Solinas prime

In mathematics, a Solinas prime, or generalized mersenne prime, is a prime number that has the form $f(2^{m})$ , where $f(x)$ is a low-degree polynomial with small integer coefficients.^[1]^[2]. These primes allow fast modular reduction algorithms and are widely used in cryptography.

This class of numbers encompasses a few other categories of prime numbers:

Mersenne primes, which have the form $2^{k}-1$
Crandall or pseudo-mersenne primes, which have the form $2^{k}-c$ for small odd $c$ ^[3]

Modular Reduction Algorithm

Suppose there is a Solinas prime $p=f(2^{m})$ , where $f(t)=t^{d}-c_{d-1}t^{d-1}-...-c_{0}$ and a number $n$ such that $n<p^{2}$ . In this case, $n$ has up to $2md$ bits; we want to find a number congruent to $n$ mod $p$ with only as many bits as $p$ --that is, up to $md$ bits.

First, represent $n$ in base $2^{m}$ :

$n=\sum _{j=0}^{2d-1}A_{j}2^{mj}$

Next, generate a $d$ -by- $d$ matrix $X$ by stepping a $d$ -bit linear feedback shift register, starting with the values $0,0,...,0,1$ , for $d$ steps, so $X_{i,j}$ is the value of the $j$ th register on the $i$ th step. Then compute the matrix $B$ with the following equation:

$(B_{0}...B_{d-1})=(A_{0}...A_{d-1})+(A_{d}...A_{2d-1})X$

It turns out that

$\sum _{j=0}^{d-1}B_{j}2^{mj}\equiv \sum _{j=0}^{2d-1}A_{j}2^{mj}\mod p$

so we have found a smaller number congruent to $n$ . Since this algorithm does not have divisions, it is very efficient compared to the naive modular reduction algorithm ( $n-p\cdot (n/p)$ ).

Examples of Solinas primes

Four of the recommended primes in NIST's document "Recommended Elliptic Curves for Federal Government Use" are Solinas primes:

p-192 $2^{192}-2^{64}-1$
p-224 $2^{224}-2^{96}+1$
p-256 $2^{256}-2^{224}+2^{192}+2^{96}-1$
p-384 $2^{384}-2^{128}-2^{96}+2^{32}-1$

Curve448 uses the Solinas prime $2^{448}-2^{224}-1$

References

↑ Solinas, Jerome (1999). Generalized Mersenne Numbers (Technical report). Center for Applied Cryptographic Research, University of Waterloo. CORR-99-39.
↑ Solinas, Jerome A. (1 January 2011). Tilborg, Henk C. A. van; Jajodia, Sushil, eds. Encyclopedia of Cryptography and Security. Springer US. pp. 509–510. doi:10.1007/978-1-4419-5906-5_32.
↑ US patent 5159632, Richard E. Crandall, "Method and apparatus for public key exchange in a cryptographic system", issued 1992-10-27, assigned to NeXT Computer, Inc.

External links

Jerome A. Solinas, "Generalized Mersenne Numbers" (pdf)

This article is issued from Wikipedia. The text is licensed under Creative Commons - Attribution - Sharealike. Additional terms may apply for the media files.

[1] Solinas, Jerome (1999). Generalized Mersenne Numbers (Technical report). Center for Applied Cryptographic Research, University of Waterloo. CORR-99-39.

[2] Solinas, Jerome A. (1 January 2011). Tilborg, Henk C. A. van; Jajodia, Sushil, eds. Encyclopedia of Cryptography and Security. Springer US. pp. 509–510. doi:10.1007/978-1-4419-5906-5_32.

[3] US patent 5159632, Richard E. Crandall, "Method and apparatus for public key exchange in a cryptographic system", issued 1992-10-27, assigned to NeXT Computer, Inc.

Prime number classes
By formula	Fermat ( $2 2 n + 1$ ) Mersenne ( $2 p - 1$ ) Double Mersenne ( $2 2 p -1 - 1$ ) Wagstaff $(2 p + 1)/3$ Proth ( $k \cdot2 n + 1$ ) Factorial ( $n! \pm 1$ ) Primorial ( $p n # \pm 1$ ) Euclid ( $p n # + 1$ ) Pythagorean ( $4 n + 1$ ) Pierpont ( $2 m \cdot3 n + 1$ ) Quartan ( $x 4 + y 4$ ) Solinas ( $2 m \pm 2 n \pm 1$ ) Cullen ( $n \cdot2 n + 1$ ) Woodall ( $n \cdot2 n - 1$ ) Cuban ( $x 3 - y 3)/(x - y$ ) Carol $(2 n - 1) 2 - 2$ Kynea $(2 n + 1) 2 - 2$ Leyland ( $x y + y x$ ) Thabit ( $3\cdot2 n - 1$ ) Williams ( $(b -1)\cdot b n - 1$ ) Mills ( $⌊ A 3 n ⌋$ )
By integer sequence	Fibonacci Lucas Pell Newman–Shanks–Williams Perrin Partitions Bell Motzkin
By property	Wieferich (pair) Wall–Sun–Sun Wolstenholme Wilson Lucky Fortunate Ramanujan Pillai Regular Strong Stern Supersingular (elliptic curve) Supersingular (moonshine theory) Good Super Higgs Highly cototient
Base-dependent	Happy Dihedral Palindromic Emirp Repunit $(10 n - 1)/9$ Permutable Circular Truncatable Strobogrammatic Minimal Weakly Full reptend Unique Primeval Self Smarandache–Wellin
Patterns	Twin ( $p, p + 2$ ) Bi-twin chain ( $n - 1, n + 1, 2 n - 1, 2 n + 1, \dots$ ) Triplet ( $p, p + 2 or p + 4, p + 6$ ) Quadruplet ( $p, p + 2, p + 6, p + 8$ ) k−Tuple Cousin ( $p, p + 4$ ) Sexy ( $p, p + 6$ ) Chen Sophie Germain ( $p, 2 p + 1$ ) Cunningham ( $p, 2 p \pm 1, 4 p \pm 3, 8 p \pm 7, ...$ ) Safe ( $p, (p - 1)/2$ ) Arithmetic progression ( $p + a\cdotn, n = 0, 1, 2, 3, ...$ ) Balanced ( $consecutive p - n, p, p + n$ )
By size	Titanic (1,000+ digits) Gigantic (10,000+ digits) Mega (1,000,000+ digits) Largest known
Complex numbers	Eisenstein prime Gaussian prime
Composite numbers	Pseudoprime Catalan Elliptic Euler Euler–Jacobi Fermat Frobenius Lucas Somer–Lucas Strong Carmichael number Almost prime Semiprime Interprime Pernicious
Related topics	Probable prime Industrial-grade prime Illegal prime Formula for primes Prime gap
First 60 primes	2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 61 67 71 73 79 83 89 97 101 103 107 109 113 127 131 137 139 149 151 157 163 167 173 179 181 191 193 197 199 211 223 227 229 233 239 241 251 257 263 269 271 277 281
List of prime numbers Mathematics portal

Solinas prime

Modular Reduction Algorithm

Examples of Solinas primes

See also

References

External links