Are 7 and 12 equivalent in mode 5 operation

/ / Published in Cybersecurity, EITC/IS/CCF Classical Cryptography Fundamentals, History of cryptography, Modular arithmetic and historical ciphers

In the context of modular arithmetic, which is a fundamental concept in classical cryptography, the question of whether the numbers 7 and 12 are equivalent in mode 5 operation can be addressed by examining their equivalence under modulo 5.

Modular arithmetic is a system of arithmetic for integers, where numbers "wrap around" upon reaching a certain value, the modulus. In this case, the modulus is 5. The notation "a ≡ b (mod m)" is used to express that a and b are congruent modulo m, meaning that when a is divided by m, it leaves the same remainder as when b is divided by m.

To determine if 7 and 12 are equivalent in mode 5, we need to compute the remainders when each number is divided by 5.

1. Calculating the Remainder of 7 Modulo 5:
– When 7 is divided by 5, the quotient is 1 and the remainder is 2.
– This can be written as: 7 = 5 * 1 + 2.
– Hence, 7 ≡ 2 (mod 5).

2. Calculating the Remainder of 12 Modulo 5:
– When 12 is divided by 5, the quotient is 2 and the remainder is also 2.
– This can be written as: 12 = 5 * 2 + 2.
– Hence, 12 ≡ 2 (mod 5).

Since both 7 and 12 leave the same remainder (2) when divided by 5, they are congruent modulo 5. Therefore, in mode 5 operation, 7 and 12 are equivalent. This can be formally stated as:

    \[ 7 ≡ 12 \, (\text{mod} \, 5) \]

Didactic Value and Historical Context

Understanding modular arithmetic is important for grasping many classical cryptographic techniques. Historically, modular arithmetic has been used in various cryptographic systems, including the famous Caesar cipher and the more complex Vigenère cipher.

Caesar Cipher
The Caesar cipher is one of the simplest and most well-known encryption techniques, attributed to Julius Caesar, who reportedly used it to communicate with his officials. It is a substitution cipher where each letter in the plaintext is shifted a certain number of places down or up the alphabet. For example, with a shift of 3:

– Plaintext: A B C D E
– Ciphertext: D E F G H

The encryption can be mathematically expressed using modular arithmetic. Let each letter be represented by its position in the alphabet (A = 0, B = 1, …, Z = 25). The encryption function can be defined as:

    \[ E(x) = (x + k) \mod 26 \]

where x is the position of the letter in the alphabet and k is the shift. For decryption, the function is:

    \[ D(x) = (x - k) \mod 26 \]

Vigenère Cipher
The Vigenère cipher, developed in the 16th century, is a method of encrypting alphabetic text by using a simple form of polyalphabetic substitution. A keyword is repeated to match the length of the plaintext, and each letter of the plaintext is shifted according to the corresponding letter of the keyword. The Vigenère cipher can be broken down into a series of Caesar ciphers.

For example, if the keyword is "KEY" and the plaintext is "HELLO":
– The keyword repeated to match the length of the plaintext: KEYKE
– The shift for each letter: K (10), E (4), Y (24)

Encryption is performed as follows:
– H (7) + K (10) = R (17)
– E (4) + E (4) = I (8)
– L (11) + Y (24) = J (9)
– L (11) + K (10) = V (21)
– O (14) + E (4) = S (18)

Thus, the ciphertext is "RIJVS".

Practical Examples in Modern Cryptography

In modern cryptography, modular arithmetic is foundational for algorithms such as RSA and Diffie-Hellman key exchange.

RSA Algorithm
RSA (Rivest-Shamir-Adleman) is a widely used public-key cryptosystem that relies on the difficulty of factoring large composite numbers. The RSA algorithm involves key generation, encryption, and decryption processes, all of which use modular arithmetic.

Key generation:
1. Choose two distinct large prime numbers p and q.
2. Compute n = pq and \phi(n) = (p-1)(q-1).
3. Choose an integer e such that 1 < e < \phi(n) and \gcd(e, \phi(n)) = 1.
4. Determine d as d \equiv e^{-1} \mod \phi(n).

The public key is (e, n) and the private key is (d, n).

Encryption:

    \[ c \equiv m^e \mod n \]

where m is the plaintext message and c is the ciphertext.

Decryption:

    \[ m \equiv c^d \mod n \]

Diffie-Hellman Key Exchange
The Diffie-Hellman key exchange protocol allows two parties to securely share a secret key over a public channel. It relies on the difficulty of computing discrete logarithms in a finite field, another application of modular arithmetic.

1. Both parties agree on a large prime p and a base g.
2. Each party selects a private key (a for Alice and b for Bob).
3. Alice computes A = g^a \mod p and Bob computes B = g^b \mod p.
4. They exchange the values of A and B.
5. Alice computes the shared secret as s = B^a \mod p.
6. Bob computes the shared secret as s = A^b \mod p.

Both parties now share the secret s.

The equivalence of 7 and 12 in mode 5 operation is a straightforward application of modular arithmetic, demonstrating that both numbers leave the same remainder when divided by 5. This concept is not only fundamental in classical cryptography but also underpins many modern cryptographic algorithms and protocols. Understanding modular arithmetic enables the comprehension of more complex cryptographic systems and their security properties.

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