Is modular exponentiation a one way function?
Another well-known possibly-one-way function is modular exponentiation, whose inverse, the discrete logarithm, computes x such that bx≡y(modn), given y.
Why is modular arithmetic used in cryptography?
6 Answers. One major reason is that modular arithmetic allows us to easily create groups, rings and fields which are fundamental building blocks of most modern public-key cryptosystems. For example, Diffie-Hellman uses the multiplicative group of integers modulo a prime p.
What is used for fast modular exponential calculation?
We can compute c using the “squares” method – this allows for fast computation of large positive integer powers of a number. For example, this allows a⁸, can be represented as ((a²)²)². … 7 multiplications are required (the exponent – 1).
What does MOD mean in math?
modulus
Given two positive numbers a and n, a modulo n (abbreviated as a mod n) is the remainder of the Euclidean division of a by n, where a is the dividend and n is the divisor. The modulo operation is to be distinguished from the symbol mod, which refers to the modulus (or divisor) one is operating from.
How do you reduce modulo?
a = q × d + r where q is the quotient, d is the divisor and r is the remainder. The reduction modulo d of an integer is, loosely speaking, its remainder in the division by d. Definition Let d be a non-zero integer. Two integers a and b are congruent modulo d if d | (a − b).
Is mod a one way function?
Theorem 13 (Rabin, 1979) Let N = pq where p and q are distinct odd primes. The function fN (x) ≡ x2 (mod N) is a 1-way function if and only if factoring N cannot be done in polynomial time.
What is the time complexity of recursive function solution analysis of modular exponentiation?
Time Complexity of above solution is O(Log y).
Is modular arithmetic important?
Modular arithmetic is important in number theory, where it is a fundamental tool in the solution of Diophantine equations (particularly those restricted to integer solutions).
How do you do modular arithmetic cryptography?
To do modular addition, you first add the two numbers normally, then divide by the modulus and take the remainder. Thus, (17+20) mod 7 = (37) mod 7 = 2.
What is fast exponentiation algorithm?
Basic algorithm Write the exponent n in binary. Read the binary representation from left to right, starting with the second bit from the left. Start with the number a, and every time you read a 0 bit, square what you’ve got. Every time you read a 1 bit, square what you’ve got and multiply by a.
What does it mean to do modular exponentiation?
Doing a “modular exponentiation” means calculating the remainder when dividing by a positive integer m (called the modulus) a positive integer b (called the base) raised to the e-th power (e is called the exponent). In other words, problems take the form where given base b, exponent e, and modulus m, one wishes to calculate c such that:
When do you use modular exponentiation in cryptography?
It is particularly useful in computer science, especially in the field of cryptography . Doing a “modular exponentiation” means calculating the remainder when dividing by a positive integer m (called the modulus) a positive integer b (called the base) raised to the e-th power (e is called the exponent).
How to perform modular exponentiation in pseudocode?
In pseudocode, this method can be performed the following way: function modular_pow (base, exponent, modulus) is if modulus = 1 then return 0 c := 1 for e_prime = 0 to exponent-1 do c := (c * base) mod modulus return c
How many times does the loop execute in modular exponentiation?
Because exponent is four binary digits in length, the loop executes only four times: Upon entering the loop for the first time, variables base = 4, exponent = 1101 (binary), and result = 1.