An Introduction to Binary Finite Fields GF2 m

An Introduction to Binary Finite Fields GF(2 m) By Francisco Rodríguez Henríquez. Aritmética Computacional Francisco Rodríguez Henríquez

What is a Field? • A field is a set of elements with two custom-defined arithmetic operations: most commonly, addition and multiplication. The elements of the field are an additive abelian group, and the non-zero elements of the field are a multiplicative abelian group. This means that all elements of the field have an additive inverse, and all non-zero elements have a multiplicative inverse. • A field is called finite if it has a finite number of elements. The most commonly used finite fields in cryptography are the field Fp (where p is a prime number) and the field F 2 m. Aritmética Computacional Francisco Rodríguez Henríquez

Finite Fields • A finite field or Galois field denoted by GF(q=pn), is a field with characteristic p, and a number q of elements. As we have seen, such a finite field exists for every prime p and positive integer n, and contains a subfield having p elements. This subfield is called ground field of the original field. • For the rest of this class, we will consider only the two most used cases in cryptography: q=p, with p a prime and q=2 m. The former case, GF(p), is denoted as the prime field, whereas the latter, GF(2 m), is known as the finite field of characteristic two or simply binary field. Aritmética Computacional Francisco Rodríguez Henríquez

Finite Fields • A finite field is a field with a finite number of elements. The number of elements in a finite field is called the order of the field. Fields of the same order are isomorphic: they display exactly the same algebraic structure differing only in the representation of the elements. Aritmética Computacional Francisco Rodríguez Henríquez

The field F 2 m ‘Plegaria del Codificador teórico: Juro por Galois que seré leal a las nobles tradiciones de la teoría de códigos; que hablaré de ella en el secreto lenguaje sólo conocido por los contados iniciados; y que celosamente vigilaré la sagrada teoría de aquellos que quisieran profanarla para usarla en aplicaciones mundanas”. J. L. Massey Although the description of the field F 2 m is complicated, this field is extremely beautiful and also quite useful, because its computations can be done efficiently when implemented in hardware. There are several ways to describe arithmetic in F 2 m; the most common one is the so-called polynomial representation. Aritmética Computacional Francisco Rodríguez Henríquez

Some definitions Here, we restrict our discussion to the numbers that belongs to the finite field F=GF(2 m) over K=GF(2). K is also known as the characteristic field. The elements of F are polynomials of degree less than m, with coefficients in K; that is, {am-1 xm-1+am-2 xm-2+. . . +a 2 x 2+a 1 x+a 0|ai= 0 or 1}. These elements are frequently written in vector form as (am-1. . . a 1 a 0). F has exactly 2 m-1 nonzero elements plus the zero element. Aritmética Computacional Francisco Rodríguez Henríquez

The Binary Field F 2 m A polynomial p in GF(2 m) is irreducible if p is not a unit element and if p=fg then f or g must be a unit, that is, a constant polynomial. Let us consider a finite field F=GF(2 m) over K=GF(2). Elements of F: Polynomials of degree less than m, with coefficients in K, such that, {am-1 xm-1+am-2 xm-2+. . . +a 2 x 2+a 1 x+a 0|ai= 0 or 1}. Fact: The field F has exactly q-1=2 m-1 nonzero elements plus the zero element. Aritmética Computacional Francisco Rodríguez Henríquez

Generating polynomial The finite field F=GF(2 m) is completely described by a monic irreducible polynomial, often called generating polynomial, of the form Where ki GF(2) for i=0, 1, …, m-1. Let be a root of the monic irreducible polynomial in (0), i. e. , f( ) = 0, Then, taking advantage of the fact that over GF(2) addition is equivalent to subtraction, we get the important relation Aritmética Computacional Francisco Rodríguez Henríquez

Generating polynomial and polynomial basis Then, we define the polynomial or canonical basis of GF(2 m) over GF(2) using the primitive element and its m first powers {1, , 2, …, m-1}, which happen to be linearly independent over GF(2). Aritmética Computacional Francisco Rodríguez Henríquez

Polynomial representation Using the canonical basis we can uniquely represent any number A F=GF(2 m) as Sometimes, it is more convenient to represent a field element using the so-called coordinate representation, Aritmética Computacional Francisco Rodríguez Henríquez

Element’s Representation By using the polynomial basis given in last equation, we can represent any number A F=GF(2 m) uniquely by Where all the coefficients a. I's belong to the characteristic field GF(2). Elements of the field are m-bit strings. The rules for arithmetic in F can be defined by polynomial representation. Since F operates on bit strings, computers can perform arithmetic in this field very efficiently. Aritmética Computacional Francisco Rodríguez Henríquez

Order definition • The order of an element in F, is defined as the smallest positive integer k such that k=1. Any finite field always contains at least one element, called a primitive element, which has order q-1. We say that f(x) is a primitive polynomial, if any one of its roots, say , is a primitive element in F. If f(x) is primitive, then all the q elements of F, can be expressed as the union of the zero element and the set of the first q-1 powers of , • In fact, this is always the case for any finite field F=GF(2 m) where we can always define the so-called polynomial basis of GF(2 m) over GF(2) as as the linearly independent set of the first m powers of {1, , 2, …, m-1} Aritmética Computacional Francisco Rodríguez Henríquez

An example Example. Let K = GF(24), F = GF(2), with defining primitive polynomial f(x) given by f(x) = x 4 + x + 1 Then, if is a root of f(x), we have f( )=0, which implies that f( ) = 4 + + 1 = 0 This equation over GF(2), means that satisfies the following equation 4 = + 1. Using the above equation, one can now express each one of the 15 nonzero elements of K over F as is shown in the next table. Aritmética Computacional Francisco Rodríguez Henríquez

Discrete log table Aritmética Computacional Francisco Rodríguez Henríquez

Finite fields: definitions and operations F 2 m finite field operations : Addition, Squaring, multiplication and inversion Aritmética Computacional Francisco Rodríguez Henríquez

Arithmetic in the field F 2 m The irreducible generating polynomial used for these sample calculations is again f(x) =x 4+x+1. Notice that all the coefficients are reduced modulo 2!! Addition (0110)+(0101)=(0011). Multiplication (1101) (1001) = (x 3+x 2+1) (x 3+1) mod f(x) = x 6+x 5+2 x 3+x 2+1 mod f(x) = x 6+x 5+x 2+1 mod f(x) = (x 4+x+1)(x 2+x)+(x 3+x 2+x+1) mod f(x) = x 3+x 2+x+1 = (1111). Aritmética Computacional Francisco Rodríguez Henríquez

Arithmetic in the field F 2 m Exponentiation To compute (0010)4, first find (0010)2 = (0010) = x x mod f(x) = x 2 = (0100). Then (0010)4 = (0010)2 = (0100) = x 2 mod f(x) = (x 4+x+1)(1)+(x+1) mod f(x) =x+1 = (0011). Aritmética Computacional Francisco Rodríguez Henríquez

Arithmetic in the field F 2 m Multiplicative Inversion The multiplicative identity for the field is 0 = (0001). The multiplicative inverse of 7 = (1011) is -7 mod 15= 8 mod 15=(0101). To verify this, see that, (1011) (0101)= (x 3+x+1) (x 2+1) mod f(x) = x 5+x 2+x+1 mod f(x) = (x 4+x+1)(x)+(1) mod f(x) =1 = (0001) Which is the multiplicative identity Aritmética Computacional . Francisco Rodríguez Henríquez

Field multipliers Aritmética Computacional Francisco Rodríguez Henríquez

Two-steps Multipliers In most algorithms the modular product is computed in two steps: polynomial multiplication followed by modular reduction. Let A(x), B(x) and (x) GF(2 m) and P(x) be the irreducible field generator polynomial. • In order to compute the modular product we first obtain the product polynomial C(x), of degree at most 2 m-2, as Polynomial product 2 m-1 coordinates • Then, in the second step, a reduction operation is performed in order to obtain the m-1 degree polynomial C’(x) is defined as Reduction step m coordinates Aritmética Computacional Francisco Rodríguez Henríquez

Squaring over Aritmética Computacional m GF(2 ) Francisco Rodríguez Henríquez

GF(2 m) Squarer In most algorithms the modular product is computed in two steps: polynomial multiplication followed by modular reduction. Let A(x) GF(2 m) be an arbitrary element in the field and P(x) be the irreducible field generator polynomial. • In order to compute the modular square of the element A(x) we first obtain the polynomial product C(x), of degree at most 2 m-2, as Polynomial product 2 m-1 coordinates • Then, in a second step, a reduction operation is performed in order to obtain the m-1 degree polynomial C’(x) defined as Reduction step m coordinates Aritmética Computacional Francisco Rodríguez Henríquez

Squaring: Example • Let A be an element of the finite field F=GF(25). Then, the square of A is given as, a 4 a 3 a 2 a 1 a 0 * a 4 a 3 a 2 a 1 a 0 a 4 0 a 3 0 a 2 0 a 1 0 a 0 In general, for an arbitrary element A in the field F=GF(25), we have, Aritmética Computacional Francisco Rodríguez Henríquez

Squaring: Software Solution rct_word sqr_table_low[256] = { 0, 1, 4, 5, 16, 17, 20, 21, 64 65, 68, 69, 80, 81, 84, 85, 256, 257, 260, 261, 272, 273, 276, 277, 320, 321, 324, 325, 336, 337, 340, 341, 1024, 1025, 1028, 1029, 1040, 1041, 1044, 1045, 1088, 1089, 1092, 1093, 1104, 1105, 1108, 1109, 1280, 1281, 1284, 1285, 1296, 1297, 1300, 1301, 1344, 1345, 1348, 1349, 1360, 1361, 1364, 1365, 4096, 4097, 4100, 4101, 4112, 4113, 4116, 4117, 4160, 4161, 4164, 4165, 4176, 4177, 4180, 4181, 4352, 4353, 4356, 4357, 4368, 4369, 4372, 4373, 4416, 4417, 4420, 4421, 4432, 4433, 4436, 4437, 5120, 5121, 5124, 5125, 5136, 5137, 5140, 5141, 5184, 5185, 5188, 5189, 5200, 5201, 5204, 5205, 5376, 5377, 5380, 5381, 5392, 5393, 5396, 5397, 5440, 5441, 5444, 5445, 5456, 5457, 5460, 5461, 16384, 16385, 16388, 16389, 16400, 16401, 16404, 16405, 16448, 16449, 16452, 16453, 16464, 16465, 16468, 16469, 16640, 16641, 16644, 16645, 16656, 16657, 16660, 16661, 16704, 16705, 16708, 16709, 16720, 16721, 16724, 16725, 17408, 17409, 17412, 17413, 17424, 17425, 17428, 17429, 17472, 17473, 17476, 17477, 17488, 17489, 17492, 17493, 17664, 17665, 17668, 17669, 17680, 17681, 17684, 17685, 17728, 17729, 17732, 17733, 17744, 17745, 17748, 17749, 20480, 20481, 20484, 20485, 20496, 20497, 20500, 20501, 20544, 20545, 20548, 20549, 20560, 20561, 20564, 20565, 20736, 20737, 20740, 20741, 20752, 20753, 20756, 20757, 20800, 20801, 20804, 20805, 20816, 20817, 20820, 20821, 21504, 21505, 21508, 21509, 21520, 21521, 21524, 21525, 21568, 21569, 21572, 21573, 21584, 21585, 21588, 21589, 21760, 21761, 21764, 21765, 21776, 21777, 21780, 21781, 21824, 21825, 21828, 21829, 21840, 21841, 21844, 21845 }; Aritmética Computacional Francisco Rodríguez Henríquez

Squaring: Software Implementation void rce_Field. Sqr 2 k_Random(rct_word *ax, rct_word rct_octet *offsetptr) { rct_index i; rct_word C, S; rct_index wlen, blen_p; rct_word *tmp; *tx, rce_context *cntxt, wlen = cntxt->ecp->wlen; blen_p = cntxt->ecp->blen_p; tmp = (rct_word *) offsetptr; tmp[0]=0; tmp[1]=0; for (i=0; i<wlen; i++) { S = sqr_table_low[(ax[i]&0 xff)]; S ^= (sqr_table_low[(ax[i]>>8)&0 xff]<<16); C = sqr_table_low[(ax[i]>>16)&0 xff]; C ^= (sqr_table_low[(ax[i]>>24)&0 xff]<<16); } tmp[i*2] = S; tmp[i*2+1] = C; RCE_FIELD_REDUC 2 K(cntxt) (tmp, blen_p, cntxt->ecp->poly); //rce_residue 2 k(tmp, blen_p, cntxt->ecp->poly); } for (i=0; i<wlen; i++) tx[i] = tmp[i]; Aritmética Computacional Francisco Rodríguez Henríquez

Second step: reduction • Problem: Given the polynomial product C(x) with at most, 2 m-1, obtain the modular product C' with m coordinates, using the generating irreducible polynomial P(x). Notice that since we are interested in the polynomial reminder of the above equation, we can safely add any multiple of P(x) to C(x) without altering the desired result. This simple observation suggest the following algorithm that can reduce k bits of the polynomial product C at once. Aritmética Computacional Francisco Rodríguez Henríquez

Second step: reduction • Let us assume that the m+1 and 2 m-1 coordinates of P(x) and C(x), respectively, are distributed as follows: • Then, there always exists a k-bit constant scalar S, such that where 0 < k <m. Notice that all the k MSB of SP become identical to the corresponding ones of the number C. By left shifting the number SP exactly Shift = 2 m-2 -k-1 positions, we effectively reduce the number in C by k bit. Aritmética Computacional Francisco Rodríguez Henríquez

Software reduction implementation 2 m-1 coordinates Addition operations < 4 wlen; SHIFT operations < 4 wlen; Comparisons = 2 wlen. Aritmética Computacional Francisco Rodríguez Henríquez

Squaring: Polynomial Multiplication Step FPGA Implementation [by Nazar Saqib] A = 1111 A 2= 1010101 Aritmética Computacional Francisco Rodríguez Henríquez

Squaring: Reduction Step FPGA Implementation [by Nazar Saqib] Aritmética Computacional Francisco Rodríguez Henríquez

Full Parallel Multipliers over GF(2 m) Aritmética Computacional Francisco Rodríguez Henríquez

Modular multiplication for software applications 1. Polynomial multiplication: • Look-up tables • Karatsuba/Look-up tables Modular Multiplication Software 2. Reduction step: • Standard reduction • trinomials & pentanomials • General irreducible polynomials • Montgomery reduction • trinomials & pentanomials • General irreducible polynomials Aritmética Computacional Francisco Rodríguez Henríquez

Polynomial multiplication: classical algorithm AND gates = m 2 XOR gates = (m-1)2 Time delay = Aritmética Computacional Francisco Rodríguez Henríquez

Polynomial multiplication: Karatsuba Multipliers Karatsuba's algorithm is based on the idea that the polynomial product C=AB can be written as, • It can be computed with 3 poly mults and 4 poly additions. • By using this idea recursively, one can obtain O(mlog 23) space complexities. • Best results obtained by using a combination of classic and Karatsuba strategies. Aritmética Computacional Francisco Rodríguez Henríquez

2 kn-bit Karatsuba Multipliers Aritmética Computacional Francisco Rodríguez Henríquez

2 kn-bit Karatsuba Multipliers There are some asymptotically faster methods for polynomial multiplications, such as the Karatsuba-Ofman algorithm. Discovered in 1962, it was the first algorithm able to accomplish polynomial multiplication under O(m 2) operations. Karatsuba's algorithm is based on the idea that the polynomial product C=AB can be written as, Aritmética Computacional Francisco Rodríguez Henríquez

2 kn-bit Karatsuba Multipliers • last equation can be carried out at the cost of only 3 polynomial multiplications and four polynomial additions. • Of course, Karatsuba strategy can be applied recursively to the three polynomial multiplications of last equation. • By applying this strategy recursively, it is possible to achieve a polynomial complexity of • Best results can be obtained by combining classical method with Karatsuba strategy. Aritmética Computacional Francisco Rodríguez Henríquez

• Procedure Kmul 2 k(C, A, B) • Input: Two elements A , B ЄGF(2 m) with m=rn=2 kn, and where A, B can be expressed as, • Output: A polynomial C=AB with up to 2 m-1 coordinates, where C=xm. CH+CL. . February 2000 Francisco Rodríguez Henríquez

2 kn-bit Karatsuba Multipliers It can be shown that the space and time complexities of a m=2 kn-bit Karatsuba multiplier combined with a classical method are given as, Aritmética Computacional Francisco Rodríguez Henríquez

Space and Time complexities m r n AND gates 1 1 0 TA 2 1 2 4 1 TX+TA 7. 2 4 16 9 2 TX+TA 40. 0 8 2 4 48 55 6 TX+TA 181. 5 16 4 4 144 225 10 TX+TA 676. 4 32 8 4 432 799 14 TX+TA 2302. 1 64 16 4 1296 2649 18 TX+TA 7460. 8 128 32 4 3888 8455 22 TX+TA 23499. 9 256 64 4 11664 26385 26 TX+TA 72743. 6 512 128 4 34992 81199 30 TX+TA 222727. 7 Aritmética Computacional XOR gates Time Delay Area (NAND units) 1. 26 Francisco Rodríguez Henríquez

Space complexity of hybrid Karatsuba multipliers for arbitrary m using n=1, 2, 3 Aritmética Computacional Francisco Rodríguez Henríquez

Binary Karatsuba Multipliers Aritmética Computacional Francisco Rodríguez Henríquez

Binary Karatsuba Multipliers • Problem: Find an efficient Karatsuba strategy for the multiplication of two polynomials A, B GF(2 m), such that m = 2 k + d, d 0. • Basic Idea: Pretend that both operands are polynomials with degree m’ = 2(k+1), and use normal Karatsuba approach for two of the three required polynomial multiplications, i. e. , given Aritmética Computacional Francisco Rodríguez Henríquez

Binary Karatsuba Multipliers • Compute the two 2 k-bit polynomial multiplications: • While the remaining d-bit polynomial multiplication AHBH can be computed using a -bit Karatsuba multiplier in a recursive manner (since the leftover d bits can be expressed as, d = 2 k 1+d 1). Aritmética Computacional Francisco Rodríguez Henríquez

Binary Karatsuba Multipliers • The above outlined strategy yields a Binary Karatsuba scheme where the hamming weight of the original m will determine the number of recursive iterations to be used by the algorithm. Aritmética Computacional Francisco Rodríguez Henríquez

An Example Aritmética Computacional Francisco Rodríguez Henríquez

An Example • As a design example, let us consider the polynomial multiplication of the elements A and B GF(2193). Since (193)2 = 11000001, the Hamming weight of m is h = 3. • This will imply that we need a total of three iterations in order to compute the multiplication using the generalized m-bit binary Karatsuba multiplier. Additionally we notice that for this case, m = 193 =27+65. Aritmética Computacional Francisco Rodríguez Henríquez

193 -bit binary Karatsuba Multiplier XOR gates = 20524 AND gates = 9201 Time delay = 13. 5 n. S Aritmética Computacional Francisco Rodríguez Henríquez

An Example • Where we have assumed that the above circuit has been implemented using a 1. 2 CMOS technology, where we have that the time delays associated to the AND, XOR logic gates are given as: TA Tx=0. 5 n. S. • Next slide shows a comparison between the proposed binary Karatsuba approach and the more traditional hybrid approach discussed previously. Aritmética Computacional Francisco Rodríguez Henríquez

Field Multiplication Preliminary results yield a time delay of 50 -70 Sec and 9 K Slices of hardware resources utilization. Aritmética Computacional Francisco Rodríguez Henríquez

Binary and hybrid Karatsuba multipliers’ area complexity Aritmética Computacional Francisco Rodríguez Henríquez

Second step: reduction • Problem: Given the polynomial product C(x) with at most, 2 m-1, obtain the modular product C' with m coordinates, using the generating irreducible polynomial P(x). • The computational complexity of the reduction operation is linearly proportional to the Hamming weight (the number of nonzero terms) of the generating irreducible polynomial. Aritmética Computacional Francisco Rodríguez Henríquez

Field multipliers using special irreducible polynomials Field multipliers • Equally-spaced polynomials trinomials There exist for only 81 degrees m, less than 1024 ( 8%) There exist for only 468 degrees m, less than 1024 ( 45%) Aritmética Computacional pentanomials There exists at least one for any degree m>3 Francisco Rodríguez Henríquez

Performance criteria and element representation • The most important measures of the performance for software implementations of the arithmetic operations in the Galois • field GF(2 m) are, • the total time required for execution (speed) and; • the amount of memory required for the algorithm (memory requirements) Aritmética Computacional Francisco Rodríguez Henríquez

Second step: reduction • Problem: Given the polynomial product C(x) with at most, 2 m-1, obtain the modular product C' with m coordinates, using the generating irreducible polynomial P(x). Notice that since we are interested in the polynomial reminder of the above equation, we can safely add any multiple of P(x) to C(x) without altering the desired result. This simple observation suggest the following algorithm that can reduce k bits of the polynomial product C at once. Aritmética Computacional Francisco Rodríguez Henríquez

Second step: reduction • Let us assume that the m+1 and 2 m-1 coordinates of P(x) and C(x), respectively, are distributed as follows: • Then, there always exists a k-bit constant scalar S, such that where 0 < k <m. Notice that all the k MSB of SP become identical to the corresponding ones of the number C. By left shifting the number SP exactly Shift = 2 m-2 -k-1 positions, we effectively reduce the number in C by k bit. Aritmética Computacional Francisco Rodríguez Henríquez

Standard reduction for trinomials and pentanomials 2 m-1 coordinates Addition operations < 4 wlen; SHIFT operations < 4 wlen; Comparisons = 2 wlen. Aritmética Computacional Francisco Rodríguez Henríquez

Exercises 0) Consider the polynomial Find if F=GF(55) constructed using f as a generating polynomial, is a field or not. 1) Consider the polynomial a) b) c) Show that P(x) forms a field in GF(2 m). Find whether P( ) is a primitive root or not. Find a primitive element in the field. Aritmética Computacional Francisco Rodríguez Henríquez

Exercises 2) Consider the polynomial a) b) c) d) Show that P(x) forms a field in GF(2 m). Is P(x) a primitive polynomial? Find 47 as a polynomial of degree less or equal to 5. Find the positive number k that satisfies: Aritmética Computacional Francisco Rodríguez Henríquez