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applied cryptography - protocols, algorithms, and source code in c

Details can be found in 863 or in any of the number

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Unformatted text preview: ong y, unsigned long n) { unsigned long s,t,u; int i; s = 1; t = x; u = y; while(u) { if(u&amp1) s = (s* t)%n; u>>=1; t = (t* t)%n; } return(s); } Another, recursive, algorithm is: unsigned long fast_exp(unsigned long x, unsigned long y, unsigned long N) { unsigned long tmp; if(y==1) return(x % N); if ((y&amp1)==0) { tmp = fast_exp(x,y/2,N); return ((tmp* tmp)%N); } else { tmp = fast_exp(x,(y-1)/2,N); tmp = (tmp* tmp)%N; tmp = (tmp* x)%N; return (tmp); } } This technique reduces the operation to, on the average, 1.5*k operations, if k is the length of the number x in bits. Finding the calculation with the fewest operations is a hard problem (it has been proven that the sequence must contain at least k- 1 operations), but it is not too hard to get the number of operations down to 1.1*k or better, as k grows. An efficient way to do modular reductions many times using the same n is Montgomery’s method [1111]. Another method is called Barrett’s algorithm [87]. The software performance of these two algorithms and the algorithm previously discussed is in [210]: The algorithm I’ve discussed is the best choice for singular modular reductions; Barrett’s algorithm is the best choice for small arguments; and Montgomery’s method is the best choice for general modular exponentiations. (Montgomery’s method can also take advantage of small exponents, using something called mixed arithmetic.) The inverse of exponentiation modulo n is calculating a discrete logarithm . I’ll discuss this shortly. Prime Numbers A prime number is an integer greater than 1 whose only factors are 1 and itself: No other number evenly divides it. Two is a prime number. So are 73, 2521, 2365347734339, and 2756839 - 1. There are an infinite number of primes. Cryptography, especially public-key cryptography, uses large primes (512 bits and even larger) often. Evangelos Kranakis wrote an excellent book on number theory, prime numbers, and their applications to cryptography [896]. Paulo Ribenboim wrote two excellent references on prime numbers in general [1307, 1308]. Greatest Common Divisor Two numbers are relatively prime when they share no factors in common other than 1. In other words, if the greatest common divisor of a and n is equal to 1. This is written: gcd(a,n) = 1 The numbers 15 and 28 are relatively prime, 15 and 27 are not, and 13 and 500 are. A prime number is relatively prime to all...
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