ext/ipp/sources/ippcp/src/pcpgfpxmethod_binom_epid2.c - platform/external/epid-sdk - Git at Google

 /*############################################################################
   # Copyright 1999-2018 Intel Corporation
   #
   # Licensed under the Apache License, Version 2.0 (the "License");
   # you may not use this file except in compliance with the License.
   # You may obtain a copy of the License at
   #
   #     http://www.apache.org/licenses/LICENSE-2.0
   #
   # Unless required by applicable law or agreed to in writing, software
   # distributed under the License is distributed on an "AS IS" BASIS,
   # WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
   # See the License for the specific language governing permissions and
   # limitations under the License.
   ############################################################################*/

 /*
 //     Intel(R) Performance Primitives. Cryptography Primitives.
 //     GF(p^d) methods, if binomial generator
 //
 */
 #include "owncp.h"

 #include "pcpgfpxstuff.h"
 #include "pcpgfpxmethod_com.h"

 //gres: temporary excluded: #include <assert.h>

 /*
 // EPID 2.0 specific.
 //
 // EPID 2.0 uses the following finite field hierarchy:
 //
 // 1) prime field GF(p),
 //    p = 0xFFFFFFFFFFFCF0CD46E5F25EEE71A49F0CDC65FB12980A82D3292DDBAED33013
 //
 // 2) 2-degree extension of GF(p): GF(p^2) == GF(p)[x]/g(x), g(x) = x^2 -beta,
 //    beta =-1 mod p, so "beta" represents as {1}
 //
 // 3) 3-degree extension of GF(p^2) ~ GF(p^6): GF((p^2)^3) == GF(p)[v]/g(v), g(v) = v^3 -xi,
 //    xi belongs GF(p^2), xi=x+2, so "xi" represents as {2,1} ---- "2" is low- and "1" is high-order coefficients
 //
 // 4) 2-degree extension of GF((p^2)^3) ~ GF(p^12): GF(((p^2)^3)^2) == GF(p)[w]/g(w), g(w) = w^2 -vi,
 //    psi belongs GF((p^2)^3), vi=0*v^2 +1*v +0, so "vi" represents as {0,1,0}---- "0", '1" and "0" are low-, middle- and high-order coefficients
 //
 // See representations in t_gfpparam.cpp
 //
 */

 /*
 // Multiplication case: mul(a, xi) over GF(p^2),
 // where:
 //    a, belongs to GF(p^2)
 //    xi belongs to GF(p^2), xi={2,1}
 //
 // The case is important in GF((p^2)^3) arithmetic for EPID 2.0.
 //
 */
 __INLINE BNU_CHUNK_T* cpFq2Mul_xi(BNU_CHUNK_T* pR, const BNU_CHUNK_T* pA, gsEngine* pGFEx)
 {
    gsEngine* pGroundGFE = GFP_PARENT(pGFEx);
    mod_mul addF = GFP_METHOD(pGroundGFE)->add;
    mod_sub subF = GFP_METHOD(pGroundGFE)->sub;

    int termLen = GFP_FELEN(pGroundGFE);
    BNU_CHUNK_T* t0 = cpGFpGetPool(2, pGroundGFE);
    BNU_CHUNK_T* t1 = t0+termLen;

    const BNU_CHUNK_T* pA0 = pA;
    const BNU_CHUNK_T* pA1 = pA+termLen;
    BNU_CHUNK_T* pR0 = pR;
    BNU_CHUNK_T* pR1 = pR+termLen;

    //gres: temporary excluded: assert(NULL!=t0);
    addF(t0, pA0, pA0, pGroundGFE);
    addF(t1, pA0, pA1, pGroundGFE);
    subF(pR0, t0, pA1, pGroundGFE);
    addF(pR1, t1, pA1, pGroundGFE);

    cpGFpReleasePool(2, pGroundGFE);
    return pR;
 }

 /*
 // Multiplication case: mul(a, vi) over GF((p^2)^3),
 // where:
 //    a, belongs to GF((p^2)^3)
 //    xi belongs to GF((p^2)^3), vi={0,1,0}
 //
 // The case is important in GF(((p^2)^3)^2) arithmetic for EPID 2.0.
 //
 */
 __INLINE BNU_CHUNK_T* cpFq6Mul_vi(BNU_CHUNK_T* pR, const BNU_CHUNK_T* pA, gsEngine* pGFEx)
 {
    gsEngine* pGroundGFE = GFP_PARENT(pGFEx);
    int termLen = GFP_FELEN(pGroundGFE);

    const BNU_CHUNK_T* pA0 = pA;
    const BNU_CHUNK_T* pA1 = pA+termLen;
    const BNU_CHUNK_T* pA2 = pA+termLen*2;
    BNU_CHUNK_T* pR0 = pR;
    BNU_CHUNK_T* pR1 = pR+termLen;
    BNU_CHUNK_T* pR2 = pR+termLen*2;

    BNU_CHUNK_T* t = cpGFpGetPool(1, pGroundGFE);
    //gres: temporary excluded: assert(NULL!=t);

    cpFq2Mul_xi(t, pA2, pGroundGFE);
    cpGFpElementCopy(pR2, pA1, termLen);
    cpGFpElementCopy(pR1, pA0, termLen);
    cpGFpElementCopy(pR0, t, termLen);

    cpGFpReleasePool(1, pGroundGFE);

    return pR;
 }

 /*
 // Multiplication case: mul(a, g0) over GF(()),
 // where:
 //    a and g0 belongs to GF(()) - field is being extension
 //
 // The case is important in GF(()^d) arithmetic if constructed polynomial is generic binomial g(t) = t^d +g0.
 //
 */
 static BNU_CHUNK_T* cpGFpxMul_G0(BNU_CHUNK_T* pR, const BNU_CHUNK_T* pA, gsEngine* pGFEx)
 {
    gsEngine* pGroundGFE = GFP_PARENT(pGFEx);
    BNU_CHUNK_T* pGFpolynomial = GFP_MODULUS(pGFEx); /* g(x) = t^d + g0 */
    return GFP_METHOD(pGroundGFE)->mul(pR, pA, pGFpolynomial, pGroundGFE);
 }

 /*
 // EPID20 specific
 // ~~~~~~~~~~~~~~~
 //
 // Multiplication over GF(p^2)
 //    - field polynomial: g(x) = x^2 - beta  => binominal with specific value of "beta"
 //    - beta = p-1
 //
 // Multiplication over GF(((p^2)^3)^2)    ~ GF(p^12)
 //    - field polynomial: g(w) = w^2 - vi   => binominal with specific value of "vi"
 //    - vi = 0*v^2 + 1*v + 0 - i.e vi={0,1,0} belongs to GF((p^2)^3)
 */
 static BNU_CHUNK_T* cpGFpxMul_p2_binom_epid2(BNU_CHUNK_T* pR, const BNU_CHUNK_T* pA, const BNU_CHUNK_T* pB, gsEngine* pGFEx)
 {
    gsEngine* pGroundGFE = GFP_PARENT(pGFEx);
    mod_mul mulF = GFP_METHOD(pGroundGFE)->mul;
    mod_add addF = GFP_METHOD(pGroundGFE)->add;
    mod_sub subF = GFP_METHOD(pGroundGFE)->sub;

    int groundElemLen = GFP_FELEN(pGroundGFE);

    const BNU_CHUNK_T* pA0 = pA;
    const BNU_CHUNK_T* pA1 = pA+groundElemLen;

    const BNU_CHUNK_T* pB0 = pB;
    const BNU_CHUNK_T* pB1 = pB+groundElemLen;

    BNU_CHUNK_T* pR0 = pR;
    BNU_CHUNK_T* pR1 = pR+groundElemLen;

    BNU_CHUNK_T* t0 = cpGFpGetPool(4, pGroundGFE);
    BNU_CHUNK_T* t1 = t0+groundElemLen;
    BNU_CHUNK_T* t2 = t1+groundElemLen;
    BNU_CHUNK_T* t3 = t2+groundElemLen;
    //gres: temporary excluded: assert(NULL!=t0);

    mulF(t0, pA0, pB0, pGroundGFE);    /* t0 = a[0]*b[0] */
    mulF(t1, pA1, pB1, pGroundGFE);    /* t1 = a[1]*b[1] */
    addF(t2, pA0, pA1, pGroundGFE);    /* t2 = a[0]+a[1] */
    addF(t3, pB0, pB1, pGroundGFE);    /* t3 = b[0]+b[1] */

    mulF(pR1, t2,  t3, pGroundGFE);    /* r[1] = (a[0]+a[1]) * (b[0]+b[1]) */
    subF(pR1, pR1, t0, pGroundGFE);    /* r[1] -= a[0]*b[0]) + a[1]*b[1] */
    subF(pR1, pR1, t1, pGroundGFE);

    /* EPID2 specific */
    {
       int basicExtDegree = cpGFpBasicDegreeExtension(pGFEx);

       /* deal with GF(p^2) */
       if(basicExtDegree==2) {
          subF(pR0, t0, t1, pGroundGFE);
       }
       /* deal with GF(p^6^2) */
       else if(basicExtDegree==12) {
          cpFq6Mul_vi(t1, t1, pGroundGFE);
          addF(pR0, t0, t1, pGroundGFE);
       }
       /* deal with GF(p^x^2) - it's not EPID2 case, just a case */
       else {
          cpGFpxMul_G0(t1, t1, pGFEx);
          subF(pR0, t0, t1, pGroundGFE);
       }
    }

    cpGFpReleasePool(4, pGroundGFE);
    return pR;
 }

 /*
 // EPID20 specific
 // ~~~~~~~~~~~~~~~
 //
 // Squaring over GF(p^2)
 //    - field polynomial: g(x) = x^2 - beta  => binominal with specific value of "beta"
 //    - beta = p-1
 //
 // Squaring in GF(((p^2)^3)^2)      ~ GF(p^12)
 //    - field polynomial: g(w) = w^2 - vi   => binominal with specific value of "vi"
 //    - vi = 0*v^2 + 1*v + 0 - i.e vi={0,1,0} belongs to GF((p^2)^3)
 */
 static BNU_CHUNK_T* cpGFpxSqr_p2_binom_epid2(BNU_CHUNK_T* pR, const BNU_CHUNK_T* pA, gsEngine* pGFEx)
 {
    gsEngine* pGroundGFE = GFP_PARENT(pGFEx);
    mod_mul mulF = GFP_METHOD(pGroundGFE)->mul;
    mod_sqr sqrF = GFP_METHOD(pGroundGFE)->sqr;
    mod_add addF = GFP_METHOD(pGroundGFE)->add;
    mod_sub subF = GFP_METHOD(pGroundGFE)->sub;

    int groundElemLen = GFP_FELEN(pGroundGFE);

    const BNU_CHUNK_T* pA0 = pA;
    const BNU_CHUNK_T* pA1 = pA+groundElemLen;

    BNU_CHUNK_T* pR0 = pR;
    BNU_CHUNK_T* pR1 = pR+groundElemLen;

    BNU_CHUNK_T* t0 = cpGFpGetPool(3, pGroundGFE);
    BNU_CHUNK_T* t1 = t0+groundElemLen;
    BNU_CHUNK_T* u0 = t1+groundElemLen;
    //gres: temporary excluded: assert(NULL!=t0);

    mulF(u0, pA0, pA1, pGroundGFE); /* u0 = a[0]*a[1] */

    /* EPID2 specific */
    {
       int basicExtDegree = cpGFpBasicDegreeExtension(pGFEx);

       /* deal with GF(p^2) */
       if(basicExtDegree==2) {
          addF(t0, pA0, pA1, pGroundGFE);
          subF(t1, pA0, pA1, pGroundGFE);
          mulF(pR0, t0, t1,  pGroundGFE);
          addF(pR1, u0, u0,  pGroundGFE);  /* r[1] = 2*a[0]*a[1] */
       }
       /* deal with GF(p^6^2) */
       else if(basicExtDegree==12) {
          subF(t0, pA0, pA1, pGroundGFE);
          cpFq6Mul_vi(t1, pA1, pGroundGFE);
          subF(t1, pA0, t1, pGroundGFE);
          mulF(t0, t0, t1, pGroundGFE);
          addF(t0, t0, u0, pGroundGFE);
          cpFq6Mul_vi(t1, u0, pGroundGFE);
          addF(pR0, t0, t1, pGroundGFE);
          addF(pR1, u0, u0, pGroundGFE);
       }
       /* just a case */
       else {
          sqrF(t0, pA0, pGroundGFE);      /* t0 = a[0]*a[0] */
          sqrF(t1, pA1, pGroundGFE);      /* t1 = a[1]*a[1] */
          cpGFpxMul_G0(t1, t1, pGFEx);
          subF(pR0, t0, t1, pGroundGFE);
          addF(pR1, u0, u0, pGroundGFE);  /* r[1] = 2*a[0]*a[1] */
       }
    }

    cpGFpReleasePool(3, pGroundGFE);
    return pR;
 }

 /*
 // return specific polynomi alarith methods
 // polynomial - deg 2 binomial (EPID 2.0)
 */
 static gsModMethod* gsPolyArith_binom2_epid2(void)
 {
    static gsModMethod m = {
       cpGFpxEncode_com,
       cpGFpxDecode_com,
       cpGFpxMul_p2_binom_epid2,
       cpGFpxSqr_p2_binom_epid2,
       NULL,
       cpGFpxAdd_com,
       cpGFpxSub_com,
       cpGFpxNeg_com,
       cpGFpxDiv2_com,
       cpGFpxMul2_com,
       cpGFpxMul3_com,
       //cpGFpxInv
    };
    return &m;
 }

 IPPFUN( const IppsGFpMethod*, ippsGFpxMethod_binom2_epid2, (void) )
 {
    static IppsGFpMethod method = {
       cpID_Binom2_epid20,
       2,
       NULL,
       NULL
    };
    method.arith = gsPolyArith_binom2_epid2();
    return &method;
 }


 /*
 // EPID20 specific
 // ~~~~~~~~~~~~~~~
 //
 // Multiplication over GF((p^2)^3)
 //    - field polynomial: g(v) = v^3 - xi  => binominal with specific value of "xi"
 //    - xi = x+2
 */
 static BNU_CHUNK_T* cpGFpxMul_p3_binom_epid2(BNU_CHUNK_T* pR, const BNU_CHUNK_T* pA, const BNU_CHUNK_T* pB, gsEngine* pGFEx)
 {
    gsEngine* pGroundGFE = GFP_PARENT(pGFEx);
    int groundElemLen = GFP_FELEN(pGroundGFE);

    mod_mul mulF = GFP_METHOD(pGroundGFE)->mul;
    mod_add addF = GFP_METHOD(pGroundGFE)->add;
    mod_sub subF = GFP_METHOD(pGroundGFE)->sub;

    const BNU_CHUNK_T* pA0 = pA;
    const BNU_CHUNK_T* pA1 = pA+groundElemLen;
    const BNU_CHUNK_T* pA2 = pA+groundElemLen*2;

    const BNU_CHUNK_T* pB0 = pB;
    const BNU_CHUNK_T* pB1 = pB+groundElemLen;
    const BNU_CHUNK_T* pB2 = pB+groundElemLen*2;

    BNU_CHUNK_T* pR0 = pR;
    BNU_CHUNK_T* pR1 = pR+groundElemLen;
    BNU_CHUNK_T* pR2 = pR+groundElemLen*2;

    BNU_CHUNK_T* t0 = cpGFpGetPool(6, pGroundGFE);
    BNU_CHUNK_T* t1 = t0+groundElemLen;
    BNU_CHUNK_T* t2 = t1+groundElemLen;
    BNU_CHUNK_T* u0 = t2+groundElemLen;
    BNU_CHUNK_T* u1 = u0+groundElemLen;
    BNU_CHUNK_T* u2 = u1+groundElemLen;
    //gres: temporary excluded: assert(NULL!=t0);

    addF(u0 ,pA0, pA1, pGroundGFE);    /* u0 = a[0]+a[1] */
    addF(t0 ,pB0, pB1, pGroundGFE);    /* t0 = b[0]+b[1] */
    mulF(u0, u0,  t0,  pGroundGFE);    /* u0 = (a[0]+a[1])*(b[0]+b[1]) */
    mulF(t0, pA0, pB0, pGroundGFE);    /* t0 = a[0]*b[0] */

    addF(u1 ,pA1, pA2, pGroundGFE);    /* u1 = a[1]+a[2] */
    addF(t1 ,pB1, pB2, pGroundGFE);    /* t1 = b[1]+b[2] */
    mulF(u1, u1,  t1,  pGroundGFE);    /* u1 = (a[1]+a[2])*(b[1]+b[2]) */
    mulF(t1, pA1, pB1, pGroundGFE);    /* t1 = a[1]*b[1] */

    addF(u2 ,pA2, pA0, pGroundGFE);    /* u2 = a[2]+a[0] */
    addF(t2 ,pB2, pB0, pGroundGFE);    /* t2 = b[2]+b[0] */
    mulF(u2, u2,  t2,  pGroundGFE);    /* u2 = (a[2]+a[0])*(b[2]+b[0]) */
    mulF(t2, pA2, pB2, pGroundGFE);    /* t2 = a[2]*b[2] */

    subF(u0, u0,  t0,  pGroundGFE);    /* u0 = a[0]*b[1]+a[1]*b[0] */
    subF(u0, u0,  t1,  pGroundGFE);
    subF(u1, u1,  t1,  pGroundGFE);    /* u1 = a[1]*b[2]+a[2]*b[1] */
    subF(u1, u1,  t2,  pGroundGFE);
    subF(u2, u2,  t2,  pGroundGFE);    /* u2 = a[2]*b[0]+a[0]*b[2] */
    subF(u2, u2,  t0,  pGroundGFE);

    /* EPID2 specific */
    {
       int basicExtDegree = cpGFpBasicDegreeExtension(pGFEx);

       /* deal with GF(p^2^3) */
       if(basicExtDegree==6) {
          cpFq2Mul_xi(u1, u1, pGroundGFE);
          cpFq2Mul_xi(t2, t2, pGroundGFE);
          addF(pR0, t0, u1,  pGroundGFE);  /* r[0] = a[0]*b[0] - (a[2]*b[1]+a[1]*b[2])*beta */
          addF(pR1, u0, t2,  pGroundGFE);  /* r[1] = a[1]*b[0] + a[0]*b[1] - a[2]*b[2]*beta */
       }
       /* just  a case */
       else {
          cpGFpxMul_G0(u1, u1, pGFEx);     /* u1 = (a[1]*b[2]+a[2]*b[1]) * beta */
          cpGFpxMul_G0(t2, t2, pGFEx);     /* t2 = a[2]*b[2] * beta */
          subF(pR0, t0, u1,  pGroundGFE);  /* r[0] = a[0]*b[0] - (a[2]*b[1]+a[1]*b[2])*beta */
          subF(pR1, u0, t2,  pGroundGFE);  /* r[1] = a[1]*b[0] + a[0]*b[1] - a[2]*b[2]*beta */
       }
    }

    addF(pR2, u2, t1,  pGroundGFE);       /* r[2] = a[2]*b[0] + a[1]*b[1] + a[0]*b[2] */

    cpGFpReleasePool(6, pGroundGFE);
    return pR;
 }

 /*
 // EPID20 specific
 // ~~~~~~~~~~~~~~~
 //
 // Squaring over GF((p^2)^3)
 //    - field polynomial: g(v) = v^3 - xi  => binominal with specific value of "xi"
 //    - xi = x+2
 */
 static BNU_CHUNK_T* cpGFpxSqr_p3_binom_epid2(BNU_CHUNK_T* pR, const BNU_CHUNK_T* pA, gsEngine* pGFEx)
 {
    gsEngine* pGroundGFE = GFP_PARENT(pGFEx);
    int groundElemLen = GFP_FELEN(pGroundGFE);

    mod_mul mulF = GFP_METHOD(pGroundGFE)->mul;
    mod_sqr sqrF = GFP_METHOD(pGroundGFE)->sqr;
    mod_add addF = GFP_METHOD(pGroundGFE)->add;
    mod_sub subF = GFP_METHOD(pGroundGFE)->sub;

    const BNU_CHUNK_T* pA0 = pA;
    const BNU_CHUNK_T* pA1 = pA+groundElemLen;
    const BNU_CHUNK_T* pA2 = pA+groundElemLen*2;

    BNU_CHUNK_T* pR0 = pR;
    BNU_CHUNK_T* pR1 = pR+groundElemLen;
    BNU_CHUNK_T* pR2 = pR+groundElemLen*2;

    BNU_CHUNK_T* s0 = cpGFpGetPool(5, pGroundGFE);
    BNU_CHUNK_T* s1 = s0+groundElemLen;
    BNU_CHUNK_T* s2 = s1+groundElemLen;
    BNU_CHUNK_T* s3 = s2+groundElemLen;
    BNU_CHUNK_T* s4 = s3+groundElemLen;

    addF(s2, pA0, pA2, pGroundGFE);
    subF(s2,  s2, pA1, pGroundGFE);
    sqrF(s2,  s2, pGroundGFE);
    sqrF(s0, pA0, pGroundGFE);
    sqrF(s4, pA2, pGroundGFE);
    mulF(s1, pA0, pA1, pGroundGFE);
    mulF(s3, pA1, pA2, pGroundGFE);
    addF(s1,  s1,  s1, pGroundGFE);
    addF(s3,  s3,  s3, pGroundGFE);

    addF(pR2,  s1, s2, pGroundGFE);
    addF(pR2, pR2, s3, pGroundGFE);
    subF(pR2, pR2, s0, pGroundGFE);
    subF(pR2, pR2, s4, pGroundGFE);

    /* EPID2 specific */
    {
       int basicExtDegree = cpGFpBasicDegreeExtension(pGFEx);

       /* deal with GF(p^2^3) */
       if(basicExtDegree==6) {
          cpFq2Mul_xi(s4, s4, pGroundGFE);
          cpFq2Mul_xi(s3, s3, pGroundGFE);
          addF(pR1, s1,  s4, pGroundGFE);
          addF(pR0, s0,  s3, pGroundGFE);
       }
       /* just a case */
       else {
          cpGFpxMul_G0(s4, s4, pGFEx);
          cpGFpxMul_G0(s3, s3, pGFEx);
          subF(pR1, s1,  s4, pGroundGFE);
          subF(pR0, s0,  s3, pGroundGFE);
       }
    }

    cpGFpReleasePool(5, pGroundGFE);
    return pR;
 }

 /*
 // return specific polynomi alarith methods
 // polynomial - deg 3 binomial (EPID 2.0)
 */
 static gsModMethod* gsPolyArith_binom3_epid2(void)
 {
    static gsModMethod m = {
       cpGFpxEncode_com,
       cpGFpxDecode_com,
       cpGFpxMul_p3_binom_epid2,
       cpGFpxSqr_p3_binom_epid2,
       NULL,
       cpGFpxAdd_com,
       cpGFpxSub_com,
       cpGFpxNeg_com,
       cpGFpxDiv2_com,
       cpGFpxMul2_com,
       cpGFpxMul3_com,
       //cpGFpxInv
    };
    return &m;
 }

 IPPFUN( const IppsGFpMethod*, ippsGFpxMethod_binom3_epid2, (void) )
 {
    static IppsGFpMethod method = {
       cpID_Binom3_epid20,
       3,
       NULL,
       NULL
    };
    method.arith = gsPolyArith_binom3_epid2();
    return &method;
 }
	/*############################################################################
	# Copyright 1999-2018 Intel Corporation
	#
	# Licensed under the Apache License, Version 2.0 (the "License");
	# you may not use this file except in compliance with the License.
	# You may obtain a copy of the License at
	#
	# http://www.apache.org/licenses/LICENSE-2.0
	#
	# Unless required by applicable law or agreed to in writing, software
	# distributed under the License is distributed on an "AS IS" BASIS,
	# WITHOUT WARRANTIES OR CONDITIONS OF ANY KIND, either express or implied.
	# See the License for the specific language governing permissions and
	# limitations under the License.
	############################################################################*/

	/*
	// Intel(R) Performance Primitives. Cryptography Primitives.
	// GF(p^d) methods, if binomial generator
	//
	*/
	#include "owncp.h"

	#include "pcpgfpxstuff.h"
	#include "pcpgfpxmethod_com.h"

	//gres: temporary excluded: #include <assert.h>

	/*
	// EPID 2.0 specific.
	//
	// EPID 2.0 uses the following finite field hierarchy:
	//
	// 1) prime field GF(p),
	// p = 0xFFFFFFFFFFFCF0CD46E5F25EEE71A49F0CDC65FB12980A82D3292DDBAED33013
	//
	// 2) 2-degree extension of GF(p): GF(p^2) == GF(p)[x]/g(x), g(x) = x^2 -beta,
	// beta =-1 mod p, so "beta" represents as {1}
	//
	// 3) 3-degree extension of GF(p^2) ~ GF(p^6): GF((p^2)^3) == GF(p)[v]/g(v), g(v) = v^3 -xi,
	// xi belongs GF(p^2), xi=x+2, so "xi" represents as {2,1} ---- "2" is low- and "1" is high-order coefficients
	//
	// 4) 2-degree extension of GF((p^2)^3) ~ GF(p^12): GF(((p^2)^3)^2) == GF(p)[w]/g(w), g(w) = w^2 -vi,
	// psi belongs GF((p^2)^3), vi=0v^2 +1v +0, so "vi" represents as {0,1,0}---- "0", '1" and "0" are low-, middle- and high-order coefficients
	//
	// See representations in t_gfpparam.cpp
	//
	*/

	/*
	// Multiplication case: mul(a, xi) over GF(p^2),
	// where:
	// a, belongs to GF(p^2)
	// xi belongs to GF(p^2), xi={2,1}
	//
	// The case is important in GF((p^2)^3) arithmetic for EPID 2.0.
	//
	*/
	__INLINE BNU_CHUNK_T* cpFq2Mul_xi(BNU_CHUNK_T* pR, const BNU_CHUNK_T* pA, gsEngine* pGFEx)
	{
	gsEngine* pGroundGFE = GFP_PARENT(pGFEx);
	mod_mul addF = GFP_METHOD(pGroundGFE)->add;
	mod_sub subF = GFP_METHOD(pGroundGFE)->sub;

	int termLen = GFP_FELEN(pGroundGFE);
	BNU_CHUNK_T* t0 = cpGFpGetPool(2, pGroundGFE);
	BNU_CHUNK_T* t1 = t0+termLen;

	const BNU_CHUNK_T* pA0 = pA;
	const BNU_CHUNK_T* pA1 = pA+termLen;
	BNU_CHUNK_T* pR0 = pR;
	BNU_CHUNK_T* pR1 = pR+termLen;

	//gres: temporary excluded: assert(NULL!=t0);
	addF(t0, pA0, pA0, pGroundGFE);
	addF(t1, pA0, pA1, pGroundGFE);
	subF(pR0, t0, pA1, pGroundGFE);
	addF(pR1, t1, pA1, pGroundGFE);

	cpGFpReleasePool(2, pGroundGFE);
	return pR;
	}

	/*
	// Multiplication case: mul(a, vi) over GF((p^2)^3),
	// where:
	// a, belongs to GF((p^2)^3)
	// xi belongs to GF((p^2)^3), vi={0,1,0}
	//
	// The case is important in GF(((p^2)^3)^2) arithmetic for EPID 2.0.
	//
	*/
	__INLINE BNU_CHUNK_T* cpFq6Mul_vi(BNU_CHUNK_T* pR, const BNU_CHUNK_T* pA, gsEngine* pGFEx)
	{
	gsEngine* pGroundGFE = GFP_PARENT(pGFEx);
	int termLen = GFP_FELEN(pGroundGFE);

	const BNU_CHUNK_T* pA0 = pA;
	const BNU_CHUNK_T* pA1 = pA+termLen;
	const BNU_CHUNK_T* pA2 = pA+termLen*2;
	BNU_CHUNK_T* pR0 = pR;
	BNU_CHUNK_T* pR1 = pR+termLen;
	BNU_CHUNK_T* pR2 = pR+termLen*2;

	BNU_CHUNK_T* t = cpGFpGetPool(1, pGroundGFE);
	//gres: temporary excluded: assert(NULL!=t);

	cpFq2Mul_xi(t, pA2, pGroundGFE);
	cpGFpElementCopy(pR2, pA1, termLen);
	cpGFpElementCopy(pR1, pA0, termLen);
	cpGFpElementCopy(pR0, t, termLen);

	cpGFpReleasePool(1, pGroundGFE);

	return pR;
	}

	/*
	// Multiplication case: mul(a, g0) over GF(()),
	// where:
	// a and g0 belongs to GF(()) - field is being extension
	//
	// The case is important in GF(()^d) arithmetic if constructed polynomial is generic binomial g(t) = t^d +g0.
	//
	*/
	static BNU_CHUNK_T* cpGFpxMul_G0(BNU_CHUNK_T* pR, const BNU_CHUNK_T* pA, gsEngine* pGFEx)
	{
	gsEngine* pGroundGFE = GFP_PARENT(pGFEx);
	BNU_CHUNK_T* pGFpolynomial = GFP_MODULUS(pGFEx); /* g(x) = t^d + g0 */
	return GFP_METHOD(pGroundGFE)->mul(pR, pA, pGFpolynomial, pGroundGFE);
	}

	/*
	// EPID20 specific
	// ~~~~~~~~~~~~~~~
	//
	// Multiplication over GF(p^2)
	// - field polynomial: g(x) = x^2 - beta => binominal with specific value of "beta"
	// - beta = p-1
	//
	// Multiplication over GF(((p^2)^3)^2) ~ GF(p^12)
	// - field polynomial: g(w) = w^2 - vi => binominal with specific value of "vi"
	// - vi = 0v^2 + 1v + 0 - i.e vi={0,1,0} belongs to GF((p^2)^3)
	*/
	static BNU_CHUNK_T* cpGFpxMul_p2_binom_epid2(BNU_CHUNK_T* pR, const BNU_CHUNK_T* pA, const BNU_CHUNK_T* pB, gsEngine* pGFEx)
	{
	gsEngine* pGroundGFE = GFP_PARENT(pGFEx);
	mod_mul mulF = GFP_METHOD(pGroundGFE)->mul;
	mod_add addF = GFP_METHOD(pGroundGFE)->add;
	mod_sub subF = GFP_METHOD(pGroundGFE)->sub;

	int groundElemLen = GFP_FELEN(pGroundGFE);

	const BNU_CHUNK_T* pA0 = pA;
	const BNU_CHUNK_T* pA1 = pA+groundElemLen;

	const BNU_CHUNK_T* pB0 = pB;
	const BNU_CHUNK_T* pB1 = pB+groundElemLen;

	BNU_CHUNK_T* pR0 = pR;
	BNU_CHUNK_T* pR1 = pR+groundElemLen;

	BNU_CHUNK_T* t0 = cpGFpGetPool(4, pGroundGFE);
	BNU_CHUNK_T* t1 = t0+groundElemLen;
	BNU_CHUNK_T* t2 = t1+groundElemLen;
	BNU_CHUNK_T* t3 = t2+groundElemLen;
	//gres: temporary excluded: assert(NULL!=t0);

	mulF(t0, pA0, pB0, pGroundGFE); /* t0 = a[0]b[0] /
	mulF(t1, pA1, pB1, pGroundGFE); /* t1 = a[1]b[1] /
	addF(t2, pA0, pA1, pGroundGFE); /* t2 = a[0]+a[1] */
	addF(t3, pB0, pB1, pGroundGFE); /* t3 = b[0]+b[1] */

	mulF(pR1, t2, t3, pGroundGFE); /* r[1] = (a[0]+a[1]) * (b[0]+b[1]) */
	subF(pR1, pR1, t0, pGroundGFE); /* r[1] -= a[0]b[0]) + a[1]b[1] */
	subF(pR1, pR1, t1, pGroundGFE);

	/* EPID2 specific */
	{
	int basicExtDegree = cpGFpBasicDegreeExtension(pGFEx);

	/* deal with GF(p^2) */
	if(basicExtDegree==2) {
	subF(pR0, t0, t1, pGroundGFE);
	}
	/* deal with GF(p^6^2) */
	else if(basicExtDegree==12) {
	cpFq6Mul_vi(t1, t1, pGroundGFE);
	addF(pR0, t0, t1, pGroundGFE);
	}
	/* deal with GF(p^x^2) - it's not EPID2 case, just a case */
	else {
	cpGFpxMul_G0(t1, t1, pGFEx);
	subF(pR0, t0, t1, pGroundGFE);
	}
	}

	cpGFpReleasePool(4, pGroundGFE);
	return pR;
	}

	/*
	// EPID20 specific
	// ~~~~~~~~~~~~~~~
	//
	// Squaring over GF(p^2)
	// - field polynomial: g(x) = x^2 - beta => binominal with specific value of "beta"
	// - beta = p-1
	//
	// Squaring in GF(((p^2)^3)^2) ~ GF(p^12)
	// - field polynomial: g(w) = w^2 - vi => binominal with specific value of "vi"
	// - vi = 0v^2 + 1v + 0 - i.e vi={0,1,0} belongs to GF((p^2)^3)
	*/
	static BNU_CHUNK_T* cpGFpxSqr_p2_binom_epid2(BNU_CHUNK_T* pR, const BNU_CHUNK_T* pA, gsEngine* pGFEx)
	{
	gsEngine* pGroundGFE = GFP_PARENT(pGFEx);
	mod_mul mulF = GFP_METHOD(pGroundGFE)->mul;
	mod_sqr sqrF = GFP_METHOD(pGroundGFE)->sqr;
	mod_add addF = GFP_METHOD(pGroundGFE)->add;
	mod_sub subF = GFP_METHOD(pGroundGFE)->sub;

	int groundElemLen = GFP_FELEN(pGroundGFE);

	const BNU_CHUNK_T* pA0 = pA;
	const BNU_CHUNK_T* pA1 = pA+groundElemLen;

	BNU_CHUNK_T* pR0 = pR;
	BNU_CHUNK_T* pR1 = pR+groundElemLen;

	BNU_CHUNK_T* t0 = cpGFpGetPool(3, pGroundGFE);
	BNU_CHUNK_T* t1 = t0+groundElemLen;
	BNU_CHUNK_T* u0 = t1+groundElemLen;
	//gres: temporary excluded: assert(NULL!=t0);

	mulF(u0, pA0, pA1, pGroundGFE); /* u0 = a[0]a[1] /

	/* EPID2 specific */
	{
	int basicExtDegree = cpGFpBasicDegreeExtension(pGFEx);

	/* deal with GF(p^2) */
	if(basicExtDegree==2) {
	addF(t0, pA0, pA1, pGroundGFE);
	subF(t1, pA0, pA1, pGroundGFE);
	mulF(pR0, t0, t1, pGroundGFE);
	addF(pR1, u0, u0, pGroundGFE); /* r[1] = 2a[0]a[1] */
	}
	/* deal with GF(p^6^2) */
	else if(basicExtDegree==12) {
	subF(t0, pA0, pA1, pGroundGFE);
	cpFq6Mul_vi(t1, pA1, pGroundGFE);
	subF(t1, pA0, t1, pGroundGFE);
	mulF(t0, t0, t1, pGroundGFE);
	addF(t0, t0, u0, pGroundGFE);
	cpFq6Mul_vi(t1, u0, pGroundGFE);
	addF(pR0, t0, t1, pGroundGFE);
	addF(pR1, u0, u0, pGroundGFE);
	}
	/* just a case */
	else {
	sqrF(t0, pA0, pGroundGFE); /* t0 = a[0]a[0] /
	sqrF(t1, pA1, pGroundGFE); /* t1 = a[1]a[1] /
	cpGFpxMul_G0(t1, t1, pGFEx);
	subF(pR0, t0, t1, pGroundGFE);
	addF(pR1, u0, u0, pGroundGFE); /* r[1] = 2a[0]a[1] */
	}
	}

	cpGFpReleasePool(3, pGroundGFE);
	return pR;
	}

	/*
	// return specific polynomi alarith methods
	// polynomial - deg 2 binomial (EPID 2.0)
	*/
	static gsModMethod* gsPolyArith_binom2_epid2(void)
	{
	static gsModMethod m = {
	cpGFpxEncode_com,
	cpGFpxDecode_com,
	cpGFpxMul_p2_binom_epid2,
	cpGFpxSqr_p2_binom_epid2,
	NULL,
	cpGFpxAdd_com,
	cpGFpxSub_com,
	cpGFpxNeg_com,
	cpGFpxDiv2_com,
	cpGFpxMul2_com,
	cpGFpxMul3_com,
	//cpGFpxInv
	};
	return &m;
	}

	IPPFUN( const IppsGFpMethod*, ippsGFpxMethod_binom2_epid2, (void) )
	{
	static IppsGFpMethod method = {
	cpID_Binom2_epid20,
	2,
	NULL,
	NULL
	};
	method.arith = gsPolyArith_binom2_epid2();
	return &method;
	}


	/*
	// EPID20 specific
	// ~~~~~~~~~~~~~~~
	//
	// Multiplication over GF((p^2)^3)
	// - field polynomial: g(v) = v^3 - xi => binominal with specific value of "xi"
	// - xi = x+2
	*/
	static BNU_CHUNK_T* cpGFpxMul_p3_binom_epid2(BNU_CHUNK_T* pR, const BNU_CHUNK_T* pA, const BNU_CHUNK_T* pB, gsEngine* pGFEx)
	{
	gsEngine* pGroundGFE = GFP_PARENT(pGFEx);
	int groundElemLen = GFP_FELEN(pGroundGFE);

	mod_mul mulF = GFP_METHOD(pGroundGFE)->mul;
	mod_add addF = GFP_METHOD(pGroundGFE)->add;
	mod_sub subF = GFP_METHOD(pGroundGFE)->sub;

	const BNU_CHUNK_T* pA0 = pA;
	const BNU_CHUNK_T* pA1 = pA+groundElemLen;
	const BNU_CHUNK_T* pA2 = pA+groundElemLen*2;

	const BNU_CHUNK_T* pB0 = pB;
	const BNU_CHUNK_T* pB1 = pB+groundElemLen;
	const BNU_CHUNK_T* pB2 = pB+groundElemLen*2;

	BNU_CHUNK_T* pR0 = pR;
	BNU_CHUNK_T* pR1 = pR+groundElemLen;
	BNU_CHUNK_T* pR2 = pR+groundElemLen*2;

	BNU_CHUNK_T* t0 = cpGFpGetPool(6, pGroundGFE);
	BNU_CHUNK_T* t1 = t0+groundElemLen;
	BNU_CHUNK_T* t2 = t1+groundElemLen;
	BNU_CHUNK_T* u0 = t2+groundElemLen;
	BNU_CHUNK_T* u1 = u0+groundElemLen;
	BNU_CHUNK_T* u2 = u1+groundElemLen;
	//gres: temporary excluded: assert(NULL!=t0);

	addF(u0 ,pA0, pA1, pGroundGFE); /* u0 = a[0]+a[1] */
	addF(t0 ,pB0, pB1, pGroundGFE); /* t0 = b[0]+b[1] */
	mulF(u0, u0, t0, pGroundGFE); /* u0 = (a[0]+a[1])(b[0]+b[1]) /
	mulF(t0, pA0, pB0, pGroundGFE); /* t0 = a[0]b[0] /

	addF(u1 ,pA1, pA2, pGroundGFE); /* u1 = a[1]+a[2] */
	addF(t1 ,pB1, pB2, pGroundGFE); /* t1 = b[1]+b[2] */
	mulF(u1, u1, t1, pGroundGFE); /* u1 = (a[1]+a[2])(b[1]+b[2]) /
	mulF(t1, pA1, pB1, pGroundGFE); /* t1 = a[1]b[1] /

	addF(u2 ,pA2, pA0, pGroundGFE); /* u2 = a[2]+a[0] */
	addF(t2 ,pB2, pB0, pGroundGFE); /* t2 = b[2]+b[0] */
	mulF(u2, u2, t2, pGroundGFE); /* u2 = (a[2]+a[0])(b[2]+b[0]) /
	mulF(t2, pA2, pB2, pGroundGFE); /* t2 = a[2]b[2] /

	subF(u0, u0, t0, pGroundGFE); /* u0 = a[0]b[1]+a[1]b[0] */
	subF(u0, u0, t1, pGroundGFE);
	subF(u1, u1, t1, pGroundGFE); /* u1 = a[1]b[2]+a[2]b[1] */
	subF(u1, u1, t2, pGroundGFE);
	subF(u2, u2, t2, pGroundGFE); /* u2 = a[2]b[0]+a[0]b[2] */
	subF(u2, u2, t0, pGroundGFE);

	/* EPID2 specific */
	{
	int basicExtDegree = cpGFpBasicDegreeExtension(pGFEx);

	/* deal with GF(p^2^3) */
	if(basicExtDegree==6) {
	cpFq2Mul_xi(u1, u1, pGroundGFE);
	cpFq2Mul_xi(t2, t2, pGroundGFE);
	addF(pR0, t0, u1, pGroundGFE); /* r[0] = a[0]b[0] - (a[2]b[1]+a[1]b[2])beta */
	addF(pR1, u0, t2, pGroundGFE); /* r[1] = a[1]b[0] + a[0]b[1] - a[2]b[2]beta */
	}
	/* just a case */
	else {
	cpGFpxMul_G0(u1, u1, pGFEx); /* u1 = (a[1]b[2]+a[2]b[1]) * beta */
	cpGFpxMul_G0(t2, t2, pGFEx); /* t2 = a[2]b[2] beta */
	subF(pR0, t0, u1, pGroundGFE); /* r[0] = a[0]b[0] - (a[2]b[1]+a[1]b[2])beta */
	subF(pR1, u0, t2, pGroundGFE); /* r[1] = a[1]b[0] + a[0]b[1] - a[2]b[2]beta */
	}
	}

	addF(pR2, u2, t1, pGroundGFE); /* r[2] = a[2]b[0] + a[1]b[1] + a[0]b[2] /

	cpGFpReleasePool(6, pGroundGFE);
	return pR;
	}

	/*
	// EPID20 specific
	// ~~~~~~~~~~~~~~~
	//
	// Squaring over GF((p^2)^3)
	// - field polynomial: g(v) = v^3 - xi => binominal with specific value of "xi"
	// - xi = x+2
	*/
	static BNU_CHUNK_T* cpGFpxSqr_p3_binom_epid2(BNU_CHUNK_T* pR, const BNU_CHUNK_T* pA, gsEngine* pGFEx)
	{
	gsEngine* pGroundGFE = GFP_PARENT(pGFEx);
	int groundElemLen = GFP_FELEN(pGroundGFE);

	mod_mul mulF = GFP_METHOD(pGroundGFE)->mul;
	mod_sqr sqrF = GFP_METHOD(pGroundGFE)->sqr;
	mod_add addF = GFP_METHOD(pGroundGFE)->add;
	mod_sub subF = GFP_METHOD(pGroundGFE)->sub;

	const BNU_CHUNK_T* pA0 = pA;
	const BNU_CHUNK_T* pA1 = pA+groundElemLen;
	const BNU_CHUNK_T* pA2 = pA+groundElemLen*2;

	BNU_CHUNK_T* pR0 = pR;
	BNU_CHUNK_T* pR1 = pR+groundElemLen;
	BNU_CHUNK_T* pR2 = pR+groundElemLen*2;

	BNU_CHUNK_T* s0 = cpGFpGetPool(5, pGroundGFE);
	BNU_CHUNK_T* s1 = s0+groundElemLen;
	BNU_CHUNK_T* s2 = s1+groundElemLen;
	BNU_CHUNK_T* s3 = s2+groundElemLen;
	BNU_CHUNK_T* s4 = s3+groundElemLen;

	addF(s2, pA0, pA2, pGroundGFE);
	subF(s2, s2, pA1, pGroundGFE);
	sqrF(s2, s2, pGroundGFE);
	sqrF(s0, pA0, pGroundGFE);
	sqrF(s4, pA2, pGroundGFE);
	mulF(s1, pA0, pA1, pGroundGFE);
	mulF(s3, pA1, pA2, pGroundGFE);
	addF(s1, s1, s1, pGroundGFE);
	addF(s3, s3, s3, pGroundGFE);

	addF(pR2, s1, s2, pGroundGFE);
	addF(pR2, pR2, s3, pGroundGFE);
	subF(pR2, pR2, s0, pGroundGFE);
	subF(pR2, pR2, s4, pGroundGFE);

	/* EPID2 specific */
	{
	int basicExtDegree = cpGFpBasicDegreeExtension(pGFEx);

	/* deal with GF(p^2^3) */
	if(basicExtDegree==6) {
	cpFq2Mul_xi(s4, s4, pGroundGFE);
	cpFq2Mul_xi(s3, s3, pGroundGFE);
	addF(pR1, s1, s4, pGroundGFE);
	addF(pR0, s0, s3, pGroundGFE);
	}
	/* just a case */
	else {
	cpGFpxMul_G0(s4, s4, pGFEx);
	cpGFpxMul_G0(s3, s3, pGFEx);
	subF(pR1, s1, s4, pGroundGFE);
	subF(pR0, s0, s3, pGroundGFE);
	}
	}

	cpGFpReleasePool(5, pGroundGFE);
	return pR;
	}

	/*
	// return specific polynomi alarith methods
	// polynomial - deg 3 binomial (EPID 2.0)
	*/
	static gsModMethod* gsPolyArith_binom3_epid2(void)
	{
	static gsModMethod m = {
	cpGFpxEncode_com,
	cpGFpxDecode_com,
	cpGFpxMul_p3_binom_epid2,
	cpGFpxSqr_p3_binom_epid2,
	NULL,
	cpGFpxAdd_com,
	cpGFpxSub_com,
	cpGFpxNeg_com,
	cpGFpxDiv2_com,
	cpGFpxMul2_com,
	cpGFpxMul3_com,
	//cpGFpxInv
	};
	return &m;
	}

	IPPFUN( const IppsGFpMethod*, ippsGFpxMethod_binom3_epid2, (void) )
	{
	static IppsGFpMethod method = {
	cpID_Binom3_epid20,
	3,
	NULL,
	NULL
	};
	method.arith = gsPolyArith_binom3_epid2();
	return &method;
	}