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Modular exponentiation with large numbers

Fast modular exponentiation is not only for computers. For example, to compute 237 %149, proceed as follows, (1;2;37) !(2;2;36) !(2;4;18) !(2;16;9) !(32;16;8)!(32; 42;4) !(32; 24;2) !(32; 20;1) !( 105 ; 20;0): 2. Fermat Pseudoprimes Fermat's Little Theorem states that for any positive integer n, if n is prime then bn 1 %n = 1 for b = 1;:::;n 1. 1. 2 LARGE PRIME NUMBERS In the other direction. Modular Exponentiation for high numbers in C++ 1210 Easy interview question got harder: given numbers 1..100, find the missing number(s) given exactly k are missin The problem with above solutions is, overflow may occur for large value of n or x. Therefore, power is generally evaluated under modulo of a large number. Below is the fundamental modular property that is used for efficiently computing power under modular arithmetic Most technological applications of modular arithmetic involve exponentials withvery large numbers. For example, a typical problem related to encryption mightinvolve solving one of the following two equations: 6793032319 ⌘a(mod 103969) (70)67930b ⌘48560 (mod 103969). (71

The exact sequence of operations is visible in the binary expansion of n. You can always prevent the numbers from getting too big by reducing, using the modulus, whenever the numbers exceed the modulus Given three numbers a, b and c, we need to find (a b) % c Now why do % c after exponentiation, because a b will be really large even for relatively small values of a, b and that is a problem because the data type of the language that we try to code the problem, will most probably not let us store such a large number. Examples: Input : a = 2312 b = 3434 c = 6789 Output : 6343 Input : a. This is a really useful function that I thought needed to be explained.When dealing with security methods, like RSA or the Diffie-Hellman, or finding primes. Free and fast online Modular Exponentiation (ModPow) calculator. Just type in the base number, exponent and modulo, and click Calculate. This Modular Exponentiation calculator can handle big numbers, with any number of digits, as long as they are positive integers. For a more comprehensive mathematical tool, see the Big Number Calculator Exponentiation by squaring still works for modulo exponentiation. Your problem isn't that 2 ^ 168277 is an exceptionally large number, it's that one of your intermediate results is a fairly large number (bigger than 2^32), because 673109 is bigger than 2^16. So I think the following will do

The operation of modular exponentiation calculates the remainder when an integer b (the base) raised to the e th power (the exponent), be, is divided by a positive integer m (the modulus). In symbols, given base b, exponent e, and modulus m, the modular exponentiation c is: c = be mod m. From the definition of c, it follows that 0 ≤ c < m In many problems, modular exponentiation |x b | m is a basic computation, often responsible for the overall time performance, as in some cryptosystems, since its implementation requires a large number of multiplications. It is known that |x b | m =|x |b| ϕ(m) | m for any x in [1,m−1] if m is prime; in this case the number of multiplications depends on ϕ(m) instead of depending on b This number is so large, we cannot possibly print out all its decimal digits. Even if every electron in the universe were a sheet of paper, there would not be enough paper in the universe to print out this number in decimal notation! But we are about to encounter numbers much bigger even than this. Modular Exponentiation steps - modular exponentiation for large numbers . Modulus power of big numbers (8) I'd suggest using gmp, it can handle arbitrarily long ints and has decent C++ bindings. afaik on current hardware/sofware long longs are 64bit, so unsigned can handle numbers up to (2**64)-1 == 18446744073709551615 which is quite a bit smaller than numbers you'd have to deal with with RSA. I am trying to.

Finally, let's explore the exponentiation property: A^B mod C = ((A mod C)^B) mod C Often we want to calculate A^B mod C for large values of B. Unfortunately, A^B becomes very large for even modest sized values for B This REXX program code has code to automatically adjust the number of decimal digits to accommodate huge. numbers which are computed when raising large numbers to some arbitrary power. /*REXX program displays the modular exponentiation of: a**b mod m */. parse arg a b m /*obtain optional args from the CL*/ Modular Exponentiation for large numbers. Input: x = 3, n = 2, m = 4 Output: 1 Explanation: 32 = 9. 9 % 4 = 1. Input: x = 2, n = 6, m = 10 Output: 4 Explanation: 26 = 64. 64 % 10 = 4. You don't need to read or print anything. Your task is to complete the function PowMod () which takes integers x, n and M as input parameters and returns xn % M Modular exponentiation is used in public key cryptography. It involves computing b to the power e (mod m):. c ← b e (mod m). You could brute-force this problem by multiplying b by itself e - 1 times and taking the answer mod m, but it is important to have fast (efficient) algorithms for this process to have any practical application.. In cryptography, the numbers involved are usually very large

RSA - Modular Exponentiation • Normal exponentiation, then take remainder (e.g. 2 = 4 mod 10) • Exponentiation repeats itself • i.e. x mod n = x mod n • e.g. 2 mod 10 = 4 = 2 mod 10 = 2 mod 10 • Exponentiation with large numbers (256 bit) computationally intensive - efficient techniques must be used 10 y y mod Φ(n) 2 6 10. With a quantum computer, the quantum Shor's algorithm is able to factorize large composite numbers in polynomial complexity. However, the Shor's algorithm does have a fraction that uses classical algorithm to factorize the composite number given a special modular exponential period value found by the quantum circuit Fast modular exponentiation of large numbers with large exponents Giuseppe Alia a,*, Enrico Martinelli b a DipartimentodiIngegneriadell'Informazione,Universita diPisa,ViaDiotisalvi,2,56100Pisa,Italy b DipartimentodiIngegneriadell'Informazione,Universita diSiena,ViaRoma,56,53100Siena,Italy Abstract In many problems, modular exponentiation jxbj m is a basic computation, often responsible for. Modular Exponentiation. Suppose we are asked to compute \(3^5\) modulo \(7\). We could calculate \(3^5 = 243\) and then reduce \(243\) mod \(7\), but a better way is to observe \(3^4 = (3^2)^2\). Since \(3^2 = 9 = 2\) we have \(3^4 = 2^2 = 4\), and lastly \[ 3^5 = 3^4\times 3 = 4 \times 3 = 5 \pmod{7}. \] The second way is better because the numbers involved are smaller. This trick, known as.

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Modular exponentiation can be performed with a negative exponent e by finding the multiplicative inverse d of b modulo m using the extended Euclidean algorithm. That is: where e < 0 and . Modular exponentiation problems similar to the one described above are considered easy to do, even if the numbers involved are enormous In many problems, modular exponentiation |xb|m is a basic computation, often responsible for the overall time performance, as in some cryptosystems, since its implementation requires a large number..

The calculator performs modular exponentiation of big numbers. Despite all the power of modern computers, this method does not suit us since we will use numbers for the exponent, even larger than standard 64-bit integers. E.g., Mersenne Prime number: 618970019642690137449562111 used as default exponent value has 89 bits (see Bit length). To safely handle such exponents, we must use fast. Modular Multiplication Using Intermediate Modulo-n ReductionsWhen multiplying numbers using modular arithmetic, we can exploiting some basic properties to keep the range of intermediate results that we have to work to a range that is strictly less than n 2.This means that we can evaluate the above expression, 9 11 mod 13, and never work with any number as large as 169, which is clearly a. Fast Modular Exponentiation The first recursive version of exponentiation shown works fine, but is very slow for very large exponents. It turns out that one prevalent method for encryption of data (such as credit card numbers) involves modular exponentiation, with very big exponents. Using the original recursive algorithm with current computation speeds, it would take thousands of years just. Three typical test or exam questions. I use three different methods. Also known as modular powers or modular high powers. See my other videoshttps://www.yout..

c++ - Modular Exponentiation with big numbers - Stack Overflo

  1. This reduction process means that numbers don't get too large as we com-pute large powers in modular arithmetic. Let's revisit the computation of 460 mod 69. This the same as computing (4)60 in Z=69Z. We'd like to compute it by hand with the least amount of e ort.
  2. Modular arithmetic is a system of arithmetic for integers, which considers the remainder. In modular arithmetic, numbers wrap around upon reaching a given fixed quantity (this given quantity is known as the modulus) to leave a remainder. Modular arithmetic is often tied to prime numbers, for instance, in Wilson's theorem, Lucas's theorem, and Hensel's lemma, and generally appears in fields.
  3. e the behaviour of the successive powers of \(3\) modulo \(7\). Note we compute each power by multiplying the previous answer by \(3\) then reducing modulo \(7\). Beyond this, the sequence repeats itself (why?): At a glance, the sequence \(3, 2, 6, 4, 5, 1\) seems to have no order or structure whatsoever. In fact, although there are things.
  4. modular exponentiation for large numbers . cpp by Fair Finch on Jun 06 2020 Donate . 0 Source: www.geeksforgeeks.org. C queries related to modular exponentiation for large numbers binary modula exponentiation formulas; apower m mod n program; power mod n java program; a pow b mod c; modular exponentiation.
  5. Modular exponentiation for VERY large numbers? 165. February 12, 2018, at 08:11 AM. I am trying to calculate something like this: a^b mod c, where all three numbers are ~400,000 bits (~120,000 digits). Things I've tried: Python's pow() function is taking hours and has yet to produce a result. (if someone could tell me how it's implemented that would be very helpful!) A right-to-left binary.

Modular Exponentiation (Power in Modular Arithmetic

  1. How to compute modular exponentiation with large operators based on the right-to-left binary algorithm Da-Zhi Sun a,*, Zhen-Fu Cao a,*, Yu Sun b a Department of Computer Science, Shanghai Jiao Tong University, 1954 HuaShan Road, P.O. Box 282, Shanghai 200030, PR China b Department of Management, Beijing Normal University, Beijing 100875, PR China Abstract.
  2. Calculator for large integer numbers with the number bases 2, 8, 10 or 16. Calculation of modular exponentiation, binomial coefficient, least common multiple lcm, greatest common divisor gcd. Calculator for large integer numbers . The calculation with large unsigned integer numbers is possible using the setting uint in the calculator. Rounding errors are impossible here because floating.
  3. Modular exponentiation, ae modulo m, is a common operation in many cryptographic functions. The modular arithmetic accelerator (MAA) in MAXQ microprocessors can perform this operation with modulus sizes of up to 2048 bits. It is easy to load the memory areas with a, e, and m, and start the operation. When the modulus is the product of two or more primes, we can use the results of the Chinese.
  4. Modular Exponentiation for high numbers in C++ As you might have already guessed, problems arise when the arguments are all exceptionally large numbers. For example, if I want to test the number 673109 for primality, I will at one point have to find: (2 ^ 168277) % 673109. now 2 ^ 168277 is an exceptionally large number, and somewhere in the process it overflows test, which results in an.

How are large prime number computed How are modular exponentiations computed. How are large prime number computed how are modular. School George Mason University; Course Title ISA 650; Uploaded By Mithilaesh. Pages 54 This preview shows page 27 - 35 out of 54 pages. •. Modular Exponentiation Full Large how to calculate (xy) mod z with 1 <= x, y <= 101000 and z any positive integer 1 <= z < 231 what I have done so far is: scan x and y as a string, get the modulo, then calculate (xy) mod z. I know this is wrong because (xy) mod z not equa.

I'm trying to do modular exponentiation with numbers larger than the size of uint16_t. Therefore, 65536 is the maximum value that can be represented in the data type. To express larger numbers, I'm using a pointer array and using 65536 as the base. Addition, subtraction, and multiplication are all implemented, so I'm able to use these in the function for modular exponentiation, I just. exponentiation operations rather than one large one. Instead of performing a modular-exponential calculation on the large modulus, modular-exponential calculations are done on the two factors of the modulus. For example, in RSA, the modulus is the product of two prime numbers, p and q. If p and q are both 1024 bits, doing two modular.

Modular Exponentiation in x86 assembly [ intro Public key cryptography uses this function for encrypting information. Due to its slow execution, it's primary purpose is for key exchange and digital signatures. [ tests long_test () in mtest.c will generate random numbers from 32-bits to 8192-bits and pass these through both BN_mod_exp () and. An added bonus is that the algorithm is deterministic for all 32-bit numbers, so I can significantly increase efficiency because I know exactly what witnesses to test for. So for low numbers, the algorithm works exceptionally well. However, part of the process relies upon modular exponentiation, that is (num ^ pow) % mod. so, for example, 3 ^ 2. Big Number Calculator. Free and fast online Big Integer Number calculator. Just type in your numbers in decimal or hexadecimal format and click any button. This calculator can handle large numbers, with any number of digits, as long as they are integers To test both algorithms I elevated every number from 1 up to 100,000,000 to the power of 30. Using the naive approach it took 7.1 seconds. Using the exponentiation by squaring one it took 3.9 seconds. We can also treat the case where b is odd by re-writing it as a^b = a * a^(b-1), and break the treatment of even powers in two steps. This makes. Modular exponentiation can be done using exponentiation by squaring by initializing the initial product to the Montgomery representation of 1, Such numbers are too large to be stored in a single machine word. Typically, the hardware performs multiplication mod some base B, so performing larger multiplications requires combining several small multiplications. The base B is typically 2 for.

A modular exponentiation with 1024 bit numbers takes about 10K Ram and 15K Rom footprint on an ARM Cortex M3 and executes in about 530ms at 48MHz clock speed. If you have time, maybe replacing Barett's reduction methode with Montgomery's approach could speed up execution for integers larger than 1024 bits Modular Exponentiation Calculator,Successive Squaring Calculator. Menu. Start Here; Our Story; Videos; Podcast; Upgrade to Math Mastery. Modular Exponentiation and Successive Squaring Calculator-- Enter Modular Exponentiation . Modular Exponentiation and Successive Squaring Video. Email: donsevcik@gmail.com Tel: 800-234-2933; Membership Exams CPC Podcast Homework Coach Math Glossary Subjects. As we've seen, exponentiation and modular exponentiation are one of those applications in which an efficient algorithm is required for feasibility. Using the trivial/naive algorithms is possible only for small cases which aren't very interesting. To process realistically large numbers (such as the ones required for cryptographic algorithms), one needs powerful methods in his toolbox. For.

and performance of modular exponentiation using a GPU with the OpenGL Shading Language to execute operations required in the RSA algorithm [22, 23]. They focused on implementing the modular multiplication using a Residue Number System (RNS) with the large number N. Fleissner also implemented an accelerated Montgomery method for modular exponentiation with General-purpose computing on graphics. Modular exponentiation is about finding the value of the equation c = b e mod m. This is the remainder when dividing b e by m. It is the inverse function of the discrete logarithm. Because modular exponentiation is easy and fast, and finding the discrete logarithm is difficult, both are used in fields such as public-key cryptography. Right-to-left binary method. When dealing with very large. Modular Exponentiation. Exponentiation. CSES - Easy. Focus Problem - read through this problem before continuing! Resources . Resources; cp-algo: Binary Exponentiation. Binary exponentiation can be used to efficently compute x n m o d m x ^ n \mod m x n mod m. To do this, let's break down x n x ^ n x n into binary components. For example, 5 10 5 ^ {10} 5 10 = 5 101 0 2 5 ^ {1010_2} 5 101 0 2.

exponentiation - Modulo arithmetic with big numbers

  1. This operation is called modular exponentiation. It is computationally inexpensive to compute even though and are typically large numbers in an RSA implementation. Example. Let us take a message and encrypt it using the public key . We will simply compute . On the first sight, this modular exponentiation looks like an atrociously hard computation. But remember that is infectious:-). Therefore.
  2. Modular exponentiation obeys all of the normal rules of ordinary exponentiation. Particularly: a b a c a (b+c) (mod n) and (a b) c a bc (mod n) Like the other operations, it is also permissible to take the modulo operator inside since. a 2 mod n = ((a)(a)) mod n . a 2 mod n = ((a mod n)(a mod n)) mod n . a 2 mod n = (a mod n) 2 mod n . One thing that we can't do is reduce the exponent modulo.
  3. Modular exponentiation. The equation for modular multiplication can be stated as: A^B mod C = ((A mod C) ^B) mod C. For large numbers, this equation of modular exponentiation is even more helpful. Here is the example of modular exponentiation: Suppose, we have the same values as in previous example. A = 12, B = 7, C = 3 (12 ^ 7) mod 3 = ((12 mod 3) ^7) mod 3 . Left side of the exponentiation.
  4. Modular exponentiation, , is a one-way function because the inverse of a modular exponentiation is a known hard problem [6-8]. To achieve a comfortable level of security, the length of the key material for these cryptosystems must be larger than 1024 bits [ 9 ], and in the near future, it is predicted that 2048-bit and 4096-bit systems will become standard [ 10 ]
  5. Modular Exponentiation Large Numbers Solve Problem Step Step 3 106 Mod 353 Q42195815Modular exponentiation with large numbers. How to solve thisproble... | assignmentaccess.co
  6. Modular exponentiation by repeated squaring. In my last post we saw how to quickly compute powers of the form by repeatedly squaring: ; then ; and so on. This is much more efficient than computing powers by repeated multiplication: for example, we need only three multiplications to compute by squaring, but we would need seven multiplications to.
  7. power-- (non-modular) exponentiation SqrRoot-- integer part of square root Jacobi, SqrRootMod-- Jacobi symbol and modular square root Most of these functions also have pure long versions as well, and as usual, there are both functional and procedural variants. There are other functions as well. See ZZ.txt for complete details

Iterative sliding window method for shorter number of operations in modular exponentiation and scalar multiplication. Adamu Muhammad Noma Faculty of Computer Science and Information Technology, Universiti Putra Malaysia, Serdang, 43400Selangor, Malaysia.View further author information, Abdullah Muhammed Faculty of Computer Science and Information Technology, Universiti Putra Malaysia, Serdang. Section 2.6 Exponentiation Algorithm. We present an algorithm for computing a power of an integer. We call this algorithm the Naive Exponentiation algorithm, since there is a more clever way of calculating powers which we will present with Algorithm 15.3.5.. Algorithm 2.6.1. Naive Exponentiation for Integers. Input

Video: Modular exponentiation (Recursive) - GeeksforGeek

It is demonstrated how the device can be used as a large number processor in the normal field of numbers. US5513133A - Compact microelectronic device for performing modular multiplication and exponentiation over large numbers - Google Patents Compact microelectronic device for performing modular multiplication and exponentiation over large numbers Download PDF Info Publication number. They carry out modular exponentiation, multiplication and inversion computations on large natural numbers, and can convert a large number between decimal and hexadecimal representations. Download the programs: bd_funcs-2.0.0.zip (167 kB). bd_ModExp computes the modular exponentiation a e mod m; bd_ModInv computes the modular inversion a-1 mod m; bd_ModMult computes the modular multiplication a.

How to Quickly Compute Large Modular Exponentiation Using

Modular exponentiation is about finding the value of the equation c = be mod m. This is the remainder when dividing be by m. It is the inverse function of the discrete logarithm. Because modular exponentiation is easy and fast, and finding the discrete logarithm is difficult, both are used in fields such as public-key cryptography In this tute, we will discuss Modular Exponentiation (Power in Modular Arithmetic) in C++. Given 3 integers a, b, and m, find (a b) % m.Let's see how to calculate (a b) % m in Time complexities O(b) and O(log 2 b).. Here, we will use two properties of modular arithmetic Justia Patents US Patent for Compact microelectronic device for performing modular multiplication and exponentiation over large numbers Patent (Patent # 5,742,530) Compact microelectronic device for performing modular multiplication and exponentiation over large numbers . Dec 28, 1995 - Fortress U&T Ltd. A compact synchronous microelectronic peripheral machine for standard microprocessors with. For modular multiplication in the prime and composite prime field of numbers, we define A and B to be the multiplicand and the multiplier, and N to be the module which is usually larger than A or B. N may in some instances be smaller than A. We define A, B, and N as m.k=n bit long operands. Each k bit group will be called a character. Then A, B, and N are each m characters long. For ease in. Modular exponentiation is the main operation to RSA-based public-key cryptosystems. It is performed using successive modular multiplications. This operation is time consuming for large operands, which is always the case in cryptography. For software or hardware fast cryptosystems, one needs thus reducing the total number of modular multiplications required. Existing methods attempt to reduce.

Modular Exponentiation Calculator Boxentri

Modular exponentiation with large numbers. How to solve this problem step by step? 3^106 mod 353 = ?? Expert Answer 100% (1 rating) Previous question Next question. It consists in an exponentiation followed by a modulus, but it exists optimized algorithms with big numbers to return a fast result without having to actually perform the calculation (called fast, thanks to mathematical simplifications). Example: 1234 ≡16 mod 56 12 34 ≡ 16 mod 56. The word power indicates the name of the operation, and.

modulo - Modular Exponentiation for high numbers in C++

A fast radix-4 modular multiplication hardware algorithm is proposed. It is efficient for modular exponentiation with a large modulus, used in public-key cryptosystems such as the RSA cryptosystem. The operands and the result of multiplication which are intermediate results in modular exponentiation are represented in a redundant representation. The computation proceeds in serial-parallel. Abstract: Modular exponentiation with a large modulus, which is usually accomplished by repeated modular multiplications, has been widely used in public key cryptosystems for secured data communications. To speed up the computation, the Montgomery modular multiplication algorithm is used to relax the process of quotient determination, and the carry-save addition (CSA) is employed to reduce the. Compute the modular exponentiation a b mod m by using powermod. The powermod function is efficient because it does not calculate the exponential a b. c = powermod(3,5,7) c = 5. Prove Fermat's Little Theorem. Fermat's little theorem states that if p is prime and a is not divisible by p, then a (p-1) mod p is 1. Test Fermat's little theorem for p = 5, a = 3. As expected, powermod returns 1. p. Modular Arithmetic. Let n be a positive integer. We denote the set [ 0.. n − 1] by Z n. We consider two integers x, y to be the same if x and y differ by a multiple of n, and we write this as x = y ( mod n), and say that x and y are congruent modulo n. We may omit ( mod n) when it is clear from context

Modular exponentiation - Wikipedi

Posted December 24, 2014. December 24, 2016. Vamsi Sangam. In competitive programming, Modular Arithmetic Properties are essential tools in solving big number problems. In the problem statement, whenever they say, print the answer , it's not that simple. You may have worked a lot to get the logic, but the output must be given as they say Now that you know how to compute the modular multiplicative inverse (to refresh, a − 1 = am − 2 mod m when m is prime), you can now define the division operator: a / b mod m = a ⋅ b − 1 mod m. This also extends the mod operator to rational numbers (i.e. fractions), as long as the denominator is coprime to m

Right-to-Left binary algorithm diagram

steps - modular exponentiation for large numbers - Code

Modular exponentiation (article) Khan Academ

4: Our CCNOT construction(PDF) Analysis of RSA algorithm using GPU programming

Modular exponentiation - Rosetta Cod

Modular Exponentiation for large numbers Practice

Modular exponentiation is a type of exponentiation performed over a modulus.It is useful in computer science, especially in the field of public-key cryptography.. The operation of modular exponentiation calculates the remainder when an integer b (the base) raised to the e th power (the exponent), b e, is divided by a positive integer m (the modulus). In symbols, given base b, exponent e, and. multiplication is invertible as long as x and N have no common factors; we would just multiply yxn modN by xr−n where r is the order of x. (Of course, we don't know what r is, but that doesn't change the fact that the multiplication is invertible in principle.) We call this unitary operator the circuit for modular exponentiation. - p. 6/ ?

Fast Modular Exponentiation Dev Note

In modular arithmetic, the modular multiplicative inverse of an integer a modulo m is an integer x such that. a x ≡ 1 ( mod m ) . {\displaystyle a\,x\equiv 1 {\pmod {m}}.} Or in other words, such that: It can be shown that such an inverse exists if and only if a and m are coprime, but we will ignore this for this task Modular exponentiation is an important operation which requires a vast amount of computations. Current fastest modular exponentiation algorithms are based on square-and-multiply method, which is described in Algorithm 1 derived from [5]. INPUT: g, p and a positive integer e = (emem−1e1e0)2. OUTPUT: ge mod p 1. A=1. 2. For i from m down to 0 do the following: 2.1 A=A2 mod p 2.2 If ei = 1. PowerMod is also known as modular exponentiation. Mathematical function, suitable for both symbolic and numerical manipulation. Typically used in modular arithmetic, cryptography, random number generation and cyclic operations in programs. PowerMod [ a, b, m] gives the remainder of a b divided by m This complexity analysis of the naive exponentiation algorithm also holds for the naive exponentiation algorithm for integers, Algorithm 2.6.1. In Chapter 2, we gave two algorithms for computing \(c^4\) for some integer \(c\text{,}\) namely Algorithm 2.2.3 and Algorithm 2.4.3.Although the output of both algorithms was the same, the number of multiplications to compute the output differed

1: Reversible gates: CNOT, CCNOT (Toffoli), control-SWAP

Composite Number Factorization Using Modular

#Calculate exponents in the Python programming language. In mathematics, an exponent of a number says how many times that number is repeatedly multiplied with itself (Wikipedia, 2019). We usually express that operation as b n, where b is the base and n is the exponent or power. We often call that type of operation b raised to the n-th power, b raised to the power of n, or most. Three new types of power analysis attacks against smartcard implementations of modular exponentiation algorithms are described. The first attack requires an adversary to exponentiate many random messages with a known and a secret exponent. The second attack assumes that the adversary can make the smartcard exponentiate using exponents of his own choosing. The last attack assumes the adversary. Modular exponentiation. We start with a simple implementation of modular exponentiation, that is, computing , via repeated squaring. This comes up occasionally in both number theory problems (unsurprisingly) and combinatorics problems (because such problems often ask for a very large answer to be given modulo or some other large prime) Modular arithmetic operations are provided to compute residues, perform exponentiation, and compute multiplicative inverses. These methods always return a non-negative result, between 0 and (modulus - 1), inclusive. Bit operations operate on a single bit of the two's-complement representation of their operand. If necessary, the operand is sign.