protected":false},"excerpt":{"rendered":"
The most spectacular successes of quantum information processing tech- niques have been in the \ufb01eld of cryptography, the science of secret message exchange. 1 The reason why Shor\u2019s algorithm shot into prominence and major players started funding quantum computing research was the challenge it of- fered to currently trusted schemes of data encryption, particularly the […]<\/p>\n","protected":false},"author":1,"featured_media":4043,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[498],"tags":[],"class_list":["post-4111","post","type-post","status-publish","format-standard","has-post-thumbnail","hentry","category-information-and-communication"],"jetpack_featured_media_url":"https:\/\/workhouse.sweetdishy.com\/wp-content\/uploads\/2024\/09\/informative.png","_links":{"self":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4111","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/comments?post=4111"}],"version-history":[{"count":2,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4111\/revisions"}],"predecessor-version":[{"id":4581,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/posts\/4111\/revisions\/4581"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media\/4043"}],"wp:attachment":[{"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/media?parent=4111"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/categories?post=4111"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/workhouse.sweetdishy.com\/index.php\/wp-json\/wp\/v2\/tags?post=4111"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}